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A positive semidefinite matrix is a Hermitian matrix $A$ with $x^* A x \ge 0$ for every vector $x$.

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path. Let $f : U \to \mathbb{C}$ be holomorphic on $U$. Let $\gamma : [a, b] \to U$ be a closed path that is continuous and piecewise continuously differentiable. Then the contour integral of $f$ along $\gamma$ is $$\oint_\gamma f(z) \, dz = 0.$$

A nondegenerate bilinear form is a bilinear form for which no nonzero vector gives zero with every vector.

One could guess that the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are equal at each point where both exist. This is false. Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by $$f(x, y) = \frac{x y (x^2 - y^2)}{x^2 + y^2} \text{ for } (x, y) \ne (0, 0), \qquad f(0, 0) = 0.$$ Both mixed second partial derivatives of $f$ exist at $(0, 0)$, but $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)(0, 0) = -1$ and $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)(0, 0) = 1$.

The hyperbolic functions are analogues of the trigonometric functions defined from the exponential function. The main ones are the hyperbolic sine, cosine and tangent. The node hyperbolic functions gives the precise definition.

Spherical coordinates describe a point of space by three numbers. They are its distance from the origin, the angle between its position vector and the positive $z$-axis, and an azimuth angle around the $z$-axis. They simplify problems with symmetry about a point, such as integrals over balls.

$$\lim_{x \to 0} \frac{\sin x}{x} = 1.$$

For every natural number $n$, $$0 + 1 + 2 + \dots + n = \sum_{k=0}^{n} k = \frac{n(n + 1)}{2}.$$

Let $G$ be a graph. Then $G$ is bipartite if and only if $G$ has no cycle of odd length.

Let $f$ be differentiable at $a$. The tangent line to the graph of $f$ at $(a, f(a))$ is the line $y = f(a) + f'(a)(x - a)$. It is the line through this point that best fits the graph near it.

Let $A$ be a symmetric $n \times n$ matrix, and let $Q(x) = x^{\mathsf{T}} A x$ be its real quadratic form on $\mathbb{R}^n$. The form $Q$ is positive definite if $Q(x) > 0$ for all $x \ne 0$. It is negative definite if $Q(x) < 0$ for all $x \ne 0$. It is positive or negative semidefinite if $Q(x) \ge 0$ or $Q(x) \le 0$ for all $x$, and indefinite if it takes both positive and negative values.

An elementary matrix is a matrix that we get from the identity matrix by one elementary row operation. Multiplying a matrix on the left by an elementary matrix performs that row operation.

A Lyapunov function for an equilibrium of a differential equation is a scalar function with three properties. It is zero at the equilibrium, positive away from it, and non-increasing along solutions. Its existence proves stability without solving the equation. It is named after Aleksandr Lyapunov. See the definition.

A residue class modulo $n$ is the set of all integers that are congruent to a given integer modulo $n$. The definition of a residue class gives the precise form.

A function is surjective when each element of the codomain is a value of the function.

The Diffie–Hellman key exchange is an example of public-key cryptography. Two parties, Alice and Bob, use it to agree on a secret key over a public channel. They agree in public on a prime $p$ and an integer $g$, often a primitive root modulo $p$. Alice chooses a secret integer $a$ and sends $A = g^a \bmod p$. Bob chooses a secret integer $b$ and sends $B = g^b \bmod p$. Alice computes $B^a \bmod p$, and Bob computes $A^b \bmod p$. Both are equal to $g^{ab} \bmod p$, the shared secret. A small case: $p = 11$, $g = 2$, $a = 3$ and $b = 4$. Then $A = 8$, $B = 16 \bmod 11 = 5$, and $5^3 \equiv 8^4 \equiv 4 \pmod{11}$.

A hyperbolic equilibrium point of a vector field is an equilibrium point at which the Jacobian matrix has no eigenvalue with real part zero. Such a point has no centre manifold. See the definition.

A countable set is a set whose elements can be listed in a finite or infinite sequence. It is finite or equinumerous with the natural numbers.

A family of vectors is orthogonal for a bilinear form when the form is zero on each pair of distinct members of the family.

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$. Suppose that $f$ is differentiable at each point of $U$, and that the derivative $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$ is differentiable at $a$. Then the second derivative $D^2 f(a)$ is a symmetric bilinear map: $D^2 f(a)(u, v) = D^2 f(a)(v, u)$ for all $u, v \in \mathbb{R}^n$. In particular, $\dfrac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \dfrac{\partial^2 f}{\partial x_j \, \partial x_i}(a)$ for all $i, j$.

Let $f \colon A \to B$ be a function. Then $f$ is bijective if and only if there is a function $g \colon B \to A$ with $g \circ f = \mathrm{id}_A$ and $f \circ g = \mathrm{id}_B$.

A prime element of a commutative ring is a nonzero nonunit $p$ with the following property. If $p$ divides a product, then $p$ divides one of the factors. The definition of a prime element gives the conditions.

Let $K$ be a field, and let $L$ and $L'$ be algebraic closures of $K$. Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

The Cantor set $C = \bigcap_{n \ge 0} C_n$ is a closed subset of $\mathbb{R}$.

Let $z$ and $w$ be complex numbers with $z \neq 0$ and $w \neq 0$. If $\theta$ is an argument of $z$ and $\varphi$ is an argument of $w$, then $\theta + \varphi$ is an argument of $z w$. Thus $\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}$.

Let $p$ be a prime number and let $n$ be a natural number. Let $\nu_p$ be the p-adic valuation. Then $$\nu_p(n!) = \sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor.$$ The sum is finite, because $\lfloor n / p^i \rfloor = 0$ when $p^i > n$.

Let $n$ be a positive integer, and let $F_m$ be the Fibonacci numbers. Then $$F_{n-1} F_{n+1} - F_n^2 = (-1)^n.$$

Let $a$ and $r$ be real numbers with $|r| < 1$. The geometric series $\sum_{n=0}^{\infty} a r^n$ converges, and $$\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}.$$ The sum of a finite geometric series gives the partial sums $s_N = a (1 - r^{N+1})/(1 - r)$. Since $r^{N+1} \to 0$, the partial sums tend to $a/(1 - r)$.

The moments of a random variable are the expected values of its powers. The first moment is the mean. The node moment gives the precise definition for random variables.

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$, or a method to find it, and proves $P(a)$. A non-constructive proof shows that an object exists without giving it, for example by proof by contradiction.

The limit of a function at infinity is the value that the function approaches as its argument grows without bound. It describes horizontal asymptotes of graphs. The node limit at infinity gives the precise definition.

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

Let $X$ be a random variable on $(\Omega, \mathcal{F}, P)$ with values in $(E, \mathcal{E})$ and with law $\mu_X$. Let $g \colon E \to \mathbb{R}$ be measurable. Then $g(X)$ is integrable if and only if $\int_E |g| \, d\mu_X < \infty$. In that case, $$E[g(X)] = \int_E g(x) \, \mu_X(dx).$$ In particular, if $X$ is discrete with mass function $p$, then $E[g(X)] = \sum_x g(x) \, p(x)$. If $X$ is real with density $f$, then $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx$.

Let $P$ be a subset of $\mathbb{R}^d$, and let $x$ be a point of its convex hull $\operatorname{conv}(P)$. Then there are at most $d + 1$ points $p_1, \dots, p_k$ of $P$, with $k \leq d + 1$, such that $x \in \operatorname{conv}\{p_1, \dots, p_k\}$. Equivalently, $x$ is a convex combination of at most $d + 1$ points of $P$.

Let $G$ be a finite connected graph or multigraph with at least one edge. Then $G$ has an Eulerian circuit if and only if every vertex of $G$ has even degree.

Each positive integer $n$ has a prime factorization. That is, there are prime numbers $p_1, \dots, p_k$, with $k \ge 0$, such that $n = p_1 p_2 \cdots p_k$.

The exponential of a real or complex square matrix $A$ is the sum of the convergent power series $I + A + A^2/2! + A^3/3! + \cdots$.

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size. Compact sets are exactly the totally bounded sets that are complete.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded. Then $f$ is Riemann integrable if and only if its upper and lower Darboux integrals are equal. In this case the common value is the Riemann integral $\int_a^b f(x)\, dx$.

The order of cardinals compares the sizes of sets by injections.