A Jónsson–Tarski algebra is a set with a perfect way to pack two elements into one and unpack them again. Packing $x$ and $y$ gives $x \cdot y$, and the maps $L$ and $R$ recover the left and right parts. Every element arises from exactly one pair in this way. Infinite binary sequences give an example: shuffle two sequences together like two decks of cards, and unshuffle to get them back. No finite set with more than one element can do this, since it has more pairs than elements.

A Brownian path hits each level it visits infinitely often, but the set of times it sits exactly at a level has length zero. So the plain time spent at $x$ is always zero, and it says nothing. Local time instead counts the time spent in a thin band $(x - \varepsilon, x + \varepsilon)$ and divides by the width $2\varepsilon$. As $\varepsilon$ shrinks, this ratio has a finite positive limit at levels the path has crossed. For a standard Brownian motion started at $0$, the expected local time at $0$ up to time $t$ is $\sqrt{2t/\pi}$.

A Gaussian prime is a prime element of the ring of Gaussian integers. The definition of a Gaussian prime gives the precise form.

A real hypersurface in $\C^n$ has one real dimension less than the space. Most such hypersurfaces bend in the complex directions, and the Levi form measures this bending. A Levi-flat hypersurface does not bend at all in these directions. The model is $\set{z \in \C^2 : \operatorname{Im} z_1 = 0}$, which is the union of the complex lines $\set{z_1 = t}$ for real $t$. So it looks like a stack of complex lines indexed by a real number, like the pages of a book.

If a quantity depends on many independent random inputs, and no single input can change it much, then the quantity is almost constant. McDiarmid's inequality makes this precise: the chance of a deviation $t$ decays like $e^{-2t^2 / \sum_i c_i^2}$.

At the top of a smooth hill, the tangent line is horizontal. If the slope were positive or negative, moving a little would give a higher value.

Each real quadratic form becomes a sum of squares after a rotation of the axes, possibly with a reflection. The new axes are the principal axes of the form. This is how we find the shape of an ellipse or a hyperbola from its equation.

Sylvester's criterion tests positive definiteness with $n$ determinants. We check the top left $1 \times 1$, $2 \times 2$, and so on, up to the full matrix. All of them must be positive.

Picture going down a staircase in a poset, one step at a time, never going back up. The descending chain condition says that you must stop moving after finitely many real steps. The natural numbers satisfy it, since you cannot go down from $10$ more than ten times. The integers do not: $0 > -1 > -2 > \cdots$ goes down forever.

If $g \circ f$ is homotopic to a constant map, there is a homotopy that witnesses this. If also $h \circ g$ is null, there is a second witnessing homotopy. Comparing the two witnesses for the triple composite $h \circ g \circ f$ gives a loop of maps, and this loop is a map out of a suspension. The Toda bracket is the set of all maps that arise from such choices. For example, in the stable homotopy groups of spheres the bracket $\langle 2, \eta, 2 \rangle$ is the single element $\eta^2$, where $\eta$ is the stable Hopf map.

The discriminator is an "if equal, then this, else that" operation. It looks at its first two arguments: if they are the same, it outputs the third, and if they differ, it outputs the first. On $\set{0, 1, 2}$, for example, $t(1, 1, 2) = 2$ and $t(1, 0, 2) = 1$. Many logical and algebraic structures can express this test with their own operations, and this has strong consequences.

When a group is built from a normal subgroup and a quotient, its cohomology is assembled from the cohomology of the two pieces. The spectral sequence is the bookkeeping device for this assembly. It does not always give the answer at once, but it bounds it and often determines it.

Start with some operations on a set and allow yourself to plug operations into each other, repeat variables and ignore variables. Everything you can build in this way forms a clone. For example, on $\set{0, 1}$ the operations $\wedge$ and $\neg$ generate a clone that contains $x \vee y = \neg(\neg x \wedge \neg y)$. In fact this clone contains every operation on $\set{0, 1}$. A clone records the expressive power of a family of operations, not the particular family.

Euclid's formula produces every right triangle with whole-number sides, from two numbers $m$ and $n$. The identity $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ shows that the formula always works.

Multiplying by a matrix can never create new independent directions. So the rank of a product is at most the rank of each factor. Multiplying by an invertible matrix keeps the rank.

In a Noetherian ring every ideal has finitely many generators, and the relations among the generators are also finitely generated. A coherent ring keeps only the second half for finitely generated ideals: once an ideal has finitely many generators, it has finitely many relations among them too. For example, the polynomial ring $k[x_1, x_2, x_3, \dots]$ in infinitely many variables over a field $k$ is coherent but not Noetherian.

A complex Lie group can be cut down to a real group in several ways. Inside the complex $n \times n$ matrices of determinant $1$, one can keep the matrices with real entries. One can also keep the unitary ones, which preserve lengths in $\C^n$. Both choices have half the dimension, and both "complexify" back to the whole group. Such real subgroups are the real forms.

Imagine a medium that fills half of space, such as a star's atmosphere seen as a slab of infinite depth. Each layer interacts with the others through a rule that depends only on their distance, but there are no layers beyond the surface. The equation for the state of the layers is then a convolution, cut off at the surface. A small example: the function $\varphi(x) = 1 + x$ satisfies $\varphi(x) = \int_0^\infty \frac{1}{2} e^{-\abs{x - t}} \varphi(t)\, dt$ for all $x \ge 0$.

The Radon transform of a function lists its integrals over all hyperplanes, and the formula gets the function back. First each profile is sharpened by a filter that boosts high frequencies. Then all sharpened profiles are smeared back over space and added up. The result is exactly the original function.

Multiply the first coordinate of a point by $e^{i\theta_1}$, the second by $e^{i\theta_2}$, and so on. This turns each coordinate around the origin of its own complex plane without changing its size. A Reinhardt domain contains, with each point, all the points obtained in this way. So one can draw it as a region in the "absolute value space" of points $(\abs{z_1}, \dots, \abs{z_n})$, which is a part of $[0, \infty)^n$. For example, the unit polydisc of $\C^2$ is drawn as the square $[0, 1) \times [0, 1)$, and the unit ball as a quarter of a disc.