The bell curve $e^{-\pi x^2}$ and its transform decay at the same Gaussian rate. Hardy's theorem says that no nonzero function can do better on both sides at once. If both $f$ and $\hat f$ decay faster than this matched pair of Gaussians, the function must be zero.

A varifold is a very general kind of surface, possibly with folds, crossings and corners. Suppose such a surface is almost balanced, in the sense of small mean curvature, and looks almost like a single flat sheet in a ball. Allard's theorem says that it is then a smooth sheet in a smaller ball. So roughness cannot hide inside a region where the surface has nearly the area of a flat disc.

To describe a module, one picks generators, then relations among them, then relations among the relations, and so on. If one picks as few as possible at each step, the counts do not depend on the choices. These counts are the Betti numbers. For the residue field $k$ of $k[x, y]$, one needs $1$ generator, $2$ relations $x$ and $y$, and $1$ relation among them, so the Betti numbers are $1, 2, 1$. The name comes from the Betti numbers of topology, which count holes in a space.

The extended real line adds two end points to the number line, one at each end. This lets you write $\lim_{x \to \infty}$ and $\sup$ of unbounded sets with symbols instead of words.

In the cofinite topology, a set is open when it misses only finitely many points. On an infinite set, open sets are very large, so no two nonempty open sets are disjoint.

For a composite number $n$, a factor of $n$ appears in the product $1 \cdot 2 \cdots (n - 1)$. So the product cannot leave the remainder $n - 1$. For example, $5! = 120 \equiv 0 \pmod 6$.

Keep all the objects, but glue some arrows together. For this to make sense, gluing $f$ to $f'$ must also glue $g \circ f$ to $g \circ f'$, so that composition still works on the glued arrows. The main example is the homotopy category of topological spaces, where two continuous maps become the same arrow when one can be deformed into the other. For a small example, a monoid is a category with one object, and a quotient of that category is a quotient monoid. Gluing integers of the same parity turns $\Z$ under addition into $\Z/2\Z$.

The influence of the point $s$ on the point $x$ depends only on how far apart they are, and in which direction. Shifting both points by the same amount does not change the kernel. For example, $K(x, s) = e^{-\abs{x - s}}$ is a difference kernel with $k(t) = e^{-\abs{t}}$. The Volterra equation $\varphi(x) = f(x) + \int_0^x (x - s) \varphi(s) \, ds$ has the difference kernel with $k(t) = t$.

Many physical systems change in time according to a rule that involves their whole history. When the state at each time is a function of space, such as a temperature profile, the state is a point of an infinite-dimensional space. The abstract Volterra equation writes the rule as an integral over the past. A weight $a$ says how strongly the past is remembered, and an operator $A$ acts in space. If $a = 1$, all of the past counts equally, and the equation is the integrated form of $u' = A u$.

A cycle graph is a ring of $n$ vertices, each joined to the next one and the last joined to the first. It looks like a polygon with $n$ corners.

If a set $X$ maps into a disjoint union $A \sqcup B$, then $X$ breaks into the part that lands in $A$ and the part that lands in $B$. These two parts do not overlap, and their union is $X$. An extensive category is one where every object over a sum breaks up in this way. In vector spaces this fails: the diagonal line in $\R \oplus \R$ meets both axes only at zero, yet it is not the sum of those two meetings.

An analytic function of one variable is a convergent power series $\sum a_n x^n$. For a map on a space of vectors, the monomial $a_n x^n$ is replaced by an expression of degree $n$ in the vector, such as $A(x, x, \ldots, x)$ for a multilinear map $A$. A basic example is $(1 - x)^{-1} = 1 + x + x^2 + \cdots$ for a square matrix $x$ of small norm.

Sometimes no single norm can measure the functions of interest, for example all continuous functions on the real line, which may be unbounded. Instead one measures a function by its maximum on $[-1, 1]$, on $[-2, 2]$, on $[-3, 3]$, and so on. A sequence converges when it converges uniformly on each of these intervals. With this notion of convergence, the continuous functions on $\R$ form a Fréchet algebra.

Every complex number $w$ can be written as $\varphi \cdot 1 + \psi \cdot i$ with $\varphi, \psi$ real, and $w$ is holomorphic when these real coefficients vary in a special way. A pseudoanalytic function replaces the fixed pair $(1, i)$ by a pair of functions $(F, G)$ that are independent over the reals at each point. One writes $w = \varphi F + \psi G$ and asks the real coefficients to satisfy the analogue of the Cauchy–Riemann condition. With $F = 1$ and $G = i$ this is exactly ordinary holomorphy.

A divergent series is a series whose partial sums do not have a finite limit. Summation methods give values to some divergent series. The node divergent series gives the precise definition.

Mikusiński replaced the Laplace transform by pure algebra. He made fractions of functions, with convolution as multiplication, much as one makes fractions of whole numbers. In this world the derivative is multiplication by a fixed element $s$, corrected by the initial value. Solving $y' = a y$, $y(0) = 1$ becomes $s y - 1 = a y$, so $y = 1/(s - a)$, and one checks that this fraction is the function $e^{at}$.

A congruence says which elements count as equal. It is fully invariant when this equality survives every way of mapping the algebra into itself by a homomorphism. In a term algebra, an endomorphism is the same as a substitution of terms for variables. So a fully invariant congruence there is a set of equations between terms that stays true after any substitution, which is exactly how identities like $x \cdot y = y \cdot x$ behave.

A Gaussian process is a random function whose values at any finitely many points have a joint normal distribution. One describes it by an average curve and by a rule saying how strongly the values at two points move together. For example, take a standard normal number $Z$ and set $X_t = Z t$; this random straight line through the origin is a Gaussian process with mean $0$ and covariance $st$.

Draw all the essentially different loops on a surface, one dot for each. Join two dots by an edge when the two loops can be drawn without crossing. Fill in a triangle when three loops can be drawn pairwise without crossing, and so on. The resulting shape is infinite and tangled, but it records how all loops on the surface sit with respect to each other.

Flip a fair coin at each step and move one unit up for heads and one unit down for tails. Your position follows the rule: new position equals old position plus a random step. That rule is a difference equation with a random term. It tells you right away that you stay at $0$ on average while your typical distance grows like $\sqrt{n}$. It also gives the gambler's ruin rule: starting with $j$ euros and playing fair one-euro bets, you reach $N$ euros before going broke with probability $j/N$.