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A Baire space is a topological space in which countable unions of closed sets with empty interior have empty interior. Equivalently, countable intersections of dense open sets are dense.

The conditional distribution of $Y$ given $X = x$ is the distribution of $Y$ when one knows that $X$ has the value $x$. The node conditional probability distribution gives the precise definition.

The $k$-th power sum in $n$ variables is the sum of the $k$-th powers of the variables. The definition of the power sums states it.

An isometric isomorphism is a bijective isometry between metric spaces. Two metric spaces with an isometric isomorphism between them have the same metric properties.

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.

A family of random variables is independent and identically distributed (i.i.d.) when the variables are mutually independent and all have the same distribution. It is the standard model of repeated trials of one experiment. The node i.i.d. random variables gives the precise definition.

A glide reflection is a reflection in a hyperplane followed by a translation parallel to that hyperplane. In the plane, it is a reflection in a line followed by a shift along that line.

A well-order is a total order in which each nonempty subset has a least element.

The nine-point circle of a triangle is the circle through the midpoints of its three sides. It also passes through the three feet of the altitudes and through the midpoints of the segments from the vertices to the orthocentre.

Two events are independent when the occurrence of one of them does not change the probability of the other. The node independence of two events gives the precise definition.

An absorbing Markov chain has at least one state that it never leaves. Also, each state can reach such a state. The node absorbing Markov chain gives the precise definition.

A Cauchy–Euler equation is a linear equation $\sum_k a_k x^k y^{(k)} = 0$ with constant $a_k$, in which the $k$-th derivative has the factor $x^k$. It has solutions of the form $x^m$, and the change of variable $x = e^t$ gives constant coefficients. See the definition.

The circumcircle of a triangle is the circle that passes through its three vertices. Each triangle has exactly one circumcircle. Its centre is the circumcentre and its radius is the circumradius.

A homogeneous linear system with constant coefficients is a system $x' = A x$ with a constant square matrix $A$. Its solutions are given by the matrix exponential, and the eigenvalues of $A$ control their behaviour. See the definition.

A parabola is a plane curve. Each of its points has the same distance from a fixed point, the focus, and from a fixed line, the directrix. It is symmetric about the line through the focus perpendicular to the directrix.

A random walk is a stochastic process whose position is a sum of independent random steps. The simple random walk on the integers moves one step up or down at each time. The node random walk gives the precise definition.

The perpendicular bisector of a segment passes through the midpoint of the segment at a right angle to it. In the plane it is a line, and in space it is a plane. Its points are exactly the points at the same distance from the two endpoints.

A circle is the set of all points of a plane at a given distance from a given point. The point is the centre and the distance is the radius. The region inside a circle is a disc.

A pole of a function is an isolated singularity where the modulus of the function tends to infinity, see the definition. It is the simplest kind of singularity that is not removable.

The reflection in a subspace $W$ of an inner product space fixes the vectors of $W$ and negates the vectors orthogonal to $W$.

A separable equation is a first-order equation $y' = g(x) h(y)$ whose right side is a product of a function of $x$ and a function of $y$. One solves it by separation of variables: divide by $h(y)$ and integrate both sides. See the definition.

The incircle of a triangle is the largest circle inside the triangle. It touches each of the three sides at one point. Its centre is the incentre and its radius is the inradius.

The order on the integers is the relation $\le$ on $\mathbb{Z}$ that puts the integers in the sequence $\dots < -2 < -1 < 0 < 1 < 2 < \dots$. It is a total order that agrees with addition and with multiplication by positive integers.

A linear first-order equation is an equation $y' + p(x) y = q(x)$ with given functions $p$ and $q$. It is the simplest linear differential equation, and an integrating factor solves it explicitly. See the definition.

A translation moves each point of a space by the same distance in the same direction. It adds one fixed vector to each point.

A removable singularity of a holomorphic function is an isolated singularity where one can define or redefine the function so that it becomes holomorphic there. See the definition.

A sequence of random variables converges in $r$-th mean when the expected value of $|X_n - X|^r$ tends to $0$. The node convergence in mean gives the precise definition.

The least element of a subset of a partially ordered set is an element of the subset that is below every element of the subset.

A sequence of random variables converges almost surely when, for almost every outcome, the sequence of values converges. The node almost sure convergence gives the precise definition.

The Dirac measure at a point $x$ gives a set the value $1$ if the set contains $x$, and the value $0$ if not. It is the probability measure of a random quantity that always takes the value $x$. The node Dirac measure gives the precise definition.

An essential singularity is an isolated singularity that is neither removable nor a pole, see the definition. Near an essential singularity, a function comes arbitrarily close to every complex value.

The greatest common divisor of two integers is the largest integer that divides both of them. We write it as $\gcd(a, b)$. The definition of the greatest common divisor gives the precise form and the convention for $\gcd(0, 0)$.

A Bernoulli process is an infinite sequence of independent trials with two outcomes, each with the same probability of success. A model is an infinite sequence of tosses of one coin. The node Bernoulli process gives the precise definition.

Let $M\, dx + N\, dy = 0$ be an equation that is not exact on an open set $D$. An integrating factor for it is a nowhere zero differentiable function $\mu$ on $D$ such that $\mu M\, dx + \mu N\, dy = 0$ is exact. On a rectangle, with $M$, $N$ and $\mu$ of class $C^1$, the test for exactness makes this condition $(\mu M)_y = (\mu N)_x$. The two equations have the same solutions, because $\mu$ has no zero.

A martingale is a stochastic process in which the conditional expectation of the next value, given the past, is the present value. It models the fortune of a player in a fair game. The node martingale gives the precise definition in discrete time.

An order isomorphism is a bijection between two partially ordered sets that keeps the order in both directions.

A power series is a series $\sum_{n \ge 0} a_n (z - c)^n$ with complex coefficients $a_n$, a center $c$ and a complex variable $z$, see the definition. It converges on a disc, and its sum is holomorphic there.

A total order is a partial order in which any two elements are comparable.

A transposition is a permutation that swaps two elements and fixes all other elements. In cycle notation it is a $2$-cycle $(a\ b)$.

A Galton–Watson process models a population. Each individual has a random number of children, independently of the others and with one common law. The node Galton–Watson process gives the precise definition.