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A Markov chain is a random process in which the next state depends only on the present state, and not on the earlier history. The node Markov chain gives the precise definition in discrete time with a countable state space.

An entire function is a function that is holomorphic on the whole complex plane, see the definition. Polynomials and the exponential function are entire.

Two random variables are independent when each event about the first one is independent of each event about the second one. The node independence of two random variables gives the precise definition.

A rotation is a motion of a Euclidean space that keeps at least one point fixed and keeps the orientation. In the plane, a rotation turns each point about a centre through the same angle.

The order on the natural numbers is the relation $\le$ on $\mathbb{N}$. Here $a \le b$ means that $a$ comes at or before $b$ when we count. It is a total order, and $0$ is its least element.

A Euclidean isometry is a map of a Euclidean space to itself that keeps the distance between each two points. Translations, rotations and reflections are examples. The Euclidean isometries of $\mathbb{R}^n$ form the Euclidean group.

A homogeneous linear equation is a linear differential equation $L y = 0$ with right side zero. Its solutions form a vector space whose dimension is the order of the equation. See the definition.

A stationary distribution of a Markov chain is a distribution of the state that does not change in one step of the chain. The node stationary distribution gives the precise definition.

A line is a straight figure of dimension one that extends without end in both directions. In a Euclidean space, through each two distinct points there is exactly one line.

A holomorphic function is a complex function that is complex differentiable at each point of an open subset of $\mathbb{C}$, see the definition. Holomorphic functions are the main objects of complex analysis. They are infinitely differentiable and they are equal to their Taylor series near each point.

An equilibrium point is asymptotically stable if it is Lyapunov stable and nearby solutions converge to it as time tends to infinity. See the definition.

An ordinary differential equation is a differential equation for an unknown function of one variable. It relates the function, its derivatives and the variable. See the definition.

An ordered field is a field with a total order that is compatible with addition and with multiplication by positive elements. The rational numbers and the real numbers are ordered fields.

An algebraically closed field is a field in which every nonconstant polynomial has a root. The complex numbers are the main example. The definition of an algebraically closed field gives equivalent conditions.

A Bernoulli trial is a random experiment with exactly two possible outcomes, called success and failure. The probability $p$ of success is the same each time the trial is repeated, and the failure has probability $q = 1 - p$. A sequence of Bernoulli trials usually means independent repetitions of one trial. The number of successes in $n$ such trials has the binomial distribution.

An exact equation is a first-order equation $M\, dx + N\, dy = 0$ whose left side is the total differential of a function $\psi(x, y)$. Its solutions lie on the level curves $\psi(x, y) = C$. See the definition.

A sequence of random variables converges in probability when, for each tolerance, the probability of a deviation larger than this tolerance tends to $0$. The node convergence in probability gives the precise definition.

A linear differential equation is an equation in which the unknown function and its derivatives occur linearly, with coefficients that are functions of the variable. Its solutions are a particular solution plus the solutions of the associated homogeneous equation, which form a vector space. See the definition.

A partial order is a relation that is reflexive, antisymmetric and transitive. A set with a partial order is a partially ordered set, or poset.

The supremum of a set is its least upper bound. It is an upper bound of the set that is less than or equal to every other upper bound. A set can have a supremum that is not an element of the set.

A homogeneous first-order equation is an equation $y' = F(y/x)$, in which the slope depends only on the ratio $y/x$. The substitution $y = x v$ changes it into a separable equation. See the definition.

An orthonormal family is a family of unit vectors that are pairwise orthogonal.

A Möbius transformation is a map $z \mapsto \frac{az + b}{cz + d}$ of the Riemann sphere to itself, with $ad - bc \neq 0$, see the definition. These maps are bijective and holomorphic, they preserve angles, and they map circles and lines to circles and lines.

A sequence of random variables converges in distribution when their cumulative distribution functions converge to a limit at each point where the limit is continuous. The node convergence in distribution gives the precise definition.

A linear equation with constant coefficients is a linear differential equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = b(x)$ with real constants $a_0, \dots, a_{n-1}$. The roots of its characteristic polynomial give all solutions of the homogeneous equation. See the definition.

A tangent line to a circle is a line that meets the circle in exactly one point, the point of tangency. It does not enter the inside of the circle, and it is perpendicular to the radius at the point of tangency.

A Bernoulli equation is a first-order equation $y' + P(x) y = Q(x) y^n$. The substitution $v = y^{1-n}$ changes it into a linear equation. See the definition.

The Euclidean algorithm computes the greatest common divisor of two natural numbers $a$ and $b$. It uses two rules: $\gcd(a, 0) = a$, and $\gcd(a, b) = \gcd(b, a \bmod b)$ when $b \neq 0$. It applies the second rule until the second number is $0$. Then the first number is the result. The algorithm stops, because the second number gets smaller at each step.

A plane is a flat surface of dimension two that extends without end. It is the two-dimensional analogue of a point, which has dimension zero, and of a line, which has dimension one.

An ellipse is a closed plane curve with two foci. For each point of the curve, the sum of the distances to the two foci is the same constant. A circle is the special case of an ellipse where the two foci are the same point.

A Riccati equation is a first-order equation $y' = q_0(x) + q_1(x) y + q_2(x) y^2$, with a right side quadratic in $y$. It can be changed into a second-order linear equation. See the definition.

A hyperbola is a plane curve with two foci. For each point of the curve, the difference of the distances to the two foci has the same absolute value. It has two branches that are mirror images of each other.

The greatest element of a subset of a partially ordered set is an element of the subset that is above every element of the subset.

The alternating group $A_n$ is the group of the even permutations of $n$ symbols. It is a normal subgroup of index $2$ in $S_n$ for $n \geq 2$.

A homothety is a transformation that scales the space by a fixed factor from a fixed point, its centre. It sends each figure to a figure of the same shape.

A periodic continued fraction is an infinite simple continued fraction whose terms repeat a fixed block from some point on. The definition of a periodic continued fraction gives the precise form.

A path in a graph is a walk that visits each vertex at most once. Paths measure distance and connectivity in graphs. The node path gives the precise definition.

Greedy colouring is a simple method that makes a colouring of a finite graph. Put the vertices in an order $v_1, v_2, \dots, v_n$, and use the colours $1, 2, 3, \dots$. For $i = 1, \dots, n$, give $v_i$ the smallest colour that no earlier neighbour of $v_i$ has.

The divisor function $\sigma_k$ sends a positive integer $n$ to the sum of the $k$-th powers of its positive divisors. The definition of the divisor function gives the precise form.

An Eulerian trail is a trail that uses every edge of a graph exactly once. The question of which graphs have one is the origin of graph theory. The node Eulerian trail gives the precise definition.