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The improper integral of $e^{-x}$ over $[0, \infty)$ converges, and $$\int_0^\infty e^{-x} \, dx = 1.$$ For $b > 0$, the exponential formula with $c = -1$ gives $\int_0^b e^{-x} \, dx = 1 - e^{-b}$. This tends to $1$ as $b \to \infty$.

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.

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$. Then $$f(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j + o\big((x - a)^k\big) \quad \text{as } x \to a.$$

The column space of a matrix is the span of its columns. It is the image of the linear map of the matrix.

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.

Let $n$, $a$ and $b$ be integers. If $n \mid a b$ and $n$ and $a$ are coprime, then $n \mid b$.

The modulo operation takes two integers $a$ and $n$ with $n \neq 0$, and gives the remainder of the division of $a$ by $n$. We write the result as $a \bmod n$. The definition of the modulo operation gives the convention that this graph uses.

Let $X$ be a real random variable with $E[X^2] < \infty$, and let $Y$ be a random variable on the same probability space. Then $$\operatorname{Var}(X) = E\left[\operatorname{Var}(X \mid Y)\right] + \operatorname{Var}\left(E[X \mid Y]\right).$$

A multivalued function from a set $X$ to a set $Y$ assigns to each $x \in X$ a subset $F(x)$ of $Y$. For at least one $x$, the subset $F(x)$ has two or more elements. In complex analysis, the main examples come from the inverses of functions that are not injective. For example, each $z \neq 0$ has two square roots, and it has infinitely many logarithms and arguments.

A left coset of a subgroup $H$ of a group $G$ is a set $gH$, with $g \in G$. You get it when you multiply each element of $H$ on the left by $g$. The left cosets of $H$ partition $G$.

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X$ have the normal distribution $\mathcal{N}(\mu, \sigma^2)$. Then $$\operatorname{Var}(X) = \sigma^2.$$

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 range of a function is the set of all its values.

An irreducible polynomial is a nonconstant polynomial that is not a product of two polynomials of smaller degree, over a field. More generally, it is an irreducible element of a polynomial ring. The definition of an irreducible polynomial gives the conditions.

Let $a$, $b$ and $c$ be vectors in $\mathbb{R}^3$. Let $\times$ be the cross product and let $\cdot$ be the dot product. Then $$a \times (b \times c) = (a \cdot c) \, b - (a \cdot b) \, c.$$

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open, let $g \colon U \to W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$. Suppose that $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$. Then the Jacobian matrices satisfy $$J_{f \circ g}(a) = J_f(g(a)) \, J_g(a),$$ a product of a $k \times m$ matrix and an $m \times n$ matrix.

The dimension of a vector space is the number of vectors in a basis of the space. It does not depend on the basis.

For every real $x$, $$\sin^2 x + \cos^2 x = 1.$$

The $p$-adic valuation of a nonzero integer $n$, for a prime $p$, is the exponent of the highest power of $p$ that divides $n$. The definition of the p-adic valuation gives the precise form.

An antichain in a partially ordered set is a subset in which no two different elements are comparable.

For all real numbers $a$ and $b$, $$|a + b| \le |a| + |b|,$$ where $|\cdot|$ is the absolute value.

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be differentiable on $U$, and let $x, y \in U$ be points such that the segment $\{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$. Then there is a number $c \in (0, 1)$ such that $$f(y) - f(x) = \nabla f\big(x + c (y - x)\big) \cdot (y - x).$$

A topological space is regular when each closed set and each point outside it lie in disjoint open sets.

A normed vector space is a real or complex vector space together with a norm.

Let $p$ be a prime number. Then $(p - 1)! \equiv -1 \pmod p$.

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$. Then any two norms on $V$ are equivalent.

For every real number $x$ and every integer $n$, $$(\cos x + i \sin x)^n = \cos(n x) + i \sin(n x).$$

A type I error is the rejection of a true null hypothesis, a false positive. A type II error is the failure to reject a false null hypothesis, a false negative. The node type I and type II errors gives the precise definition.

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.

The minimal polynomial of an endomorphism of a finite-dimensional space is the monic polynomial of least degree that annihilates it.

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$, with dot product $u \cdot v = u_1 v_1 + \dots + u_n v_n$. Let $\theta$ be the angle between $u$ and $v$. Then $u \cdot v = 0$ if and only if $\theta = \pi / 2$.

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, y)$ and $(x, 0) \sim (x, 1)$ for all $x, y \in [0, 1]$. The quotient space $[0, 1]^2 / {\sim}$, with the quotient topology, is the torus $T^2$.

Let $X$ and $Y$ be real random variables on one probability space with $E[X^2] < \infty$ and $E[Y^2] < \infty$. Then $XY$ is integrable and $$\left(E[XY]\right)^2 \le E[X^2] \, E[Y^2].$$ Equality holds if and only if $aX + bY = 0$ almost surely for some reals $a$ and $b$, not both $0$.

A topological space is second-countable when its topology has a countable basis. This condition limits the size of the topology and holds for Euclidean spaces.

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$. Then $A$ is invertible if and only if its determinant $\det A$ is a unit of $K$. In particular, over a field, $A$ is invertible if and only if $\det A \ne 0$.

Let $U$ be a finite set and let $A \subseteq U$. Let $A^c = U \setminus A$ be the complement of $A$ in $U$. Then $|A^c| = |U| - |A|$.

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$. Let $P_k$ be the Taylor polynomial of degree $k$ of $f$ at $a$. Then there is a function $h_k \colon I \to \mathbb{R}$ with $h_k(x) \to 0$ as $x \to a$ and $$f(x) = P_k(x) + h_k(x) (x - a)^k \quad \text{for all } x \in I.$$

Let $n$ be a natural number, and let $D_n$ be the number of derangements of an $n$-element set. Then $$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!} = \sum_{k=0}^{n} (-1)^k \frac{n!}{k!}.$$

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers with $b \ne 0$. Assume that $a_n \to a$ and $b_n \to b$ as $n \to \infty$. Then there is a natural number $N$ such that $b_n \ne 0$ for all $n \ge N$. Moreover, the sequence $(a_n / b_n)_{n \ge N}$ converges to $a / b$.

A continuous linear map between normed spaces is a linear map that is continuous for the norm topologies. Such maps are the same as the bounded linear maps, which multiply lengths by at most a fixed constant.