Definition:Differential Equation

A '''differential equation''' is a mathematical equation for an unknown function of one or several variables relating: :$(1): \quad$ The values of the function itself :$(2): \quad$ Its derivatives of various orders.


Definition:First Order Ordinary Differential Equation

A '''first order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $1$.


Definition:Dependent Variable

=== Real Function === {{:Definition:Dependent Variable/Real Function}} === Complex Function === {{:Definition:Dependent Variable/Complex Function}}


Definition:Initial Condition

Let $\Phi = \map F {x, y, y', y'', \ldots, y^{\paren n} }$ be an ordinary differential equation. An '''initial condition''' is an ordered pair $\tuple {x_0, y_0}$ which any solution of $\Phi$ must satisfy. That is, an '''initial condition''' is the additional imposition that a solution $y = \map y x$ of $\Phi$ satisfy: :$\map y {x_0} = y_0$


Definition:Initial Value Problem

Let $\map y x$ be a solution to the first order ordinary differential equation: :$\dfrac {\d y} {\d x} = \map f {x, y}$ which is subject to an initial condition: $\tuple {a, b}$. That is, it is required that $y = b$ when $x = a$, that is $\map y a = b$. The finding of the resulting particular solution is known as an '''initial value problem'''.


Definition:Homogeneous Function

Let $V$ and $W$ be two vector spaces over a field $F$. Let $f: V \to W$ be a function from $V$ to $W$. Then $f$ is '''homogeneous of degree $n$''' {{iff}}: :$f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$ for all nonzero $\mathbf v \in V$ and $\alpha \in F$.


Definition:Homogeneous Differential Equation

A '''homogeneous differential equation''' is a first order ordinary differential equation of the form: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ where both $M$ and $N$ are homogeneous functions of the same degree.


Definition:Exact Differential Equation

Let a first order ordinary differential equation be expressible in this form: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ such that $M$ and $N$ are ''not'' homogeneous functions of the same degree. However, suppose there happens to exist a function $\map f {x, y}$ such that: :$\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$ such that the second partial derivatives of $f$ exist and are continuous. Then the expression $M \rd x + N \rd y$ is called an '''exact differential''', and the differential equation is called an '''exact differential equation'''.


Definition:Integrating Factor

Consider the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$ Then from Solution to Exact Differential Equation, $(1)$ is not exact, and that method can not be used to solve it. However, suppose we can find a real function of two variables $\map \mu {x, y}$ such that: :$\map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} } = 0$ is exact. Then the solution of $(1)$ ''can'' be found by the technique defined in Solution to Exact Differential Equation. The function $\map \mu {x, y}$ is called an '''integrating factor'''.


Definition:Pervushin's Number

'''Pervushin's number''' is the Mersenne number $M_{61} = 2^{61} - 1$. {{NamedforDef|Ivan Mikheevich Pervushin|cat = Pervushin}}


Definition:Linear First Order Ordinary Differential Equation

A '''linear first order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x$ where $\map P x$ and $\map Q x$ are functions of $x$.


Definition:Bernoulli's Equation

'''Bernoulli's equation''' is a first order ordinary differential equation which can be put into the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x y^n$ where $n \ne 0$ and $n \ne 1$.


Definition:Boundary (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition from Closure and Interior === {{:Definition:Boundary (Topology)/Definition 1}}


Definition:Probability Space

A '''probability space''' is a measure space $\struct {\Omega, \Sigma, \Pr}$ in which $\map \Pr \Omega = 1$. A '''probability space''' is used to define the parameters determining the outcome of an experiment $\EE$. In this context, the elements of a '''probability space''' are generally referred to as follows: :$\Omega$ is called the sample space of $\EE$ :$\Sigma$ is called the event space of $\EE$ :$\Pr$ is called the probability measure on $\EE$.  Thus it is a measurable space $\struct {\Omega, \Sigma}$ with a probability measure $\Pr$. === Discrete Probability Space === {{:Definition:Probability Space/Discrete}} === Continuous Probability Space === {{:Definition:Probability Space/Continuous}}


Definition:Measure Space

A '''measure space''' is a triple $\struct {X, \Sigma, \mu}$ where: :$X$ is a set :$\Sigma$ is a $\sigma$-algebra on $X$ :$\mu$ is a measure on $\Sigma$. Thus it is a measurable space $\struct {X, \Sigma}$ with a measure.


Definition:Outer Measure

Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set. An '''outer measure (on $X$)''' is a mapping: : $\mu^*: \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ that satisfies the following conditions: :$(1): \quad \mu^* \left({\varnothing}\right) = 0$ :$(2): \quad \mu^* \left({A}\right) \le \mu^* \left({B}\right)$ for all $A, B \in \mathcal P \left({X}\right)$ with $A \subseteq B$ (that is, $\mu^*$ is monotone) :$(3): \quad \displaystyle \mu^* \left({\bigcup_{i \mathop \in \N} A_i}\right) \le \sum_{i \mathop = 1}^\infty \mu^* \left({A_i}\right)$ for all sequences $\left({A_i}\right)_{i \mathop \in \N} \in \mathcal P \left({X}\right)$ (that is, $\mu^*$ is countably subadditive) where $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers.


=== Definition 1 === {{:Definition:Limit Superior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Superior of Sequence of Sets/Definition 2}}


=== Definition 1 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 2}}


Definition:Limit of Sets

Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets. Let the limit superior of $\Bbb S$ be equal to the limit inferior of $\Bbb S$. Then the '''limit of $\Bbb S$''', denoted $\ds \lim_{n \mathop \to \infty} E_n$, is defined as: :$\ds \lim_{n \mathop \to \infty} E_n := \limsup_{n \mathop \to \infty} E_n$ and so also: :$\ds \lim_{n \mathop \to \infty} E_n := \liminf_{n \mathop \to \infty} E_n$ and '''$\Bbb S$ converges to the limit'''.


Definition:Convergence Almost Everywhere

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.  Let $D \in \Sigma$. Let $f: D \to \R$ be a $\Sigma$-measurable function. Let $\left({f_n}\right)_{n \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n: D \to \R$. Then $\left({f_n}\right)_{n \in \N}$ is said to '''converge almost everywhere''' (or '''converge a.e.''') on $D$ to $f$ {{iff}}: :$\mu \left({ \left\{{x \in D : f_n \left({x}\right) \text{ does not converge to } f \left({x}\right) }\right\} }\right) = 0$ and we write $f_n \stackrel{a.e.}{\to} f$. In other words, the sequence of functions converges pointwise outside of a $\mu$-null set.


Definition:Almost Uniform Convergence

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $D \in \Sigma$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions. Then $\sequence {f_n}_{n \mathop \in \N}$ is said to '''converge almost uniformly''' (or '''converge a.u.''') on $D$ {{iff}}: :For all $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ of $D$ such that: ::$(1): \quad \map \mu {E_\epsilon} < \epsilon$; ::$(2): \quad \sequence {f_n}_{n \mathop \in \N}$ converges uniformly to $f$ on $D \setminus E_\epsilon$.


Definition:Convergence in Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of $\Sigma$-measurable functions. Then $f_n$ is said to '''converge in measure''' to a measurable function $f: X \to \R$ iff: :$\displaystyle \forall \epsilon > 0: \lim_{n \to \infty} \mu \left({ \left\{ {x \in D : \left|{ f_n \left({ x }\right) - f \left({ x }\right) }\right| \ge \epsilon }\right\} }\right) = 0$ for all $D \in \Sigma$ with $\mu \left({D}\right) < + \infty$. To express that $f_n$ '''converges to $f$ in measure''' one writes $f_n \stackrel{\mu}{\longrightarrow} f$ or $\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$.


Definition:Vector Length

The '''length''' of a vector $\mathbf v$ in a vector space $\struct {G, +_G, \circ}_K$ is defined as $\norm V$, the norm of $V$.


Definition:Dimension (Topology)/Locally Euclidean Space

Let $M$ be a locally Euclidean space.  Let $\left({U, \kappa}\right)$ be a coordinate chart such that:  :$\kappa: U \to \kappa \left({U}\right) \subseteq \R^n$ for some $n \in \N$. Then the natural number $n$ is called the '''dimension of $M$'''.


Definition:Congruence (Metric Spaces)

Two subsets $A, B \subset X$ of a space $X$ are said to be '''congruent''' if there exists an isometry $f: X \to X$ such that $f \left({A}\right) = B$. Such an isometry is called a '''congruence'''. {{SUBPAGENAME}} byxjucn6es49hx80reyrkzddur0unqq


Definition:Outer Product

Given two vectors $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$, their '''outer product''' $\mathbf u \otimes \mathbf v$ is defined as: :$\mathbf u \otimes \mathbf v = A = \begin{bmatrix} u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\  u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\  \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$


Definition:Harshad Number

A '''harshad number''' is a positive integer which is divisible by the sum of its digits base $10$. Hence a '''harshad number''' is a Niven number base $10$.


Definition:Number Base

=== Integers === {{:Definition:Number Base/Integers}} === Real Numbers === {{:Definition:Number Base/Real Numbers}} === Integer Part === {{:Definition:Number Base/Integer Part}} === Fractional Part === {{:Definition:Number Base/Fractional Part}} === Radix Point === {{:Definition:Number Base/Radix Point}}


Definition:Digit

Let $n$ be a number expressed in a particular number base, $b$ for example. Then $n$ can be expressed as: :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 . r_{-1} r_{-2} \ldots}_b$ where: :$m$ is such that $b^m \le n < b^{m+1}$; :all the $r_i$ are such that $0 \le r_i < b$. Each of the $r_i$ are known as the '''digits of $n$ (base $b$)'''.


Definition:Niven Number

A '''Niven number''' (in a given number base $b$) is a positive integer which is divisible by the sum of its digits in that given base $b$. That is, $N$ is a '''Niven number base $b$''' {{iff}}: :$\displaystyle \exists A \in \Z: N = \sum_{k \mathop = 0}^m r_k b^k = A \sum_{k \mathop = 0}^m r_k$ where $\displaystyle \sum_{k \mathop = 0}^m r_k b^k$ is the representation of $N$ in base $b$ as defined according to the Basis Representation Theorem.


Definition:All-Harshad Number

An '''all-Harshad number''' (or '''all-Niven number''') is a positive integer which is a Niven number in '''all''' number bases.


Definition:Total Derivative

Let $f \left({x_1, x_2, \ldots, x_n}\right)$ be a continuous real function of multiple variables. Let each of $x_1, x_2, \ldots, x_n$ be continuous real functions of a single independent variable $t$. Then the '''total derivative of $f$ with respect to $t$''' is defined as: :$\displaystyle \frac {\mathrm d f}{\mathrm d t} = \sum_{k \mathop = 1}^n \frac {\partial f} {\partial x_k} \frac {\mathrm d x_k}{\mathrm d t} = \frac {\partial f} {\partial x_1} \frac {\mathrm d x_1}{\mathrm d t} + \frac {\partial f} {\partial x_2} \frac {\mathrm d x_2}{\mathrm d t} + \cdots + \frac {\partial f} {\partial x_n} \frac {\mathrm d x_n}{\mathrm d t}$ where $\dfrac {\partial f} {\partial x_k}$ is the partial derivative of $f$ with respect to $x_k$. Note that in the above definition, nothing precludes $t$ from being one of the instances of $x_k$ itself. So we have that the '''total derivative of $f$ with respect to $x_k$''' is defined as: :$\dfrac {\mathrm d f} {\mathrm d x_k} = \dfrac {\partial f} {\partial x_1} \dfrac {\mathrm d x_1}{\mathrm d x_k} + \dfrac {\partial f} {\partial x_2} \dfrac {\mathrm d x_2} {\mathrm d x_k} + \cdots + \dfrac {\partial f} {\partial x_k} + \cdots + \dfrac {\partial f} {\partial x_n} \dfrac {\mathrm d x_n}{\mathrm d x_k}$


Definition:Second Chebyshev Function

The '''Second Chebyshev Function''' $\psi \left({x}\right)$ is defined as follows: :$\displaystyle \psi \left({x}\right) = \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le x} \ln p$ where, for each $k$, the sum extends over all powers of prime numbers $p$ such that $p^k \le x$.


Definition:First Chebyshev Function

The '''First Chebyshev Function''' $\vartheta \left({x}\right)$, or $\theta \left({x}\right)$, is defined as:  :$\displaystyle \vartheta \left({x}\right) = \sum_{p \mathop \le x} \ln p$  where the sum extends over all prime numbers $p$ such that $p \le x$. {{NamedforDef|Pafnuty Lvovich Chebyshev|cat=Chebyshev}} Category:Definitions/Number Theory 55hzinvcscyby3ei76iikbxdag52fiq


Definition:Kronecker Product

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices. The '''Kronecker product''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \otimes \mathbf B$ and is defined as the block matrix: :$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} \mathbf B & a_{12} \mathbf B & \cdots & a_{1n} \mathbf B \\ a_{21} \mathbf B & a_{22} \mathbf B & \cdots & a_{2n} \mathbf B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} \mathbf B & a_{m2} \mathbf B & \cdots & a_{mn} \mathbf B \end{bmatrix}$ Writing this out in full: :$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\ a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq}  \end{bmatrix}$ Thus, if: :$\mathbf A$ is a matrix with order $m \times n$ :$\mathbf B$ is a matrix with order $p \times q$ then $\mathbf A \otimes \mathbf B$ is a matrix with order $m p \times n q$.


Definition:Kronecker Sum

Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_m$ be square matrices with orders $n$ and $m$ respectively. The '''Kronecker sum''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as: :$\mathbf A \oplus \mathbf B = \paren {\mathbf A \otimes \mathbf I_m} + \paren {\mathbf I_n \otimes \mathbf B}$ where: :$\otimes$ denotes the Kronecker product :$+$ denotes conventional matrix entrywise addition :$\mathbf I_m$ and $\mathbf I_n$ are the unit matrices of order $m$ and $n$ respectively.


Definition:Column Matrix

A '''column matrix''' is an $m \times 1$ matrix: :$\mathbf C = \begin {bmatrix} c_{1 1} \\ c_{2 1} \\ \vdots \\ c_{m 1} \end {bmatrix}$ That is, it is a matrix with only one column.


Definition:Matrix Direct Sum

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices. The '''matrix direct sum''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as: :$\mathbf A \oplus \mathbf B := \begin {bmatrix} \mathbf A & \mathbf 0 \\ \mathbf 0 & \mathbf B \end {bmatrix}$ where $\mathbf 0$ is a zero matrix, the upper-right $\mathbf 0$ being $m \times q$ and the lower left $\mathbf 0$ being $n \times p$. Thus, if: :$\mathbf A$ is a matrix with order $m \times n$ :$\mathbf B$ is a matrix with order $p \times q$ then $\mathbf A \oplus \mathbf B$ is a matrix with order $\paren {m + p} \times \paren {n + q}$.


Definition:Matrix Addition

=== Matrix Entrywise Addition === This is the usual operation when '''matrix addition''' is specified without qualification): {{:Definition:Matrix Entrywise Addition}} === Matrix Direct Sum === {{:Definition:Matrix Direct Sum}} === Kronecker Sum === {{:Definition:Kronecker Sum}}


Definition:Matrix Product

=== Matrix Product (Conventional) === {{:Definition:Matrix Product (Conventional)}} === Matrix Scalar Product === {{:Definition:Matrix Scalar Product}} === Commutative Matrix Product === {{:Definition:Commutative Matrix Product}} === Kronecker Product === Also known as matrix direct product: {{:Definition:Kronecker Product}} === Hadamard Product === Also known as Matrix Entrywise Product or Schur Product: {{:Definition:Hadamard Product}} === Frobenius Inner Product === {{:Definition:Frobenius Inner Product}} === Cracovian === {{:Definition:Cracovian}}


Definition:Graph (Graph Theory)/Order

Let $G = \struct {V, E}$ be a graph. The '''order''' of $G$ is the cardinality of its vertex set.


Definition:Graph (Graph Theory)/Size

Let $G = \struct {V, E}$ be a graph. The '''size''' of $G$ is the count of its edges.


Definition:Directed Graph

{{:Definition:Directed Graph/Formal Definition}}


Definition:Directed Graph/Arc

Let $G = \struct {V, E}$ be a digraph. The '''arcs''' are the elements of $E$. Informally, an '''arc''' is a line that joins one vertex to another.


Definition:Network

A '''network''' $N = \struct {G, w}$ is: :a graph or digraph $G = \struct {V, E}$ together with: :a mapping $w: E \to \R$ from the edge set $E$ of $G$ into the set $\R$ of real numbers. === Weight Function === {{:Definition:Network/Weight Function}} A general '''network''' can be denoted $N = \struct {V, E, w}$ where the elements are understood to be expressed in the order: vertex set, edge set, weight function. === Weight === {{:Definition:Network/Weight}}


Definition:Edge Set

Let $G = \struct {V, E}$ be a graph. The set $E$ of edges in $G$ is called the '''edge set'''. It consists of (unordered) pairs of elements of the vertex set $V$. It is often convenient to refer to the '''edge set''' for a given graph $G$ as $\map E G$, especially if there is at any one time more than one graph under consideration.


Definition:Loop (Graph Theory)

Let $G = \struct {V, E}$ be a loop-graph. A '''loop''' is an edge $e$ of $G$ whose endvertices are the same vertex. Thus a '''loop''' $e$ on the vertex $v$ would be written: :$e = vv$


Definition:Loop-Graph/Loop-Digraph

A '''loop-digraph''' is a directed graph which allows an arc to start and end at the same vertex: :380px


Definition:Signed Graph

A '''signed graph''' is an undirected network whose functional values are $\pm 1$. The edges of such a graph are known as '''positive edges''' and '''negative edges'''. It is usual to indicate the functional values by means of just the $+$ or $-$ sign. === Example === :320px Another way to represent a '''signed graph''' is to use two different line styles: one to represent '''positive edges''' and one for the '''negative edges'''. Thus the above example can be rendered as: :320px


Definition:Vertex Set

Let $G = \struct {V, E}$ be a graph. The set $V$ of vertices in $G$ is called the '''vertex set'''. It is often convenient to refer to the '''vertex set''' for a given graph $G$ as $\map V G$, especially if there is at any one time more than one graph under consideration.


Definition:Regular Graph

Let $G = \struct {V, E}$ be an simple graph whose vertices all have the same degree $r$. Then $G$ is called '''regular of degree $r$''', or '''$r$-regular'''.


Definition:Complete Graph

Let $G = \struct {V, E}$ be a simple graph such that every vertex is adjacent to every other vertex. Then $G$ is called '''complete'''. The '''complete graph''' of order $p$ is denoted $K_p$.


Definition:Relation Isomorphism

Let $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ be relational structures. Let there exist a bijection $\phi: S_1 \to S_2$ such that: :$(1): \quad \forall \tuple {s_1, t_1} \in \RR_1: \tuple {\map \phi {s_1}, \map \phi {t_1} } \in \RR_2$ :$(2): \quad \forall \tuple {s_2, t_2} \in \RR_2: \tuple {\map {\phi^{-1} } {s_2}, \map {\phi^{-1} } {t_2} } \in \RR_1$ Then $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ are '''isomorphic''', and this is denoted $S_1 \cong S_2$. The function $\phi$ is called a '''relation isomorphism''', or just an '''isomorphism''', from $\struct {S_1, \RR_1}$ to $\struct {S_2, \RR_2}$.


Definition:Separation (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $A$ and $B$ be open sets of $T$. $A$ and $B$ form a '''separation of $T$''' {{iff}}: :$(1): \quad A$ and $B$ are non-empty :$(2): \quad A \cup B = S$ :$(3): \quad A \cap B = \O$


Definition:Component of Graph

Let $G$ be a graph. Let $H$ be a subgraph of $G$ such that: :$H$ is connected :$H$ is not contained in any connected subgraph of $G$ which has more vertices or edges than $H$ has. Then $H$ is a '''component''' of $G$.


Definition:Cut-Vertex

Let $G = \struct {V, E}$ be a connected graph. Let $v$ be a vertex of $G$. Then $v$ is a '''cut-vertex''' of $G$ {{iff}} the vertex deletion $G - v$ is a vertex cut of $G$. That is, such that $G - v$ is disconnected. Thus, a '''cut-vertex''' of $G$ is a singleton vertex cut of $G$.


Definition:Bridge

Let $G = \struct {V, E}$ be a connected graph. Let $e \in E$ be an edge of $G$ such that the edge deletion $G - e$ is disconnected. Then $e$ is known as a '''bridge''' of $G$.


Definition:Semi-Eulerian Graph

A graph is called '''semi-Eulerian''' if it contains an Eulerian trail. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph. Note that an Eulerian graph is also '''semi-Eulerian''', as an Eulerian circuit is still a path, and therefore an Eulerian trail.


Definition:Eulerian Trail

An '''Eulerian trail''' is a trail $T$ that passes through every vertex of a graph $G$ and uses every edge of $G$ exactly once.


Definition:Loop-Graph

A '''loop-graph''' is a graph which allows an edge to start and end at the same vertex:


Definition:Multigraph/Multiple Edge

Let $G = \struct {V, E}$ be a multigraph. A '''multiple edge''' is an edge of $G$ which has another edge with the same endvertices. That is, where there is more than one edge that joins any pair of vertices, each of those edges is called a multiple edge.


Definition:Edge-Disjoint Trails

Let $G = \struct {V, E}$ be an undirected graph. Let $T_1$ and $T_2$ be trails in $G$. Then $T_1$ and $T_2$ are '''edge-disjoint''' {{iff}} they have no edges in common. Category:Definitions/Graph Theory 8469m5sgbg3g6u72kp32xrsk59hnf37


Definition:Degree Sequence

Let $G = \struct {V, E}$ be a graph. The '''degree sequence''' of $G$ is the sequence obtained by listing all the degrees of all the vertices of $G$ in ascending order, with repeats as needed.


Definition:Edgeless Graph

An '''edgeless graph''' is a graph with no edges. That is, an '''edgeless graph''' is a graph of size zero. Equivalently, an '''edgeless graph''' is a graph whose vertices are all isolated. The '''edgeless graph''' of order $n$ is denoted $N_n$ and can be referred to as the '''$n$-edgeless graph'''.


Definition:Cycle Graph

A '''cycle graph''' is a graph which consists of a single cycle. The cycle graph of order $n$ is denoted $C_n$. A cycle graph is $2$-regular.


Definition:Cubic Graph

A '''cubic graph''' is a $3$-regular graph, that is, a graph whose vertices all have degree $3$. Category:Definitions/Graph Theory pq0840pba83c94wu707geqzrc2qyyue


Definition:Platonic Graph

The '''Platonic graphs''' are a set of graphs representing the edges and vertices of the five Platonic solids. === Tetrahedron === {{:Definition:Platonic Graph/Tetrahedron}} === Octahedron === {{:Definition:Platonic Graph/Octahedron}} === Cube === {{:Definition:Platonic Graph/Cube}} === Icosahedron === {{:Definition:Platonic Graph/Icosahedron}} === Dodecahedron === {{:Definition:Platonic Graph/Dodecahedron}} Category:Definitions/Graph Theory 1jcdrumlmzh2kvngkaxwcxvdkx0qo84


Definition:Graph (Graph Theory)/Edge

Let $G = \struct {V, E}$ be a graph. The '''edges''' are the elements of $E$.


Definition:Petersen Graph

The '''Petersen graph''' is a $3$-regular graph of order $10$ and size $15$, as follows: :300px {{NamedforDef|Julius Petersen|cat=Petersen}} Category:Definitions/Graph Theory bicspd0k48dgehu31cc38obfr3smtdv


Definition:Bipartite Graph

A '''bipartite graph''' is a graph $G = \left({V, E}\right)$ where: :$V$ is partitioned into two sets $A$ and $B$ such that: :each edge is incident to a vertex in $A$ and a vertex in $B$.


Definition:Complete Bipartite Graph

A '''complete bipartite graph''' is a bipartite graph $G = \left({A \mid B, E}\right)$ in which every vertex in $A$ is adjacent to every vertex in $B$. The complete bipartite graph where $A$ has $m$ vertices and $B$ has $n$ vertices is denoted $K_{m, n}$. Note that $K_{m, n}$ is the same as $K_{n, m}$.


Definition:Path Graph

A '''path graph''' is a tree which has a path which passes through all its vertices. The path graph with $n$ vertices is denoted $P_n$.


Definition:Hamiltonian Path

A '''Hamiltonian path''' in a graph is a path (not a cycle) that contains every vertex of the graph (but not necessarily every edge).


Definition:Semi-Hamiltonian Graph

A '''semi-Hamiltonian graph''' is a graph that contains a Hamiltonian path, but ''not'' a Hamilton cycle.


Definition:Kolmogorov Space

{{:Definition:Kolmogorov Space/Definition 1}}


Definition:Balanced Signed Graph

Let $G = \left({V, E}\right)$ be a signed graph. Then $G$ is '''balanced''' if it is possible to partition $V$ into two subsets $A, B$ such that: * the positive edges have both ends in $A$ or both ends in $B$; * the negative edges have one end in $A$ and the other end $B$. It is a frequent practice to apply a different colour to the vertices in each set to enhance clarity.


Definition:Forest

A '''forest''' is a simple graph whose components are all trees. A '''connected forest''' is just a single tree.


Definition:Labeled Graph

A '''labeled graph''' is a graph whose vertices are each assigned an element from a set of symbols (letters, usually, but this is unimportant). The important thing to note is that the vertices can be distinguished one from another. Note that it is not necessary for all the vertices to be assigned different labels. A vertex-colored graph can be considered as a labeled graph, in which the labels assigned are elements from a set of colors. In this context it is then rare that all vertices are required to be assigned different colors. A graph which has no such labeling is called an '''unlabeled graph'''. Category:Definitions/Graph Theory 4b2l3t5iposhx9ll8xnw8edw2drmyzw


Definition:Binary Tree

A '''binary tree''' is a rooted tree which has at most two branches at any node. That is, every node in a '''binary tree''' has (apart from the root) degree of either $1$ (for the leaf nodes) or $3$ (one for the parent, two for the children). The branches at any particular node are frequently called the '''left-hand branch''' and the '''right-hand branch''', and a distinction is made between them.


Definition:Prüfer Sequence

A '''Prüfer sequence''' of order $n$ is a (finite) sequence of integers: :$\left({\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_{n-2}}\right)$ such that $\forall i: 1 \le i \le n-2: 1 \le \mathbf a_i \le n$. That is, it is a (finite) sequence of $n - 2$ integers between $1$ and $n$.


Definition:Sierpiński Number of the Second Kind

A '''Sierpiński number of the second kind''' is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$. That is, when $k$ is a '''Sierpiński number of the second kind''', all members of the set: :$\left\{{k 2^n + 1}\right\}$ are composite.


Definition:Center (Abstract Algebra)/Ring

The '''center of a ring''' $\struct {R, +, \circ}$, denoted $\map Z R$, is the subset of elements in $R$ that commute with every element in $R$. Symbolically: :$\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$ That is, the '''center of $R$''' is the centralizer of $R$ in $R$ itself. It is clear that the '''center of a ring''' $\struct {R, +, \circ}$ can be defined as the center of the group $\struct {R, \circ}$. {{explain|I think the group is not $R$, but $R^*$, to avoid finding an inverse for the zero under multiplication.<br>Taking this question to Discussion.}}


Definition:Center of Tree

Take a tree, and remove all the nodes whose degree is $1$, along with all their incident edges. Repeat the process until either: :One node is left or: :Two nodes are left, joined by a single edge. If one node is left, it is called the '''center''' of the tree. If two nodes are left, joined by a single edge, this is called the '''bicenter''' of the tree. === Central Tree === {{:Definition:Central Tree}} === Bicentral Tree === {{:Definition:Bicentral Tree}}


Definition:Sophie Germain Prime

A '''Sophie Germain prime''' is a prime number $p$ such that $2 p + 1$ is also prime.


Definition:Spanning Tree

Let $G$ be a connected graph. A '''spanning tree for $G$''' is a spanning subgraph of $G$ which is also a tree. {{refactor|Move the below into its own page}} Clearly a tree is its own spanning tree: As a tree $T$ of order $n$ has $n - 1$ edges, its spanning tree must also contain $n-1$ edges, and those must be the same ones as in $T$.


Definition:Minimum Spanning Tree

Let $G$ be a weighted graph. The '''minimum spanning tree''' for $G$ is a spanning tree for $G$ which has a minimum total weight. For a given $G$, the '''minimum spanning tree''' may not be unique. It can also be called the '''minimum connector''' for $G$.


Definition:Maximum Spanning Tree

Let $G$ be a weighted graph. The '''maximum spanning tree''' for $G$ is a spanning tree for $G$ which has a maximum total weight. For a given $G$, the '''maximum spanning tree''' may not be unique. It can also be called the '''maximum connector''' for $G$. Category:Definitions/Graph Theory 3lrmza7lzie58zcyrmb6czhcbeu7ssf


Definition:Greedy Algorithm

A '''greedy algorithm''' is an algorithm whose decision strategy is such that any choice made at any stage takes no consideration of any future state. The strategy is to make the choice which has the greatest short-term effect towards the long-term goal. In some problems this may not produce the optimum solution.


Definition:Adjacency Matrix

An '''adjacency matrix''' is a matrix which describes a graph by representing which vertices are adjacent to which other vertices. If $G$ is a graph of order $n$, then its '''adjacency matrix''' is a square matrix of order $n$, where each row and column corresponds to a vertex of $G$. The element $a_{i j}$ of such a matrix specifies the number of edges from vertex $i$ to vertex $j$. An '''adjacency matrix''' for a simple graph and a loop-digraph is a logical matrix, that is, one whose elements are all either $0$ or $1$. An '''adjacency matrix''' for an undirected graph is symmetrical about the main diagonal. This is because if vertex $i$ is adjacent to vertex $j$, then $j$ is adjacent to $i$. An '''adjacency matrix''' for a weighted graph or network contains the weights of the edges.


Definition:Witch of Agnesi

:620px Let $OAM$ be a circle of radius $a$ whose center is at $\tuple {0, a}$. Let $M$ be the point such that $OM$ is a diameter of $OAM$. Let $OA$ be extended to cut the tangent to the circle through $M$ at $N$. Generate $NP$ perpendicular to $MN$ and $AP$ parallel to $MN$. As $A$ moves around the circle $OAM$, the point $P$ traces the curve known as the '''Witch of Agnesi'''.


Definition:Preordering

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. === Definition 1 === {{:Definition:Preordering/Definition 1}} === Definition 2 === {{:Definition:Preordering/Definition 2}}


Definition:Partial Preordering

Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is a '''partial preordering''' on $S$ iff $\precsim$ is not connected. That is, iff there is at least one pair of elements of $S$ which is non-comparable: :$\exists x, y \in S: x \not \precsim y \land y \not \precsim x$ Category:Definitions/Preorder Theory 5bfsqd25kqx0ntuksh9o4xk4bvw0d32


Definition:Total Preordering

Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is a '''total preordering''' on $S$ {{iff}} $\precsim$ is connected. That is, {{iff}} there is no pair of elements of $S$ which is non-comparable: :$\forall x, y \in S: x \precsim y \lor y \precsim x$


Definition:Trichotomy

Let $S$ be a set. A '''trichotomy''' on $S$ is a relation $\RR$ on $S$ such that for every pair of elements $a, b \in S$, '''exactly''' one of the following three conditions applies: :$a \mathrel \RR b$ :$a = b$ :$b \mathrel \RR a$


Definition:Right-Total Relation

Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation in $S$ to $T$. Then $\mathcal R$ is '''right-total''' {{iff}}: :$\forall t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R$ That is, {{iff}} every element of $T$ is related to by some element of $S$. That is, {{iff}}: :$\Img {\mathcal R} = T$ where $\Img {\mathcal R}$ denotes the image of $\mathcal R$.


Definition:Left-Total Relation

Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation in $S$ to $T$. Then $\mathcal R$ is '''left-total''' {{iff}}: :$\forall s \in S: \exists t \in T: \tuple {s, t} \in \mathcal R$ That is, {{iff}} every element of $S$ relates to some element of $T$.


Definition:Endorelation

Let $S \times S$ be the cartesian product of a set or class $S$ with itself. Let $\RR$ be a relation on $S \times S$. Then $\RR$ is referred to as an '''endorelation on $S$'''.


Definition:Correspondence

A relation $\RR \subseteq S \times T$ is a '''correspondence''' {{iff}} $\RR$ is both left-total and right-total.


Definition:Coreflexive Relation

{{:Definition:Coreflexive Relation/Definition 1}}


Definition:Reflexive Relation

{{:Definition:Reflexive Relation/Definition 1}}


Definition:Antireflexive Relation

$\RR$ is '''antireflexive''' {{iff}}: :$\forall x \in S: \tuple {x, x} \notin \RR$


Definition:Symmetric Relation

{{:Definition:Symmetric Relation/Definition 1}}


Definition:Asymmetric Relation

{{:Definition:Asymmetric Relation/Definition 1}}


Definition:Antisymmetric Relation

{{:Definition:Antisymmetric Relation/Definition 1}}


Definition:Transitive Relation

{{:Definition:Transitive Relation/Definition 1}}


Definition:Antitransitive Relation

$\mathcal R$ is '''antitransitive''' {{iff}}: :$\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \notin \mathcal R$ that is: : $\left\{ {\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \implies \left({x, z}\right) \notin \mathcal R$


Definition:Euclidean Relation

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.


Definition:Serial Relation

Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is '''serial''' {{iff}}: :$\forall x \in S: \exists y \in S: \tuple {x, y} \in \mathcal R$


Definition:Totally Ordered Ring

Let $\struct {R, +, \circ, \preceq}$ be an ordered ring. If the ordering $\preceq$ is a total ordering, then $\struct {R, +, \circ, \preceq}$ is a '''totally ordered ring'''.


Definition:Isolated Vertex

If the degree of $v$ is zero, then $v$ is an '''isolated vertex'''.


Definition:Locally Finite Graph

A '''locally finite graph''' $G$ is an infinite graph where every vertex of $G$ has finite degree.


Definition:Infinite Graph

A graph which has either an infinite number of edges or vertices is called an '''infinite graph'''.


Definition:Graph (Graph Theory)/Edge/Endvertex

Let $G = \struct {V, E}$ be a graph or digraph. Let $e = u v$ be an edge of $G$, that is, $e \in E$. The '''endvertices''' of $e$ are the vertices $u$ and $v$.


Definition:Link (Graph Theory)

Let $G = \left({V, E}\right)$ be a graph. A '''link''' is a edge of $G$ which is ''not'' a loop. That is, a '''link''' is an edge whose endvertices are  distinct. When $G$ is not a loop-graph, then ''all'' edges are '''links'''. Category:Definitions/Graph Theory Category:Definitions/Loop-Graphs f9wb2ctf7wc7s87eboen67avjnjs8w5


Definition:Complement (Graph Theory)

=== Simple Graph === {{:Definition:Complement (Graph Theory)/Simple Graph}} === Loop-Graph === {{:Definition:Complement (Graph Theory)/Loop-Graph}}


Definition:Null Graph

The '''null graph''' is the graph which has no vertices. That is, the '''null graph''' is the graph of order zero. It is called '''''the'' null graph''' because, from Empty Set is Unique, there is only one such entity.


Definition:Group Homomorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in G$: :$\map \phi {a \circ b} = \map \phi a * \map \phi b$ Then $\phi: \struct {G, \circ} \to \struct {H, *}$ is a group homomorphism.


Definition:Ring Homomorphism

Let $\struct {R, +, \circ}$ and $\struct{S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$. That is, $\forall a, b \in R$: {{begin-eqn}} {{eqn | n = 1       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | n = 2       | l = \map \phi {a \circ b}       | r = \map \phi a * \map \phi b }} {{end-eqn}} Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.


Definition:R-Algebraic Structure Homomorphism

Let $R$ be a ring. Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures. Let $\phi: S \to T$ be a mapping. Then $\phi$ is an '''$R$-algebraic structure homomorphism''' {{iff}}: :$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$ :$(2): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$ where $\closedint 1 n = \set {1, 2, \ldots, n}$ denotes an integer interval. Note that this definition also applies to modules and vector spaces.


Definition:Ring Monomorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a (ring) homomorphism. Then $\phi$ is a ring monomorphism {{iff}} $\phi$ is an injection.


Definition:Group Monomorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a (group) homomorphism. Then $\phi$ is a group monomorphism {{iff}} $\phi$ is an injection.


Definition:Ring Epimorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a (ring) homomorphism. Then $\phi$ is a ring epimorphism {{iff}} $\phi$ is a surjection.


Definition:Totally Ordered Field

Let $\struct {F, +, \circ, \preceq}$ be an ordered ring. Let $\struct {F, +, \circ}$ be a field. Let the ordering $\preceq$ be a total ordering. Then $\struct {F, +, \circ, \preceq}$ is a '''totally ordered field'''.


Definition:Group Epimorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a (group) homomorphism. Then $\phi$ is a group epimorphism {{iff}} $\phi$ is a surjection.


Definition:F-Homomorphism

Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Then a ring homomorphism $\varphi: R \to S$ is called an '''$F$-homomorphism''' if: :$\forall a \in F: \map \phi a = a$ That is, $\phi \restriction_F = I_F$ where: :$\phi \restriction_F$ is the restriction of $\phi$ to $F$ :$I_F$ is the identity mapping on $F$.


Definition:Homomorphism (Graph Theory)

Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs. Let there exist a mapping $F: \map V G \to \map V H$ such that: :for each edge $\set {u, v} \in \map E G$ :there exists an edge $\set {\map F u, \map F v} \in \map E H$. Then $G$ and $H$ are '''homomorphic'''. The mapping $F$ is called a '''homomorphism''' from $G$ to $H$.


Definition:Spanning Subgraph

Let $G$ be a graph. A '''spanning subgraph for $G$''' is a subgraph of $G$ which contains every vertex of $G$. This is also called a '''factor of $G$'''. {{questionable|Should it not also be specified that $G$ should be connected, and so should such a subgraph?}} Category:Definitions/Graph Theory 5q0yh81fvqx7br0i540r006k03z4scq


Definition:Induced Subgraph

Let $G = \struct {\map V G, \map E G}$ be a graph. Let $V' \subseteq V$ be a subset of vertices of $G$. The '''subgraph of $G$ induced by $V'$''' is the subgraph $G' = \struct {\map {V'} {G'}, \map {E'} {G'} }$ of $G$ such that: :$G'$ has the vertex set $V'$ :For all $u, v \in V'$, $e = u v \in \map E G$ {{iff}} $e \in \map {E'} {G'}$. That is, it contains all the edges of $G$ that connect elements of the given subset of the vertex set of $G$, and only those edges. Category:Definitions/Graph Theory s3rv5i6adxz714n5952txh488944nv0


Definition:Universal Graph

Let $\Bbb G$ be a set of graphs. A '''universal graph in $\Bbb G$''' is a graph $K$ in which every element of $\Bbb G$ can be embedded in $K$. Note that $K$ need not itself be an element of $\Bbb G$. {{SUBPAGENAME}} psjyq6jsu0vkx9s97sgljejl6sxgyxr


Definition:Undirected Graph

An '''undirected graph''' is a graph whose edge set considered as a relation is symmetric. That is, if $G$ is an '''undirected graph''' and $e = u v$ is an edge in $G$, then $v u$ is also an edge in $G$. In other words, if you can "go from $u$ to $v$", you can also "go from $v$ to $u$".


Definition:Acyclic Graph

An '''acyclic graph''' is a graph or digraph with no cycles. An acyclic connected undirected graph is a tree. An acyclic disconnected undirected graph is a forest. Category:Definitions/Graph Theory msa04pz6xvmbm8pnm72qmm2q3eqybl5


Definition:Girth

Let $G$ be a graph. The '''girth''' of $G$ is the smallest length of any cycle in $G$. An acyclic graph is defined as having a girth of infinity. Category:Definitions/Graph Theory lx5d97tx9bvd2oei5nq2w1ezn00wsae


Definition:Circumference (Graph Theory)

Let $G$ be a graph. The '''circumference''' of $G$ is the longest length of any cycle in $G$. An acyclic graph is defined as having a circumference of infinity. Category:Definitions/Graph Theory if2yu8uzvefxxd38tq9v3qkbii32gnx


Definition:Strict Ordering

=== Definition 1 === {{:Definition:Strict Ordering/Asymmetric and Transitive}} === Definition 2 === {{:Definition:Strict Ordering/Antireflexive and Transitive}}


Definition:Strict Partial Ordering

Let $\left({S, \prec}\right)$ be a relational structure. Let $\prec$ be a strict ordering. Then $\prec$ is a '''strict partial ordering''' on $S$ iff $\prec$ is not connected. That is, iff $\left({S, \prec}\right)$ has at least one pair which is non-comparable: :$\exists x, y \in S: x \not \prec y \land y \not \prec x$


Definition:Strict Total Ordering

Let $\struct {S, \prec}$ be a relational structure. Let $\prec$ be a strict ordering. Then $\prec$ is a '''strict total ordering''' on $S$ {{iff}} $\struct {S, \prec}$ has no non-comparable pairs: :$\forall x, y \in S: x \ne y \implies x \prec y \lor y \prec x$ That is, {{iff}} $\prec$ is connected.


Definition:Strict Weak Ordering

A '''strict weak ordering''' on a set $S$ is a relation $\mathcal R$ such that: :$(1): \quad \mathcal R$ is a strict partial ordering :$(2): \quad$ The incomparability relation $\mathcal R'$ defined as: ::::$a \mathrel {\mathcal R'} b  := \neg \left({a \mathrel {\mathcal R} b}\right) \land \neg \left({b \mathrel {\mathcal R} a}\right)$ :::is transitive.


Definition:Heronian Mean

The '''Heronian mean''' of two numbers $x$ and $y$ is defined as: :$\displaystyle H = \frac {x + \sqrt{xy} + y} 3$ It can also be defined as: :$\displaystyle H = \frac 2 3 \left({\frac {x + y} 2}\right) + \frac 1 3 \sqrt {x y}$ Thus it is seen to be a weighted mean of their arithmetic mean and geometric mean. {{NamedforDef|Heron of Alexandria|cat=Heron}} Category:Definitions/Algebra 31ngnwoz9cx69aihedjxw56an5tqgp2


Definition:Weighted Mean

Let $S = \sequence {x_1, x_2, \ldots, x_n}$ be a sequence of real numbers. Let $\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted mean''' of $S$ is defined as: :$\bar x := \dfrac {\displaystyle \sum_{i \mathop = 1}^n \map W {x_i} x_i} {\displaystyle \sum_{i \mathop = 1}^n \map W {x_i} }$ This means that elements of $S$ with a larger weight contribute more to the '''weighted mean''' than those with a smaller weight. If we write: :$\forall i: 1 \le i \le n: w_i = \map W {x_i}$ we can write this '''weighted mean''' as: :$\bar x := \dfrac {w_1 x_1 + w_2 x_2 + \cdots + w_n x_n} {w_1 + w_2 + \cdots + w_n}$ From the definition of the weight function, none of the weights can be negative. While some of the weights may be zero, not ''all'' of them can, otherwise we would be dividing by zero. === Normalized Weighted Mean === {{:Definition:Weighted Mean/Normalized}}


Definition:Reflexive Closure

=== Definition 1 === {{:Definition:Reflexive Closure/Union with Diagonal}} === Definition 2 === {{:Definition:Reflexive Closure/Smallest Reflexive Superset}} === Definition 3 === {{:Definition:Reflexive Closure/Intersection of Reflexive Supersets}}


Definition:Reflexive Reduction

Let $\RR$ be a relation on a set $S$. The '''reflexive reduction''' of $\RR$ is denoted $\RR^\ne$, and is defined as: :$\RR^\ne := \RR \setminus \set {\tuple {x, x}: x \in S}$


=== Definition 1 === {{:Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset}} === Definition 2 === {{:Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets}} === Definition 3 === {{:Definition:Transitive Closure (Relation Theory)/Finite Chain}} === Definition 4 === {{:Definition:Transitive Closure (Relation Theory)/Union of Compositions}}


Definition:Symmetric Closure

Let $\mathcal R$ be a relation on a set $S$. === Definition 1 === {{:Definition:Symmetric Closure/Definition 1}} === Definition 2 === {{:Definition:Symmetric Closure/Definition 2}}


Definition:Reflexive Transitive Closure

Let $\mathcal R$ be a relation on a set $S$. === Smallest Reflexive Transitive Superset === {{:Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset}} === Reflexive Closure of Transitive Closure === {{:Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure}} === Transitive Closure of Reflexive Closure === {{:Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure}}


Definition:Continuous Real Function

=== Continuity at a Point === {{Definition:Continuous Real Function/Point}} === Continuous Everywhere === {{Definition:Continuous Real Function/Everywhere}} === Continuity on a Subset of Domain === {{Definition:Continuous Real Function/Subset}}


Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$. == Continuous at a Point == {{Definition:Continuous Mapping (Topology)/Point}} == Continuous on a Set == {{Definition:Continuous Mapping (Topology)/Set}} == Continuous Everywhere == {{Definition:Continuous Mapping (Topology)/Everywhere}}


Definition:Discontinuous Mapping

=== Discontinuous Real Function === {{:Definition:Discontinuous Mapping/Real Function}} === Discontinuous Topological Space === {{:Definition:Discontinuous Mapping/Topological Space}}


Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Metric Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Metric Space)/Space}}


Definition:Universe (Set Theory)

Sets are considered to be subsets of some large '''universal set''', also called the '''universe'''. Exactly what this '''universe''' is will vary depending on the subject and context. When discussing particular sets, it should be made clear just what that '''universe''' is. However, note that from There Exists No Universal Set, this '''universe''' cannot be ''everything that there is''. The traditional symbol used to signify the '''universe''' is $\mathfrak A$. However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\mathbb U$ or just $U$ instead. With this notation, this definition can be put into symbols as: :$\forall S: S \subseteq \mathbb U$ The use of $\mathbb U$ or a variant is not universal: some sources use $X$.


Definition:Ring of Sets

{{:Definition:Ring of Sets/Definition 1}}


Definition:Ring (Abstract Algebra)/Ring Axioms

A ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in R         | m = a * b \in R         | lc= Closure under addition }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text A 4         | q = \forall a \in R: \exists a' \in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''ring axioms'''.


Definition:Group Axioms

A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G 2         | lc= Identity         | q = \exists e \in G: \forall a \in G         | m = e \circ a = a = a \circ e }} {{axiom | n = \text G 3         | lc= Inverse         | q = \forall a \in G: \exists b \in G         | m = a \circ b = e = b \circ a }} {{end-axiom}} These four stipulations are called the '''group axioms'''.


Definition:Field Axioms

The properties of a field are as follows. For a given field $\struct {F, +, \circ}$, these statements hold true: {{begin-axiom}} {{axiom | n = \text A 0         | lc= Closure under addition         | q = \forall x, y \in F         | m = x + y \in F }} {{axiom | n = \text A 1         | lc= Associativity of addition         | q = \forall x, y, z \in F         | m = \paren {x + y} + z = x + \paren {y + z} }} {{axiom | n = \text A 2         | lc= Commutativity of addition         | q = \forall x, y \in F         | m = x + y = y + x }} {{axiom | n = \text A 3         | lc= Identity element for addition         | q = \exists 0_F \in F: \forall x \in F         | m = x + 0_F = x = 0_F + x         | rc= $0_F$ is called the zero }} {{axiom | n = \text A 4         | lc= Inverse elements for addition         | q = \forall x: \exists x' \in F         | m = x + x' = 0_F = x' + x         | rc= $x'$ is called a negative element }} {{axiom | n = \text M 0         | lc= Closure under product         | q = \forall x, y \in F         | m = x \circ y \in F }} {{axiom | n = \text M 1         | lc= Associativity of product         | q = \forall x, y, z \in F         | m = \paren {x \circ y} \circ z = x \circ \paren {y \circ z} }} {{axiom | n = \text M 2         | lc= Commutativity of product         | q = \forall x, y \in F         | m = x \circ y = y \circ x }} {{axiom | n = \text M 3         | lc= Identity element for product         | q = \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F         | m = x \circ 1_F = x = 1_F \circ x         | rc= $1_F$ is called the unity }} {{axiom | n = \text M 4         | lc= Inverse elements for product         | q = \forall x \in F^*: \exists x^{-1} \in F^*         | m = x \circ x^{-1} = 1_F = x^{-1} \circ x }} {{axiom | n = \text D         | lc= Product is distributive over addition         | q = \forall x, y, z \in F         | m = x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z} }} {{end-axiom}} These are called the '''field axioms'''.


Definition:Unit of System of Sets

Let $\mathcal S$ be a system of sets. Let $U \in \mathcal S$ such that: :$\forall A \in \mathcal S: A \cap U = A$ Then $U$ is '''the unit of $\mathcal S$'''. Note that, for a given system of sets, if $U$ exists then it is unique. {{SUBPAGENAME}} 72s55wkujcn21d1czz8k1faomsc1q94


Definition:Semiring of Sets

A '''semiring of sets''' or '''semi-ring of sets''' is a system of sets $\SS$, subject to: :$(1):\quad \O \in \SS$ :$(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable :$(3):\quad$ If $A, A_1 \in \SS$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \SS$ such that: ::$(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ ::$(3b):\quad$ The $A_k$ are pairwise disjoint Alternatively, criterion $(3)$ can be replaced by: :$(3'):\quad$ If $A, B \in \SS$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.


Definition:Continuous Complex Function

As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.


Definition:Triangle (Geometry)/Isosceles

An '''isosceles triangle''' is a triangle in which two sides are the same length. :300px


Definition:Collinear

Three (or more) points are '''collinear''' {{iff}} there exists a straight line that passes through all the points.


Definition:Triangle (Geometry)/Right-Angled

A '''right-angled triangle''' is a triangle in which one of the vertices is a right angle. :450px Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.


Definition:Chord

=== Chord of a Circle === :300px {{:Definition:Chord of Circle}} === Chord of a Polygon === {{:Definition:Chord of Polygon}} === Chord of a Parabola === {{:Definition:Chord of Parabola}} === Chord of a general Curve === {{:Definition:Chord of Curve}} There are other sorts of chords, still to be documented.


Definition:Row Matrix

A '''row matrix''' is a $1 \times n$ matrix: :$\mathbf R = \begin{bmatrix} r_{1 1} & r_{1 2} & \cdots & r_{1 n} \end{bmatrix}$ That is, it is a matrix with only one row.


Definition:Complements of Parallelograms

Let $ABDC$ and $EFHG$ be two parallelograms with the same angles, which share a diagonal, such that $ABDC \cap EFHG \ne \O$. :400px Then the two parallelograms $CIGK$ and $BJHL$ are known as the '''complements''' of the parallelograms $ABDC$ and $EFHG$.


Definition:Diameter of Parallelogram

Let $ABCD$ be a parallelogram: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices.


Definition:Conjugate Angles

The '''conjugate''' of an angle $\theta$ is the angle $\phi$ such that: :$\theta + \phi = 2 \pi$ where $\theta$ and $\pi$ are expressed in radians. That is, it is the angle that makes the given angle equal to a full angle. Equivalently, the '''conjugate''' of an angle $\theta$ is the angle $\phi$ such that: :$\theta + \phi = 360 \degrees$ where $\theta$ and $\pi$ are expressed in degrees. Thus, '''conjugate angles''' are two angles whose measures add up to the measure of $4$ right angles. That is, their measurements add up to $360$ degrees or $2 \pi$ radians.


Definition:Porism

The '''porism''' is a term whose precise definition is difficult to pin down. It arises from the works of {{AuthorRef|Euclid}}, and appears to have two different meanings: # A result which follows directly, or in passing, during the course of proving something else; that is, a corollary. # A proposition that asserts that there may be conditions under which a particular problem has either no solutions or an indeterminate number of them.


Definition:Perpendicular Bisector

Let $AB$ be a line segment. The '''perpendicular bisector''' of $AB$ is the straight line which: :is perpendicular to $AB$ :passes through the point which bisects $AB$. :300px


Definition:Concentric Circles

Circles are said to be '''concentric''' if they both have the same point as the center. :300px


Definition:Baire Space (Topology)

Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Baire Space (Topology)/Definition 1}} === Definition 2 === {{:Definition:Baire Space (Topology)/Definition 2}} === Definition 3 === {{:Definition:Baire Space (Topology)/Definition 3}} === Definition 4 === {{:Definition:Baire Space (Topology)/Definition 4}}


Definition:Sequence/Infinite Sequence

An '''infinite sequence''' is a sequence whose domain is infinite.


Definition:Baire Space (Set Theory)

The '''Baire space''' $\mathbf B$ is defined as the set of all infinite sequences of natural numbers. It can also be defined as the Cartesian product of a countably infinite number of copies of the set of natural numbers.


Definition:Lindelöf Space

A topological space $T = \struct {S, \tau}$ is a '''Lindelöf space''' {{iff}} every open cover of $S$ has a countable subcover.


Definition:Concyclic Points

A set of $4$ or more points $S = \set {P_1, P_2, \ldots, P_n}$ is '''concyclic''' if they all lie on the circumference of a circle. :320px In the above diagram, points $A, B, C, D$ are '''concyclic''', as they all lie on the circumference of the circle centered at $O$.


Definition:Ramsey Number

Ramsey's Theorem states that in any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. More precisely, for any given number of colors $c$, and any given integers $n_1, \ldots, n_c$, there is a number $\map R {n_1, \ldots, n_c}$ such that: :if the edges of a complete graph of order $\map R {n_1, \ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\map R {n_1, \ldots, n_c}$ is called the '''Ramsey number''' for $n_1, \ldots, n_c$. {{NamedforDef|Frank Plumpton Ramsey|cat = Ramsey}} Category:Definitions/Ramsey Theory ss1epz5od4r49erb7jxpvryp77w1lej


Definition:Triangle (Graph Theory)

The complete graph $K_3$ of order $3$ is called a '''triangle'''. :250px Category:Definitions/Graph Theory tarxgbvw4ub9izysid9scvqkk2vm13d


Definition:Claw

The complete bipartite graph $K_{1, 3}$ is known as a '''claw'''. :250px Category:Definitions/Examples of Graphs Category:Definitions/Examples of Trees Category:Definitions/Star Graphs if50yz7bco8hmj3w5ybm8q8ue795blv


Definition:Group Product

=== Group Law === {{:Definition:Group Product/Group Law}} === Product Element === {{:Definition:Group Product/Product Element}}


Definition:Coloring

=== Vertex Coloring === {{:Definition:Coloring/Vertex Coloring}} === Edge Coloring === {{:Definition:Coloring/Edge Coloring}}


Definition:Incident (Graph Theory)/Planar Graph

Let $G = \left({V, E}\right)$ be a planar graph: Then a face of $G$ is '''incident to''' an edge $e$ of $G$ if $e$ is one of those which surrounds the face. Similarly, a face of $G$ is '''incident to''' a vertex $v$ of $G$ if $v$ is at the end of one of those incident edges. In the above graph, for example, the face $ABHC$ is incident to: : the edges $AB, BH, HC, CA$ : the vertices $A, B, H, C$.


Definition:Monochromatic Graph

A '''monochromatic graph''' is a colored graph (either vertex-colored or edge-colored, depending on the context) in which each of the vertices or edges is assigned the same color. Category:Definitions/Graph Theory Category:Definitions/Ramsey Theory 3ooml52187mqsm3jya2goo6e8dlmogt


Definition:Indeterminate Equation

An '''indeterminate equation''' is an equation for which there is more than one unknown. Hence, unless other restrictions are imposed, there may be an infinite number of solutions.


Definition:Equation

An '''equation''' is a mathematical statement that states that two expressions are equal. For expressions $A$ and $B$, this would usually be portrayed: :$A = B$ where $A$ is known as the {{LHS}} and $B$ the {{RHS}}.


Definition:Expression

An '''expression''' is a well-formed word of mathematical symbols which together can be considered as a single object. {{expand|Match to natural language use of the word}} Category:Definitions/Language Definitions rneabjlvkc4874mwc2gbp8s0rlip1tf


Definition:Algebraic Number

An algebraic number is an algebraic element of the field extension $\C / \Q$. That is, it is a complex number that is a root of a polynomial with rational coefficients.


Definition:Cyclic Quadrilateral

A '''cyclic quadrilateral''' is a quadrilateral which can be circumscribed: :300px


Definition:Recursive Sequence

A '''recursive sequence''' is a sequence where each term is defined from earlier terms in the sequence. A famous example of a recursive sequence is the Fibonacci sequence: :$F_n = F_{n-1} + F_{n-2}$ The equation which defines this sequence is called a '''recurrence relation''' or '''difference equation'''. === Initial Terms === {{:Definition:Recursive Sequence/Initial Terms}}


Definition:Lebesgue Space

Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \R$, $p \ge 1$. The '''(real) Lebesgue $p$-space of $\mu$''' is defined as: :$\map {\mathcal L^p} \mu := \set {f: X \to \R: f \in \map {\mathcal M} \Sigma, \displaystyle \int \size f^p \rd \mu < \infty}$ where $\map {\mathcal M} \Sigma$ denotes the space of $\Sigma$-measurable functions. On $\map {\mathcal L^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by: :$\forall f \in \mathcal L^p: \norm f_p := \paren {\displaystyle \int \size f^p \rd \mu}^{1 / p}$ Next, define the equivalence $\sim$ by: :$f \sim g \iff \norm {f - g}_p = 0$ The resulting quotient space: :$\map {L^p} \mu := \map {\mathcal L^p} \mu / \sim$ is also called '''(real) Lebesgue $p$-space'''.


Definition:Closed-Form Expression

An expression is a '''closed-form expression''' {{iff}} it can be expressed in terms of a bounded number of elementary functions. Informally is it in apposition to a recurrence relation, which defines a sequence of terms from earlier terms in that sequence.


Definition:Closed-Form Solution

A system of equations has a '''closed-form solution''' {{iff}} at least one solution can be expressed as a closed-form expression. Category:Definitions/Analysis Category:Definitions/Number Theory 5lp48p7qalkeo0oz6hglvonfh7wk32e


Definition:Lexicographic Order

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets. The '''lexicographic order on $S_1 \times S_2$''' is the relation $\preccurlyeq$ defined on $S_1 \times S_2$ as: :$\tuple {x_1, x_2} \preccurlyeq \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preceq_2 y_2}$


Definition:Basis Expansion

=== Positive Real Numbers === {{:Definition:Basis Expansion/Positive Real Numbers}} === Negative Real Numbers === {{:Definition:Basis Expansion/Negative Real Numbers}}


Definition:Decimal Notation

'''Decimal notation''' is the quotidian technique of expressing numbers in base $10$. Every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 10^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$


Definition:Decimal Expansion

Let $x \in \R$ be a real number. The '''decimal expansion''' of $x$ is the expansion of $x$ in base $10$. $x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {10^j}$: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$ where: :$s = \floor x$, the floor of $x$ :it is not the case that there exists $m \in \N$ such that $d_M = 9$ for all $M \ge m$. (That is, the sequence of digits does not end with an infinite sequence of $9$s.)


Definition:Neo-Pythagoreanism

'''Neo-Pythagoreanism''' was a school of philosophy which revived the teachings of {{AuthorRef|Pythagoras of Samos|Pythagoras}}. It flourished mainly in the 1st and 2nd centuries C.E. Its influence on mathematics was to associate the numbers with mystical properties, effectively adding a religious aspect to the subject. Its influence upon the Western mathematical tradition was considerably greater than it might have been, in part because of the popularity of {{AuthorRef|Nicomachus of Gerasa|Nicomachus's}} {{BookLink|Introduction to Arithmetic|Nicomachus of Gerasa}}. Category:Definitions/Schools 5sx9x7pn4a8u0bmron9i6upxur2md08


Definition:Iverson's Convention

'''Iverson's Convention''' is a notation which allows a compact means of assigning a value of $1$ or $0$ to a proposition $P$, depending on whether $P$ is true or false: :$\sqbrk P = \begin{cases} 1 & : P \ \text { is true} \\ 0 & : P \ \text { is false} \end{cases}$ It is sometimes seen specified as: :$\sqbrk P = \begin{cases} 1 & : P \ \text { is true} \\ 0 & : P \ \text { otherwise} \end{cases}$ which can be useful in fields of mathematics where the Law of the Excluded Middle does not apply.


Definition:Combinatorial Matrix

A '''combinatorial matrix''' is a square matrix specified as: :$a_{ij} = y + \delta_{ij} x$ where $\delta_{ij}$ is the Kronecker delta.


Definition:Vandermonde Matrix

The '''Vandermonde matrix''' of order $n$ is a square matrix specified variously as: :$a_{ij} = x_i^{j - 1}$ :$a_{ij} = x_j^i$ :$a_{ij} = x_i^{n - j}$ etc.


Definition:Cauchy Matrix

The '''Cauchy matrix''', commonly denoted $C_n$, can be found defined in two forms. The '''Cauchy matrix''' is an $m \times n$ matrix whose elements are in the form: : either $a_{ij} = \dfrac 1 {x_i + y_j}$ : or $a_{ij} = \dfrac 1 {x_i - y_j}$. where $x_1, x_2, \ldots, x_m$ and $y_1, y_2, \ldots, y_n$ are elements of a field $F$.


Definition:Cauchy Determinant

A '''Cauchy determinant of order $n$''' is the determinant of a square Cauchy matrix of order $n$: :$\det \left({C_n}\right) = \begin{vmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{vmatrix}$ Its value is given by: :$\det \left({C_n}\right) = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i + y_j}\right)}$ If $C_n$ is given by: :$\det \left({C_n}\right) = \begin{bmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{bmatrix}$ then its value is given by: :$\det \left({C_n}\right) = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i - y_j}\right)}$


Definition:Ordered Semigroup

An '''ordered semigroup''' is an ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.


Definition:Ordered Monoid

An '''ordered monoid''' is an ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a monoid.


Definition:Ordered Group

An '''ordered group''' is an ordered structure $\left({G, \circ, \preceq}\right)$ such that $\left({G, \circ}\right)$ is a group.


Definition:Unitary Module

Let $\struct {R, +_R, \times_R}$ be a ring with unity whose unity is $1_R$. Let $\struct {G, +_G}$ be an abelian group. A unitary module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$  which satisfies the unitary module axioms: {{:Definition:Unitary Module Axioms}}


Definition:Scalar Ring

Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations :$\circ: R \times S \to S$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$. If the scalar ring is understood, then $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ can be rendered $\struct {S, *_1, *_2, \ldots, *_n, \circ}$. === Scalar === {{:Definition:Scalar/R-Algebraic Structure}} === Zero Scalar === {{:Definition:Scalar Ring/Zero Scalar}}


Definition:Zero Vector

The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the '''zero vector''': :$\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$ Note that on occasion it is advantageous to denote the '''zero vector''' differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, in order to highlight the fact that the '''zero vector''' is not the same object as the zero scalar.


Definition:Ones Matrix

A '''ones matrix''' is a matrix all of whose elements are $1$. === Square Ones Matrix === {{:Definition:Ones Matrix/Square}} Category:Definitions/Examples of Matrices 5cjlislse9yofpn5hab0ex9o4j10tl3


Definition:Infinite Hilbert Matrix

The '''infinite Hilbert matrix''' is the infinite matrix whose elements are defined as: :$a_{ij} = \dfrac 1 {i + j - 1}$ In full, it appears as: :$\begin{bmatrix} 1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \ldots \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \ldots \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \ldots \\ \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \ldots \\ \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \ldots \\ \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots  \end{bmatrix}$ {{NamedforDef|David Hilbert|cat = Hilbert}}


Definition:Submatrix

Let $\mathbf A$ be a matrix with $m$ rows and $n$ columns. A '''submatrix''' of $\mathbf A$ is a matrix formed by selecting from $\mathbf A$: :a subset of the rows and: :a subset of the columns and forming a new matrix by using those entries, in the same relative positions, that appear in both the rows and columns of those selected.


Definition:Infinite Matrix

An '''infinite matrix''' is a matrix whose order is infinite. Category:Definitions/Matrices Category:Definitions/Infinite Sets ewbagrpivov0hm03vovayq0jx2rchdd


Definition:Hilbert Matrix

A '''Hilbert matrix''' is an order $n$ square submatrix of the infinite Hilbert matrix, consisting of the elements in the first $n$ rows and columns of that matrix. Thus it is an $n \times n$ matrix whose elements are defined as: :$a_{i j} = \dfrac 1 {i + j - 1}$


Definition:Closure Operator/Power Set

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. A '''closure operator''' on $S$ is a mapping: :$\cl: \powerset S \to \powerset S$ which satisfies the following conditions for all sets $X, Y \subseteq S$: {{begin-axiom}} {{axiom | n = 1         | lc= $\cl$ is inflationary         | q = \forall X \subseteq S         | ml= X         | mo= \subseteq         | mr= \map \cl X }} {{axiom | n = 2         | lc= $\cl$ is increasing         | q = \forall X, Y \subseteq S         | ml= X \subseteq Y         | mo= \implies         | mr= \map \cl X \subseteq \map \cl Y }} {{axiom | n = 3         | lc= $\cl$ is idempotent         | q = \forall X \subseteq S         | ml= \map \cl {\map \cl X}         | mo= =         | mr= \map \cl X }} {{end-axiom}}


Definition:Prime Factor

Let $n \in \Z$ be an integer. Then $p$ is a '''prime factor''' of $n$ {{iff}}: : $(1): \quad p$ is a prime number : $(2): \quad p$ is a divisor (that is, factor) of $n$.


Definition:Modulo Operation

Let $x, y \in \R$ be real numbers. Then the '''modulo operation''' is defined and denoted as: :$x \bmod y := \begin{cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\, \cdot \,}$ denotes the floor function.


Definition:Integral Multiple

=== Rings and Fields === {{:Definition:Integral Multiple/Rings and Fields}} === Real Numbers === This concept is often seen when $F$ is the set of real numbers $\R$. {{:Definition:Integral Multiple/Real Numbers}}


Definition:Quotient (Algebra)

Let $a, b \in \Z$ be integers such that $b \ne 0$. From the Division Theorem: :$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$ The value $q$ is defined as the '''quotient of $a$ on division by $b$''', or the '''quotient of $\dfrac a b$'''. === Real Arguments === When $x, y \in \R$ the quotient is still defined: {{:Definition:Quotient (Algebra)/Real}}


Definition:Pre-Socratic

The term '''pre-Socratic''' refers to various schools of Greek philosophy before {{AuthorRef|Socrates}}.


Definition:Eleatic School

The '''Eleatic School''' was a pre-Socratic school of Greek philosophy. Its guiding tenet was the concept of Oneness. God was the eternal Unity, who permeated the universe and guided it with his thought.


Definition:Steiner Inellipse

Let $\triangle ABC$ be a triangle whose sides have midpoints $D, E, F$. Then the unique ellipse which can be drawn wholly inside $\triangle ABC$ tangent to $D, E, F$ is called the '''Steiner inellipse''': :400px {{namedfor|Jakob Steiner}} Category:Definitions/Geometry gnl34qb94fbxkknnk6j0c2nbp18fsw3


Definition:Boundary (Graph Theory)

=== Simple Graph === Let $G = \tuple {V, E}$ be a simple graph. Let $v \in V$ be a vertex of $G$. Then the '''boundary''' of $v$ is the set of all vertices of $G$ which are adjacent to $v$: :$\map B v = \set {u \in V: \set {u, v} \in E}$ {{NoSources}} Category:Definitions/Graph Theory 0gncwb2r43l2nyhgh8hawis4u4igswy


Definition:Rectifiable Curve

A '''rectifiable curve''' is a curve of finite length.


Definition:Ray (Topology)

Let $T = \struct {S, \tau}$ be a topological space. A '''ray''' in $T$ is an embedding $R_{>0} \to S$. Category:Definitions/Topology h89tok1dvfpmc6d0xrmzp40eszkad2b


Definition:Weight Function

A '''weight function''' on a set $S$ is a mapping from $S$ to the real numbers: :$w: S \to \R$ It is common for the requirements of a specific application under discussion for the codomain of $w$ to be restricted to the positive reals: :$w: S \to \R_{\ge 0}$ The thing that determines whether a given mapping is a '''weight function''' depends more on how it is used.


Definition:Rising Factorial

Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ rising''' is defined as: :$\displaystyle x^{\overline n} := \prod_{j \mathop = 0}^{n - 1} \paren {x + j} = x \paren {x + 1} \cdots \paren {x + n - 1}$


Definition:Falling Factorial

Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ falling''' is: :$\displaystyle x^{\underline n} := \prod_{j \mathop = 0}^{n - 1} \left({x - j}\right) = x \left({x - 1}\right) \cdots \left({x - n + 1}\right)$


=== Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}}


Definition:Stirling Numbers

'''Stirling numbers''' come in various forms. In the below: : $\delta_{n k}$ is the Kronecker delta : $n$ and $k$ are non-negative integers. === Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}} === Stirling Numbers of the Second Kind === {{:Definition:Stirling Numbers of the Second Kind|Stirling Numbers of the Second Kind}}


{{:Definition:Stirling Numbers of the Second Kind/Definition 1}}


Definition:Pascal's Triangle

:$\begin{array}{r|rrrrrrrrrr} n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 & \binom n {10} & \binom n {11} & \binom n {12} \\ \hline 0  & 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 1  & 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 2  & 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 3  & 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 4  & 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 5  & 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 6  & 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 &  0  &  0 &  0 & 0 \\ 7  & 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 &  0  &  0 &  0 & 0 \\ 8  & 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 &  0  &  0 &  0 & 0 \\ 9  & 1 &  9 & 36 &  84 & 126 & 126 &  84 &  36 &   9 &  1  &  0 &  0 & 0 \\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 &  45 &  10 &  1 &  0 & 0 \\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 &  55 & 11 &  1 & 0 \\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\ \end{array}$


Definition:Stirling's Triangles

'''Stirling's Triangles''' are the arrays formed by arranging Stirling's Numbers of the first and second kind, as follows:


Definition:Array

An '''array''' is an arrangement (usually rectangular) of objects (usually numbers) such that the relations between the entries borne by their relative positions in that arrangement have a particular significance.


Definition:Congruence (Number Theory)

Let $z \in \R$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Modulo Operation}} === Definition by Integer Multiple === {{:Definition:Congruence (Number Theory)/Integer Multiple}}


Definition:Filter on Set

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter on Set/Definition 1}} === Definition 2 === {{:Definition:Filter on Set/Definition 2}} === Filtered Set === {{:Definition:Filter on Set/Filtered Set}} === Trivial Filter === {{:Definition:Filter on Set/Trivial Filter}}


Definition:Ultrafilter on Set

=== Definition 1 === {{:Definition:Ultrafilter on Set/Definition 1}} === Definition 2 === {{:Definition:Ultrafilter on Set/Definition 2}} === Definition 3 === {{:Definition:Ultrafilter on Set/Definition 3}} === Definition 4 === {{:Definition:Ultrafilter on Set/Definition 4}}


Definition:Filter Basis

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Filter Basis/Definition 2}}


Definition:Addition/Modulo Addition

{{:Definition:Addition/Modulo Addition/Definition 1}}


{{:Definition:Multiplication/Modulo Multiplication/Definition 1}}


Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a synthetic sub-basis on $X$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 2}}


Definition:Initial Topology

Let $X$ be a set. Let $I$ be an indexing set. Let $\left \langle {\left({Y_i, \tau_i}\right)} \right \rangle_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$. Let $\left \langle {f_i: X \to Y_i} \right \rangle_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.


Definition:Image Filter

Let $X, Y$ be sets. Let $\mathcal P \left({X}\right)$ and $\mathcal P \left({Y}\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \to Y$ a mapping. Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$. Then :$f \left({\mathcal F}\right) := \left\{{U \subseteq Y: f^{-1} \left({U}\right) \in \mathcal F}\right\}$ is a filter on $Y$, called the '''image filter''' of $\mathcal F$ with respect to $f$.


Definition:Open Mapping

Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping.  Then $f$ is said to be an '''open mapping''' {{iff}}: :$\forall U \in \tau_1: f \sqbrk U \in \tau_2$ where $f \sqbrk U$ denotes the image of $U$ under $f$.


Definition:Replicative Function

A '''replicative function''' is a real function $f$ such that: :$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^{n - 1} f \left({x + \frac k n}\right) = f \left({n x}\right)$ where $\sum$ denotes indexed summation.


Definition:Dimension (Geometry)

The '''dimension''' of a (geometrical) space is the minimum number of coordinates needed to specify a point in it.


Definition:Infinity

Informally, the term '''infinity''' is used to mean '''some infinite number''', but this concept falls very far short of a usable definition. The symbol $\infty$ (supposedly invented by {{AuthorRef|John Wallis}}) is often used in this context to mean '''an infinite number'''. However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about. It is defined as having the following properties: :$\forall n \in \Z: n < \infty$ :$\forall n \in \Z: n + \infty = \infty$ :$\forall n \in \Z: n \times \infty = \infty$ :$\infty^2 = \infty$ Similarly, the quantity written as $-\infty$ is defined as having the following properties: :$\forall n \in \Z: -\infty< n$ :$\forall n \in \Z: -\infty + n = -\infty$ :$\forall n \in \Z: -\infty \times n = -\infty$ :$\paren {-\infty}^2 = -\infty$ The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.


Definition:Veridical Paradox

A '''veridical paradox''' is a counter-intuitive result which can be demonstrated to be true.


Definition:Falsidical Paradox

A '''falsidical paradox''' is a result which, as well as being absurd on the surface, is the result of faulty reasoning and so is genuinely false. Such apparent paradoxes can be classified as fallacies.


Definition:Antinomy

An '''antinomy''' is a self-contradictory statement which arises as the result of a valid argument from a set of premises which contains an inherent ambiguity or contradiction. Such apparent paradoxes can also often be classified as fallacies.


Definition:Dialetheia

A '''dialetheia''' is a statement which is paradoxical through the fact that it is both true and false at the same time. As such, it is incompatible with {{AuthorRef|Aristotle}}'s Law of the Excluded Middle and Principle of Non-Contradiction, and therefore does not in general appear in the context of conventional mathematics.


Definition:Finite Difference Operator

Let $f: \R \to \R$ be a real function. The '''(finite) difference operator''' on $f$ comes in a number of forms, as follows.


Definition:Chen Prime

A '''Chen Prime''' is a prime number $p$ such that $p + 2$ is either a prime number or a semiprime. The even integer $2 p + 2$ then satisfies Chen's Theorem. The first few '''Chen primes''' are :$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, \ldots$ {{OEIS|A109611}} The first few '''non-Chen primes''' are :$43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, \ldots$ {{OEIS|A102540}}


Definition:Semiprime Number

A '''semiprime (number)''' is an integer which is the product of two (not necessarily distinct) primes.


Definition:Sufficiently Large

Let $P$ be a property of real numbers. Then: : '''$P \left({x}\right)$ holds for all sufficiently large $x$''' {{iff}}: :$\exists a \in \R: \forall x \in \R: x \ge a: P \left({x}\right)$ That is: :''There exists a real number $a$ such that for every (real) number not less than $a$, the property $P$ holds.'' It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $a$ actually needs to be known, merely that such a value can be demonstrated to exist.


Definition:Inverse Trigonometric Function

As there are six basic trigonometric functions, so each of these has its inverse functions.


Definition:Harmonic Numbers

The '''harmonic numbers''' are denoted $H_n$ and are defined for positive integers $n$: :$\displaystyle \forall n \in \Z, n \ge 0: H_n = \sum_{k \mathop = 1}^n \frac 1 k$


Definition:Vector Cross Product

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: :$\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$ :$\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$ where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$.


Definition:Digit Sum

Let $n \in \Z: n \ge 0$. The '''digit sum of $n$ to base $b$''' is the sum of all the digits of $n$ when expressed in base $b$. That is, if: :$\displaystyle n = \sum_{k \mathop \ge 0} r_k b^k$ where $0 \le r_k < b$, then: :$\displaystyle s_b \left({n}\right) = \sum_{k \mathop \ge 0} r_k$


Definition:Digital Root

Let $n \in \Z: n \ge 0$. Let $n$ be expressed in base $b$ notation. Let $n_1 = \map {s_b} n$ be the digit sum of $n$ to base $b$. Then let $n_2 = \map {s_b} {n_1}$ be the digit sum of $n_1$ to base $b$. Repeat the process, until $n_m$ has only one digit, that is, that $1 \le n_m < b$. Then $n_m$ is the '''digital root of $n$ to the base $b$'''.


Definition:Gaussian Integer

A '''Gaussian integer''' is a complex number whose real and imaginary parts are both integers. That is, a '''Gaussian integer''' is a number in the form: :$a + b i: a, b \in \Z$ The set of all '''Gaussian integers''' can be denoted $\Z \sqbrk i$, and hence can be defined as: :$\Z \sqbrk i = \set {a + b i: a, b \in \Z}$


Definition:Gaussian Rational

A '''Gaussian rational''' is a complex number whose real and imaginary parts are both rational numbers. That is, a '''Gaussian rational''' is a number in the form: :$a + b i: a, b \in \Q$ The set of all '''Gaussian rationals''' can be denoted $\Q \sqbrk i$, and hence can be defined as: :$\Q \sqbrk i = \set {a + b i: a, b \in \Q}$


Definition:Field Automorphism

Let $\struct {F, +, \circ}$ be a field. Let $\phi: F \to F$ be a (field) isomorphism from $F$ to itself. Then $\phi$ is a field automorphism.


Definition:Field Homomorphism

Let $\struct {F, +, \times}$ and $\struct {K, \oplus, \otimes}$ be fields. Let $\phi: F \to K$ be a mapping such that both $+$ and $\times$ have the morphism property under $\phi$. That is, $\forall a, b \in F$: {{begin-eqn}} {{eqn | n = 1       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | n = 2       | l = \map \phi {a \times b}       | r = \map \phi a \otimes \map \phi b }} {{end-eqn}} Then $\phi: \struct {F, +, \times} \to \struct {K, \oplus, \otimes}$ is a field homomorphism.


Definition:Monic Polynomial

Let $R$ be a commutative ring with unity. Let $f \in R \sqbrk x$ be a polynomial in one variable over $R$. Then $f$ is '''monic''' {{iff}} $f$ is nonzero and its leading coefficient is $1$.


Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}$ be a subring of $R$. Let $\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ be a polynomial in $x$ over $S$. The coefficient $a_n \ne 0_R$ is called the '''leading coefficient''' of $f$. === Polynomial Form === {{:Definition:Leading Coefficient of Polynomial/Polynomial Form}}


Definition:Experiment

An '''experiment''' can be defined in natural language as a '''test to see what happens'''. === Informal Definition === {{:Definition:Experiment/Informal Definition}} === Formal Definition === {{:Definition:Experiment/Formal Definition}}


Definition:Sample Space

Let $\EE$ be an experiment. The '''sample space''' of $\EE$ is usually denoted $\Omega$ (Greek capital '''omega'''), and is defined as '''the set of all possible outcomes of $\EE$'''.


Definition:Elementary Event

Let $\EE$ be an experiment. An '''elementary event''' of $\EE$, often denoted $\omega$ (Greek lowercase '''omega''') is one of the elements of the sample space $\Omega$ (Greek capital '''omega''') of $\EE$.


Definition:Event Space

Let $\EE$ be an experiment whose probability space is $\struct {\Omega, \Sigma, \Pr}$. The '''event space''' of $\EE$ is usually denoted $\Sigma$ (Greek capital '''sigma'''), and is '''the set of all outcomes of $\EE$ which are interesting'''. By definition, $\struct {\Omega, \Sigma}$ is a measurable space. Hence the '''event space''' $\Sigma$ is a sigma-algebra on $\Omega$.


Definition:Event

Let $\EE$ be an experiment. An '''event in $\EE$''' is an element of the event space $\Sigma$ of $\EE$.


Definition:Probability Measure

Let $\EE$ be an experiment. === Definition 1 === {{:Definition:Probability Measure/Definition 1}} === Definition 2 === {{:Definition:Probability Measure/Definition 2}} === Definition 3 === {{:Definition:Probability Measure/Definition 3}} === Definition 4 === {{:Definition:Probability Measure/Definition 4}}


Definition:Percentage

A '''percentage''' is a convenient way of expressing (usually) a fraction as an easily-grasped number, by multiplying it by $100$ and appending the symbol $\%$ which means '''per cent'''. As conventionally understood, the number being rendered as a '''percentage''' is between $0$ and $1$, thus determining the range of a '''percentage''' to be $0 \%$ to $100 \%$. However in many contexts that range is frequently extended (in particular, in the field of rhetoric): :''I want every member of this team to give one hundred and ten percent.''


Definition:Sigma-Ring

{{:Definition:Sigma-Ring/Definition 1}}


Definition:Delta-Ring

A '''delta-ring''' (which can conveniently be written '''$\delta$-ring''') is a ring of sets which is closed under countable intersections. That is, a ring of sets $\RR$ is a '''delta-ring''' {{iff}}: :$\displaystyle A_1, A_2, \ldots \in \RR \implies \bigcap_{n \mathop = 1}^\infty A_n \in \RR$


Definition:Delta-Algebra

A '''delta-algebra''' is a delta-ring with a unit. Thus, a '''delta-algebra''' is an algebra of sets which is closed under countable intersections.


Definition:Borel Algebra

A '''Borel algebra''' is sometimes used as another name for a $\sigma$-algebra or $\delta$-algebra. Can be confused with Borel $\sigma$-Algebra, which is a specific kind of $\sigma$-algebra.


Definition:Involution (Mapping)

An '''involution''' is a mapping which is its own inverse. === Definition 1 === {{:Definition:Involution (Mapping)/Definition 1}} === Definition 2 === {{:Definition:Involution (Mapping)/Definition 2}} === Definition 3 === {{:Definition:Involution (Mapping)/Definition 3}}


Definition:Additive Function (Conventional)

Let $f: S \to S$ be a mapping on an algebraic structure $\struct {S, +}$. Then $f$ is an '''additive function''' {{iff}} it preserves the addition operation: :$\forall x, y \in S: \map f {x + y} = \map f x + \map f y$


Definition:Completely Additive Function

Let $\left({R, +, \times}\right)$ be a ring. Let $f: R \to R$ be a mapping on $R$. Then $f$ is described as '''completely additive''' {{iff}}: :$\forall m, n \in R: f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$ That is, a '''completely additive function''' is one where the value of a product of two numbers equals the sum of the value of each one individually.


Definition:Additive Function (Measure Theory)

Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined to be '''additive''' {{iff}}: :$\forall S, T \in \SS: S \cap T = \O \implies \map f {S \cup T} = \map f S + \map f T$ That is, for any two disjoint elements of $\SS$, $f$ of their union equals the sum of $f$ of the individual elements. Note from Finite Union of Sets in Additive Function that: :$\displaystyle \map f {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \map f {S_i}$ where $S_1, S_2, \ldots, S_n$ is any finite collection of pairwise disjoint elements of $\SS$.


Definition:Countably Additive Function

Let $\Sigma$ be a $\sigma$-algebra. Let $f: \Sigma \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably additive''' {{iff}}: :$\displaystyle \map f {\bigcup_{n \mathop \in \N} E_n} = \sum_{n \mathop \in \N} \map f {E_n}$ where $\left \langle {E_n} \right \rangle$ is any sequence of pairwise disjoint elements of $\Sigma$. That is, for any countably infinite set of pairwise disjoint elements of $\Sigma$, $f$ of their union equals the sum of $f$ of the individual elements.


Definition:Subadditive Function (Conventional)

Let $\left({S, +_S}\right)$ and $\left({T, +_T, \preceq}\right)$ be semigroups such that $\left({T, +_T, \preceq}\right)$ is ordered. Let $f: S \to T$ be a mapping from $S$ to $T$ which satisfies the relation: :$\forall a, b \in S: f \left({a +_S b}\right) \preceq f \left({a}\right) +_T f \left({b}\right)$ Then $f$ is defined as being '''subadditive'''. The usual context in which this is encountered is where $S$ and $T$ are both the set of real numbers $\R$ (or a subset of them).


Definition:Subadditive Function (Measure Theory)

Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers. Then $f$ is defined to be '''subadditive''' (or '''sub-additive''') iff: :$\forall S, T \in \mathcal S: f \left({S \cup T}\right) \le f \left({S}\right) + f \left({T}\right)$ That is, for any two elements of $\mathcal S$, $f$ applied to their union is not greater than the sum of $f$ of the individual elements.


Definition:Countably Subadditive Function

Let $\Sigma$ be a $\sigma$-algebra over a set $X$. Let $f: \Sigma \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably subadditive''' iff, for any sequence $\left \langle {E_n} \right \rangle_{n \in \N}$ of elements of $\Sigma$: :$\displaystyle f \left({\bigcup_{n \mathop = 0}^\infty E_n}\right) \le \sum_{n \mathop = 0}^\infty f \left({E_n}\right)$


Definition:Measurable Space

Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Then the pair $\struct {X, \Sigma}$ is called a '''measurable space'''.


Definition:Monotone (Measure Theory)

Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline \R$ be an extended real-valued function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''monotone''' or '''monotonic''' iff: :$\forall A, B \in \mathcal S: A \subseteq B \iff f \left({A}\right) \le f \left({B}\right)$ Category:Definitions/Set Systems o9ucnp6vtw43gssg2efgwqknu392hel


Definition:Disjoint Union (Set Theory)

Let $\family {S_i}_{i \mathop \in I}$ be an $I$-indexed family of sets. The '''disjoint union''' of $\family {S_i}_{i \mathop \in I}$ is defined as the set: :$\displaystyle \bigsqcup_{i \mathop \in I} S_i = \bigcup_{i \mathop \in I} \set {\tuple {x, i}: x \in S_i}$ where $\bigcup$ denotes union. Each of the sets $S_i$ is canonically embedded in the '''disjoint union''' as the set: :$S_i^* = \set {\tuple {x, i}: x \in S_i}$


Definition:Disjoint Union (Probability Theory)

Let $\mathcal C$ be a collection of pairwise disjoint sets. That is, for all sets $A, B \in \mathcal C: A \ne B \implies A \cap B = \varnothing$. Then the union of all sets in $\mathcal C$ is called their '''disjoint union''': :$\displaystyle \bigsqcup_{A \mathop \in \mathcal C} A \equiv \bigcup_{A \mathop \in \mathcal C} A$ That is, in this context the term '''disjoint union''' means '''union of sets which are pairwise disjoint'''.


Definition:Equiprobable Outcomes

Let $\struct {\Omega, \Sigma, \Pr}$ be a finite probability space. Let $\Omega = \set {\omega_1, \omega_1, \ldots, \omega_n}$. Suppose that $\map \Pr {\omega_i} = \map \Pr {\omega_j}$ for all the $\omega_i, \omega_j \in \Omega$. Then from Probability Measure on Equiprobable Outcomes: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$ Such a probability space is said to have '''equiprobable outcomes''', and is sometimes referred to as an equiprobability space.


Definition:Logical Matrix

A '''logical matrix''' (or '''boolean matrix''') is a matrix whose entries are all either $0$ or $1$.


Definition:Finite Probability Space

A '''finite probability space''' is a discrete probability space $\left({\Omega, \Sigma, \Pr}\right)$ such that the sample space $\Omega$ is finite. {{SUBPAGENAME}} 3fpbykva2knibxeo21kkyel1xvzyh8i