Latest

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.

The ring of integers $\mathbb{Z}$ is a Euclidean domain, with the absolute value $\delta(n) = |n|$ as Euclidean function. That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$, there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$.

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ if and only if the following holds. For each sequence $(x_n)$ in $D \setminus \{a\}$ with $x_n \to a$, $f(x_n) \to L$.

Let $A$ be an $m \times n$ complex matrix. Then there are a unitary matrix $U$ of size $m \times m$, a unitary matrix $V$ of size $n \times n$, and an $m \times n$ matrix $\Sigma$ with $$A = U \Sigma V^*.$$ Here $\Sigma_{ij} = 0$ for $i \ne j$, and $\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0$ are the singular values of $A$.

The quotient set of a set by an equivalence relation is the set of all equivalence classes.

A family of events is mutually independent when, for each finite subfamily, the probability that all its events occur is the product of their probabilities. The node mutual independence of events gives the precise definition.

A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.

Let $f(x) = \sum_{n=0}^{\infty} a_n (x - c)^n$ be a power series with radius of convergence $R > 0$. Then $f$ is differentiable on $(c - R, c + R)$, and $$f'(x) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1}.$$ The derived series also has radius of convergence $R$.

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable with $f'(x) = 0$ for all $x \in I$. Then $f$ is constant on $I$.

Let $(X, d)$ be a metric space, let $S \subseteq X$ and let $x \in X$. Then $x$ is in the closure $\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that converges to $x$.

A differential equation is an equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true at each point of its domain. If the unknown function depends on one variable, the equation is an ordinary differential equation. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.

A jump discontinuity is a point where both one-sided limits of a function exist but are different. The graph jumps from one height to another there. The node jump discontinuity gives the precise definition.

Let $X$ be a set, and let $\mathcal{P}(X)$ be the set of all subsets of $X$. With $A \wedge B = A \cap B$, $A \vee B = A \cup B$, $\neg A = X \setminus A$, $0 = \varnothing$ and $1 = X$, the set $\mathcal{P}(X)$ is a Boolean algebra. Its order is inclusion $\subseteq$.

Let $G$ be a finite group, let $p$ be a prime number, and let $p^n$ be the largest power of $p$ that divides $|G|$. A Sylow $p$-subgroup of $G$ is a subgroup of order $p^n$.

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable. Then $f$ is convex if and only if $f'$ is increasing on $I$.

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$. Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$. Then there is $\xi$ strictly between $a$ and $x$ with $$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a),$$ where $P_k$ is the Taylor polynomial of degree $k$ of $f$ at $a$.

Two matrices are row equivalent when one of them can be changed into the other by a finite sequence of elementary row operations.

Let $R$ be an integral domain. Then the characteristic of $R$ is $0$ or a prime number.

The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal. Its operations come from the operations of the ring. The integers modulo $n$ are the main example. The definition of the quotient ring gives the construction.

The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both of them. We write it as $\operatorname{lcm}(a, b)$. The definition of the least common multiple gives the precise form and the convention when one of the numbers is $0$.

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$. Then the dual space $V^*$ is finite-dimensional, and $$\dim V^* = \dim V.$$

A composite number is a natural number greater than $1$ that is not prime. Such a number is a product of two smaller natural numbers, each greater than $1$. The definition of a composite number gives the precise form.

The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$. It is a commutative ring in which $1 = 0$.

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

Let $n$ and $k$ be natural numbers with $n \ge 1$. Then the number of multisets of size $k$ with elements in an $n$-element set is $$\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$$ Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

A one-sided limit is the limit of a function as the argument approaches a point from the left only or from the right only. A function has a limit at a point exactly when both one-sided limits exist and agree. The node one-sided limit gives the precise definition.

The composition of two linear maps is the linear map that applies the first map and then the second.

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$. Let $B$ be the closed ball of radius $a$ around $x_0$. Then there is exactly one continuous curve $\alpha : J \to B$ such that $$\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.$$

Let $X$ be a topological space and let $K \subseteq X$. Then $K$ is compact if and only if it has the property that follows. Let $(F_i)_{i \in I}$ be a family of closed subsets of $X$ with $K \cap F_{i_1} \cap \cdots \cap F_{i_n} \neq \emptyset$ for all finitely many indices $i_1, \ldots, i_n \in I$. Then also $K \cap \bigcap_{i \in I} F_i \neq \emptyset$.

A graph is a set of vertices together with a set of edges, each edge joining two distinct vertices. Graphs model networks, relations and maps, and they are the main object of graph theory. The node graph gives the precise definition.

The Euler–Mascheroni constant is the real number $$\gamma = \lim_{n \to \infty} \big( H_n - \ln n \big) \approx 0.5772156649,$$ where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n$-th harmonic number.

Let $m$ and $n$ be natural numbers, and let $F_k$ be the Fibonacci numbers. Then $$F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.$$

A countably infinite set is a set that is equinumerous with the set of natural numbers.

For every real $a > 0$, $$\lim_{x \to \infty} \frac{\ln x}{x^a} = 0.$$ That is, $\ln x = o(x^a)$ as $x \to \infty$.

Let $G$ be a group with identity element $e$, and let $a \in G$. For an integer $n$, define $a^n$ as follows: $a^0 = e$, $a^{n+1} = a^n a$ for $n \geq 0$, and $a^{-n} = (a^{-1})^n$ for $n > 0$. Then $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$.

Let $p$ be a prime number. There is an integer $x$ with $x^2 \equiv -1 \pmod p$ if and only if $p \not\equiv 3 \pmod 4$.

The power set of a set is the set of all its subsets.

A Hasse diagram is a drawing of a finite partially ordered set. Each element is a point, and there is a line upward from $a$ to $b$ when $b$ covers $a$. This means that $a < b$ and no $c$ satisfies $a < c < b$. Then $a \le b$ exactly when one can go from $a$ to $b$ by a path that goes only upward.

A closed map is a function between topological spaces that sends closed sets to closed sets. Continuous maps from compact spaces to Hausdorff spaces are closed maps.

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers. If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a + c \equiv b + d \pmod n$.