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A zero of a holomorphic function $f$ is a point $z_0$ with $f(z_0) = 0$, see the definition. The zeros of a holomorphic function that is not identically zero on a domain are isolated, and each has a finite order.

The direct product of two groups $G$ and $H$ is the set $G \times H$ of ordered pairs, with the operation done in each component separately. It is the simplest way to build a new group from two given groups.

A conic section is a curve that a plane cuts from the surface of a double cone. In the plane, it is the set of zeros of a polynomial of degree $2$ in two variables. The non-degenerate conics are the ellipses, the parabolas and the hyperbolas.

Given an arbitrary angle, construct with straightedge and compass alone an angle one third as large.

A lower triangular matrix is a square matrix whose entries above the main diagonal are all zero.

A divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$. It uses the digits of $n$, and it does not need a full division. Most rules for base $10$ come from the congruences of the powers $10^k$ modulo $d$. For example, $10 \equiv 1 \pmod 3$, so $n$ and the sum of its digits have the same remainder modulo $3$.

For a real number $x$, let $x^{1/3}$ be its real cube root and let $x^{2/3} = \left(x^{1/3}\right)^2$. Consider the initial value problem $x'(t) = 3\, x(t)^{2/3}$, $x(0) = 0$, on $\mathbb{R}$. The constant function $x(t) = 0$ is a solution. The function $x(t) = t^3$ is also a solution, because its derivative is $3t^2 = 3 \left(t^3\right)^{2/3}$. More generally, for each $c \ge 0$, the function equal to $0$ for $t < c$ and to $(t - c)^3$ for $t \ge c$ is a solution. So the problem has infinitely many solutions on every interval that contains $0$ in its interior.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous. Let $y$ be a real number between $f(a)$ and $f(b)$. Then there is $c \in [a, b]$ with $f(c) = y$.

The determinant is the unique function of square matrices that is multilinear and alternating in the rows and is $1$ on the identity matrix. It is zero exactly for the singular matrices, over a field.

For each complex number $z$, $|\bar{z}| = |z|$. Here $\bar{z}$ is the complex conjugate of $z$ and $|\cdot|$ is the modulus.

A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors. The prime ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime and the zero ideal. The definition of a prime ideal gives the conditions.

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.

Let $F$ be a field and let $V$ be a vector space over $F$. Then $V$ has a basis.

Let $U \subseteq \mathbb{R}^n$ be open, and let $\varphi \colon U \to \mathbb{R}^n$ be injective and of class C^1 with $\det D\varphi(u) \ne 0$ for each $u \in U$. Let $f \colon \varphi(U) \to \mathbb{R}$ be continuous with compact support in $\varphi(U)$. Then $$\int_{\varphi(U)} f(v) \, dv = \int_U f(\varphi(u)) \, \lvert \det D\varphi(u) \rvert \, du.$$

A prior probability is the probability that we give to a hypothesis before we see the data. More generally, a prior distribution is the distribution that we give to an unknown quantity before we see the data. In Bayes' theorem the prior is combined with the likelihood of the data, and the result is the posterior probability.

The complex sine and cosine extend the real sine and cosine to all of $\mathbb{C}$ with the formulas $\sin z = (e^{iz} - e^{-iz})/(2i)$ and $\cos z = (e^{iz} + e^{-iz})/2$, see the definition. They are entire functions, and they are not bounded on $\mathbb{C}$.

A primitive element of a finite field is a generator of its multiplicative group. The definition of a primitive element states it.

The sine function is differentiable on $\mathbb{R}$, and $\sin'(x) = \cos x$ for all real $x$.

The series $\sum_{p} \frac{1}{p}$, where $p$ runs over the prime numbers, diverges. That is, the partial sums $\sum_{p \le x} \frac{1}{p}$ are not bounded as $x \to \infty$.

The Gram matrix of a finite family of vectors in an inner product space is the square matrix of all their pairwise inner products.

Let $k$ and $y_0$ be real numbers. The initial value problem $y'(t) = k\, y(t)$, $y(0) = y_0$ has the unique solution $$y(t) = y_0\, e^{k t}, \qquad t \in \mathbb{R}.$$ If $k > 0$ and $y_0 > 0$, the solution grows: this is exponential growth. If $k < 0$ and $y_0 > 0$, the solution decreases to $0$: this is exponential decay.

Little o notation says that one function is negligible compared with another. The statement $f = o(g)$ means that $f/g$ tends to $0$ at the point in question. The node little o notation gives the precise definition.

Let $r$ and $s$ be positive integers. Then there is a positive integer $n$ with this property. Colour each edge of the complete graph $K_n$ blue or red, in any way. Then there is a blue clique with $r$ vertices or a red clique with $s$ vertices.

Let $\alpha, \beta, \gamma, \delta > 0$ be constants. Let $x(t) > 0$ be the size of a prey population and $y(t) > 0$ the size of a predator population at time $t$. The Lotka–Volterra equations are the autonomous system $$x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y.$$ The equilibrium points in the closed quadrant $x, y \ge 0$ are $(0, 0)$ and $(\gamma / \delta, \alpha / \beta)$. The function $V(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y$ is a first integral on the open quadrant $x, y > 0$.

Let $G$ be a finite group. The following conditions are equivalent:

A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication. It is a vector space with the same operations.

Let $n \ge 1$, let $R > 0$, and let $B_n(R) = \{x \in \mathbb{R}^n : \|x\| \le R\}$. Then $$\operatorname{vol}(B_n(R)) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2} + 1\right)} \, R^n,$$ where $\Gamma(s) = \int_0^\infty t^{s - 1} e^{-t} \, dt$ is the gamma function.

The arcsine is the inverse of the sine function restricted to $[-\pi/2, \pi/2]$. It is defined on $[-1, 1]$. The node arcsine gives the precise definition.

Let $\gamma_1 \colon [a, b] \to \mathbb{R}^n$ and $\gamma_2 \colon [b, c] \to \mathbb{R}^n$ be piecewise $C^1$ curves with $\gamma_1(b) = \gamma_2(b)$. Let $\gamma \colon [a, c] \to \mathbb{R}^n$ be their concatenation: $\gamma = \gamma_1$ on $[a, b]$ and $\gamma = \gamma_2$ on $[b, c]$. For each continuous vector field $F$ and each continuous function $f$ on the image of $\gamma$, $$\int_\gamma F \cdot d\mathbf{r} = \int_{\gamma_1} F \cdot d\mathbf{r} + \int_{\gamma_2} F \cdot d\mathbf{r}, \qquad \int_\gamma f \, ds = \int_{\gamma_1} f \, ds + \int_{\gamma_2} f \, ds.$$

A topological embedding is a map that identifies a space with a subspace of another space, with the same topology. It is a homeomorphism onto its image.

Let $F$ and $F'$ be finite fields with $|F| = |F'|$. Then $F$ and $F'$ are isomorphic.

Let $p$ be a prime number, and let $f$ be a polynomial with integer coefficients, with derivative $f'$. Let $r$ be an integer such that $f(r) \equiv 0 \pmod p$ and $f'(r) \not\equiv 0 \pmod p$. Then for each positive integer $k$ there is an integer $r_k$ with $r_k \equiv r \pmod p$ and $f(r_k) \equiv 0 \pmod{p^k}$. This $r_k$ is unique modulo $p^k$.

Let $n$ be a natural number, and let $B_m$ be the Bell numbers. Then $$B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k.$$

A strictly monotone function is a function that is either strictly increasing or strictly decreasing. Such a function is injective. The node strictly monotone function gives the precise definition.

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be convex. Then $f$ is continuous on $I$.

Let $F$ be a perfect field, for example a field of characteristic $0$ or an algebraically closed field. Let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map. Then there are unique linear maps $S, N \colon V \to V$ with these properties:

For every real $x$, $$1 + x \le e^x,$$ with equality if and only if $x = 0$.

A map between metric spaces is Hölder continuous with exponent $\alpha$ when $d(f(x), f(y)) \le C \, d(x, y)^\alpha$ for some constant $C$ and all $x, y$. For $\alpha = 1$ this is the Lipschitz condition.

A reduction formula expresses an integral that depends on an integer $n$ through the same integral with a smaller $n$. Repeated use of the formula reduces the integral to a simple case. Integration by parts usually produces the formula.

Let $z \in \mathbb{C}$. Then $e^z = 1$ if and only if $z = 2 \pi i k$ for some integer $k$. Here $e^z$ is the complex exponential of $z$.