Math 2.0, the latest ideas

Vector space

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A vector space over a field is a set of vectors that can be added and multiplied by scalars, with the usual rules of arithmetic. Vector spaces are the main objects of linear algebra.

The integers are a Euclidean domain

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The ring of integers $\mathbb{Z}$ is a Euclidean domain, with the absolute value $\delta(n) = |n|$ as Euclidean function. That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$ , there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$ .

Sequential characterization of limits

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Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ , let $a$ be a limit point of $D$ , and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ if and only if the following holds. For each sequence $(x_n)$ in $D \setminus \{a\}$ with $x_n \to a$ , $f(x_n) \to L$ .

Singular value decomposition

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Let $A$ be an $m \times n$ complex matrix. Then there are a unitary matrix $U$ of size $m \times m$ , a unitary matrix $V$ of size $n \times n$ , and an $m \times n$ matrix $\Sigma$ with $A = U \Sigma V^*.$ Here $\Sigma_{ij} = 0$ for $i \ne j$ , and $\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0$ are the singular values of $A$ .

Quotient set

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The quotient set of a set by an equivalence relation is the set of all equivalence classes.

Mutual independence of events

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A family of events is mutually independent when, for each finite subfamily, the probability that all its events occur is the product of their probabilities. The node mutual independence of events gives the precise definition.

Block diagonal matrix

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A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.

Power series are differentiable inside the radius of convergence

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Let $f(x) = \sum_{n=0}^{\infty} a_n (x - c)^n$ be a power series with radius of convergence $R > 0$ . Then $f$ is differentiable on $(c - R, c + R)$ , and $f'(x) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1}.$ The derived series also has radius of convergence $R$ .

A function with zero derivative is constant

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Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable with $f'(x) = 0$ for all $x \in I$ . Then $f$ is constant on $I$ .

Closure is the set of limits of sequences

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Let $(X, d)$ be a metric space, let $S \subseteq X$ and let $x \in X$ . Then $x$ is in the closure $\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that converges to $x$ .

Differential equation

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A differential equation is an equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true at each point of its domain. If the unknown function depends on one variable, the equation is an ordinary differential equation. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.

Jump discontinuity

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A jump discontinuity is a point where both one-sided limits of a function exist but are different. The graph jumps from one height to another there. The node jump discontinuity gives the precise definition.

Power set Boolean algebra

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Let $X$ be a set, and let $\mathcal{P}(X)$ be the set of all subsets of $X$ . With $A \wedge B = A \cap B$ , $A \vee B = A \cup B$ , $\neg A = X \setminus A$ , $0 = \varnothing$ and $1 = X$ , the set $\mathcal{P}(X)$ is a Boolean algebra. Its order is inclusion $\subseteq$ .

Sylow p-subgroup

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Let $G$ be a finite group, let $p$ be a prime number, and let $p^n$ be the largest power of $p$ that divides $|G|$ . A Sylow $p$ -subgroup of $G$ is a subgroup of order $p^n$ .

A differentiable function is convex iff its derivative is increasing

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Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable. Then $f$ is convex if and only if $f'$ is increasing on $I$ .

Taylor's theorem with the Cauchy remainder

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Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$ . Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$ . Then there is $\xi$ strictly between $a$ and $x$ with $f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a),$ where $P_k$ is the Taylor polynomial of degree $k$ of $f$ at $a$ .

Row equivalence

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Two matrices are row equivalent when one of them can be changed into the other by a finite sequence of elementary row operations.

The characteristic of an integral domain is zero or prime

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Let $R$ be an integral domain. Then the characteristic of $R$ is $0$ or a prime number.

Quotient ring

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The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal. Its operations come from the operations of the ring. The integers modulo $n$ are the main example. The definition of the quotient ring gives the construction.

Least common multiple

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The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both of them. We write it as $\operatorname{lcm}(a, b)$ . The definition of the least common multiple gives the precise form and the convention when one of the numbers is $0$ .

Dimension of the dual space

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Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$ . Then the dual space $V^*$ is finite-dimensional, and $\dim V^* = \dim V.$

Composite number

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A composite number is a natural number greater than $1$ that is not prime. Such a number is a product of two smaller natural numbers, each greater than $1$ . The definition of a composite number gives the precise form.

Zero ring

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The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$ . It is a commutative ring in which $1 = 0$ .

Scaling a row scales the determinant

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Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$ , and let $c \in K$ . Let $B$ be the matrix that results when one row of $A$ is multiplied by $c$ . Then $\det B = c \det A$ , where $\det$ is the determinant.

Stars and bars

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Let $n$ and $k$ be natural numbers with $n \ge 1$ . Then the number of multisets of size $k$ with elements in an $n$ -element set is $\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$ Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

One-sided limit

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A one-sided limit is the limit of a function as the argument approaches a point from the left only or from the right only. A function has a limit at a point exactly when both one-sided limits exist and agree. The node one-sided limit gives the precise definition.

Composition of linear maps

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The composition of two linear maps is the linear map that applies the first map and then the second.

Picard–Lindelöf theorem in integral form

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Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$ , let $x_0 \in \mathbb{R}^n$ , and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$ . Let $B$ be the closed ball of radius $a$ around $x_0$ . Then there is exactly one continuous curve $\alpha : J \to B$ such that $\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.$

Finite intersection property

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Let $X$ be a topological space and let $K \subseteq X$ . Then $K$ is compact if and only if it has the property that follows. Let $(F_i)_{i \in I}$ be a family of closed subsets of $X$ with $K \cap F_{i_1} \cap \cdots \cap F_{i_n} \neq \emptyset$ for all finitely many indices $i_1, \ldots, i_n \in I$ . Then also $K \cap \bigcap_{i \in I} F_i \neq \emptyset$ .

Graph

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A graph is a set of vertices together with a set of edges, each edge joining two distinct vertices. Graphs model networks, relations and maps, and they are the main object of graph theory. The node graph gives the precise definition.

Euler–Mascheroni constant

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The Euler–Mascheroni constant is the real number $\gamma = \lim_{n \to \infty} \big( H_n - \ln n \big) \approx 0.5772156649,$ where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n$ -th harmonic number.

Addition formula for Fibonacci numbers

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Let $m$ and $n$ be natural numbers, and let $F_k$ be the Fibonacci numbers. Then $F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.$

Countably infinite set

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A countably infinite set is a set that is equinumerous with the set of natural numbers.

The logarithm grows slower than any power

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For every real $a > 0$ , $\lim_{x \to \infty} \frac{\ln x}{x^a} = 0.$ That is, $\ln x = o(x^a)$ as $x \to \infty$ .

Laws of exponents in a group

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Let $G$ be a group with identity element $e$ , and let $a \in G$ . For an integer $n$ , define $a^n$ as follows: $a^0 = e$ , $a^{n+1} = a^n a$ for $n \geq 0$ , and $a^{-n} = (a^{-1})^n$ for $n > 0$ . Then $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$ .

−1 is a square modulo p if and only if p is not 3 modulo 4

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Let $p$ be a prime number. There is an integer $x$ with $x^2 \equiv -1 \pmod p$ if and only if $p \not\equiv 3 \pmod 4$ .

Power set

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The power set of a set is the set of all its subsets.

Hasse diagram

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A Hasse diagram is a drawing of a finite partially ordered set. Each element is a point, and there is a line upward from $a$ to $b$ when $b$ covers $a$ . This means that $a < b$ and no $c$ satisfies $a < c < b$ . Then $a \le b$ exactly when one can go from $a$ to $b$ by a path that goes only upward.

Closed map

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A closed map is a function between topological spaces that sends closed sets to closed sets. Continuous maps from compact spaces to Hausdorff spaces are closed maps.

Congruence is compatible with addition

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Let $n$ be a positive integer, and let $a$ , $b$ , $c$ and $d$ be integers. If $a \equiv b \pmod n$ and $c \equiv d \pmod n$ , then $a + c \equiv b + d \pmod n$ .

Zero of a holomorphic function

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A zero of a holomorphic function $f$ is a point $z_0$ with $f(z_0) = 0$ , see the definition. The zeros of a holomorphic function that is not identically zero on a domain are isolated, and each has a finite order.

Direct product of groups

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The direct product of two groups $G$ and $H$ is the set $G \times H$ of ordered pairs, with the operation done in each component separately. It is the simplest way to build a new group from two given groups.

Conic section

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A conic section is a curve that a plane cuts from the surface of a double cone. In the plane, it is the set of zeros of a polynomial of degree $2$ in two variables. The non-degenerate conics are the ellipses, the parabolas and the hyperbolas.

Trisecting the angle

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Given an arbitrary angle, construct with straightedge and compass alone an angle one third as large.

Lower triangular matrix

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A lower triangular matrix is a square matrix whose entries above the main diagonal are all zero.

Divisibility rule

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A divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$ . It uses the digits of $n$ , and it does not need a full division. Most rules for base $10$ come from the congruences of the powers $10^k$ modulo $d$ . For example, $10 \equiv 1 \pmod 3$ , so $n$ and the sum of its digits have the same remainder modulo $3$ .

Non-uniqueness for a non-Lipschitz equation

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For a real number $x$ , let $x^{1/3}$ be its real cube root and let $x^{2/3} = \left(x^{1/3}\right)^2$ . Consider the initial value problem $x'(t) = 3\, x(t)^{2/3}$ , $x(0) = 0$ , on $\mathbb{R}$ . The constant function $x(t) = 0$ is a solution. The function $x(t) = t^3$ is also a solution, because its derivative is $3t^2 = 3 \left(t^3\right)^{2/3}$ . More generally, for each $c \ge 0$ , the function equal to $0$ for $t < c$ and to $(t - c)^3$ for $t \ge c$ is a solution. So the problem has infinitely many solutions on every interval that contains $0$ in its interior.

Intermediate value theorem

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Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous. Let $y$ be a real number between $f(a)$ and $f(b)$ . Then there is $c \in [a, b]$ with $f(c) = y$ .

Determinant

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The determinant is the unique function of square matrices that is multilinear and alternating in the rows and is $1$ on the identity matrix. It is zero exactly for the singular matrices, over a field.

Modulus of the conjugate

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For each complex number $z$ , $|\bar{z}| = |z|$ . Here $\bar{z}$ is the complex conjugate of $z$ and $|\cdot|$ is the modulus.