Definition:Laplacian

=== Real-Valued Function === {{:Definition:Laplacian/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Laplacian/Vector-Valued Function}}


Definition:Laplacian/Real-Valued Function

Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$. Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''Laplacian of $f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^2 f       | o = :=       | r = \nabla \cdot \left({\nabla f}\right) }} {{eqn | r = \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \cdot \paren {\sum_{k \mathop = 1}^n \dfrac {\partial f} {\partial x_k} \mathbf e_k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\partial^2 f} {\partial {x_k}^2}       | c =  }} {{end-eqn}}


Definition:Laplacian/Vector-Valued Function

Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$. Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$. Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''Laplacian of $\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^2 \mathbf f       | o = :=       | r = \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf f} {\partial {x_k}^2} }\right) }} {{end-eqn}}


Definition:Biharmonic Operator

Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $f \left({x_1, x_2, \ldots, x_n}\right)$ denote a real-valued function on $\R^n$. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''biharmonic operator on $f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^4 f       | o = :=       | r = \nabla^2 \left({\nabla^2 f}\right) }} {{eqn | r = \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} }\right) \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} }\right) f       | c = {{Defof|Del Operator}} }} {{end-eqn}}


Definition:Primitive (Calculus)/Vector-Valued Function

Let $U \subset \R$ be an open set in $\R$. Let $\mathbf f: U \to \R^n$ be a vector-valued function on $U$: :$\forall x \in U: \map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ where: :$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$ :$\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$. Let $\mathbf f$ be differentiable on $U$. Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ {{WRT|Differentiation}} $x$. The '''primitive of $\mathbf g$ {{WRT|Integration}} $x$''' is defined as: :$\displaystyle \int \map {\mathbf g} x \rd x := \map {\mathbf f} x + \mathbf c$ where $\mathbf c$ is an arbitrary constant vector.


Definition:Definite Integral of Vector-Valued Function

Let $I \:= \closedint a b \subset \R$ be a closed real interval. Let $\mathbf f: I \to \R^n$ be a vector-valued function on $I$: :$\forall x \in I: \map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ where: :$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$ :$\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$. Let $\mathbf f$ be differentiable on $I$. Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ {{WRT|Differentiation}} $x$. The '''definite integral of $\mathbf g$ {{WRT|Integration}} $x$ from $a$ to $b$''' is defined as: :$\displaystyle \int_a^b \map {\mathbf g} x \rd x := \map {\mathbf f} b - \map {\mathbf f} a$


Definition:Derivative of Smooth Path

=== Real Cartesian Space === {{:Definition:Derivative of Smooth Path/Real Cartesian Space}} === Complex Plane === {{:Definition:Derivative of Smooth Path/Complex Plane}} Category:Definitions/Vector Analysis q44bxnogmof4cosxrf20xduw0w1a9l2


Definition:Smooth Path

=== Real Cartesian Space === {{:Definition:Smooth Path/Real Cartesian Space}} === Complex Analysis === {{:Definition:Smooth Path/Complex}} Category:Definitions/Vector Analysis ghfzofsi4chsdwv7pk61lzmt4d9zrpl


Definition:Smooth Path/Real Cartesian Space

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a path in $\R^n$. That is, let $\rho$ be a continuous real-valued function from $\left[{a \,.\,.\, b}\right]$ to $\R^n$. For each $k \in \left\{ {1, 2, \ldots, n}\right\}$, define the real function $\rho_k: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho_k \left({t}\right) = \pr_k \left({\rho \left({t}\right)}\right)$ where $\pr_k$ denotes the $k$th projection from the image $\operatorname{Im} \left({\rho}\right)$ of $\rho$ to $\R$. Then $\rho$ is a '''smooth path (in $\R^n$)''' {{iff}}: :$(1): \quad$ all of $\pr_k$ are continuously differentiable :$(2): \quad$ for all $t \in \left[{a \,.\,.\, b}\right]$, at least one $\rho_k' \left({t}\right) \ne 0$, where $\rho_k'$ denotes the derivative of $\rho_k$ {{WRT|Differentiation}} $t$.


Definition:Smooth Path/Closed/Real Cartesian Space

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. $\rho$ is a '''closed smooth path''' {{iff}} $\rho$ is a closed path. That is, {{iff}} $\rho \left({a}\right) = \rho \left({b}\right)$.


Definition:Smooth Path/Simple/Real Cartesian Space

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. $\rho$ is a '''simple smooth path (in $\R^n$)''' {{iff}}: :$(1): \quad \rho$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$ :$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \rho \left({t}\right) \ne \rho \left({b}\right)$ That is, {{iff}} $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.


Definition:Derivative of Smooth Path/Real Cartesian Space

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. For each $k \in \left\{ {1, 2, \ldots, n}\right\}$, define the real function $\rho_k: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho_k \left({t}\right) = \pr_k \left({\rho \left({t}\right)}\right)$ where $\pr_k$ denotes the $k$th projection from the image $\operatorname{Im} \left({\rho}\right)$ of $\rho$ to $\R$. It follows from the definition of a  smooth path that $\rho_k$ is continuously differentiable for all $k$. Let $\rho_k' \left({t}\right)$ denote the derivative of $\rho_k$ {{WRT|Differentiation}} $t$. The '''derivative of $\rho$''' is the continuous vector-valued function $\rho': \left[{a \,.\,.\, b}\right] \to \R^n$ defined by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho' \left({t}\right) = \displaystyle \sum_{k \mathop = 1}^n \rho_k' \left({t}\right) \mathbf e_k$ where $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ denotes the standard ordered basis of $\R^n$.


Definition:Contour

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$. For each  $i \in \left\{ {1, \ldots, n-1}\right\}$, let the end point of $\rho_i$ equal the start point of $\rho_{i + 1}$: :$\rho_i \left({b_i}\right) = \rho_{i + 1} \left({a_{i + 1} }\right)$ Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is called a '''contour''' (in $\R^n$).


Definition:Image of Contour

=== Real Cartesian Space === {{:Definition:Image of Contour (Real Cartesian Space)}} === Complex Plane === {{:Definition:Image of Contour (Complex Plane)}} Category:Definitions/Vector Analysis 7uaz6viadxp9y8eryj3rfwpmsdr83wu


Definition:Student's t-Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \R$. $X$ is said to have a '''$t$-distribution''' with $k$ degrees of freedom {{iff}} it has probability density function:  :$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$ for some $k \in \R_{> 0}$. This is written:  :$X \sim \StudentT k$


Definition:Estimator

Let $X_1, X_2, \ldots, X_n$ be random variables.  Let the joint distribution of $X_1, X_2, \ldots, X_n$ be indexed by a population parameter $\theta$.  Let $\delta: \R^n \to \R$ be a real-valued function.  The random variable $\hat \theta = \map \delta {X_1, X_2, \ldots, X_n}$ is called an '''estimator''' of $\theta$.  A particular realization of $\hat \theta$ is called an '''estimate''' of $\theta$.


Definition:Directed Smooth Curve/Complex Plane

Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a smooth path in $\C$. The '''directed smooth curve''' with '''parameterization''' $\gamma$ is defined as an equivalence class of smooth paths as follows: A smooth path $\sigma: \left[{ a \,.\,.\, b }\right] \to \C$ belongs to the equivalence class of $\gamma$ {{iff}}: : there exists a bijective differentiable strictly increasing real function: :: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ : such that $\sigma = \gamma \circ \phi$.


Definition:Directed Smooth Curve/Parameterization

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curve in $\R^n$. Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\R^n$. Then $\rho$ is a '''parameterization''' of $C$ {{iff}} $\rho$ is an element of the equivalence class that constitutes $C$.


Definition:Parameterization

=== Parameterization of Directed Smooth Curve === {{:Definition:Parameterization of Directed Smooth Curve}} === Parameterization of Contour === {{:Definition:Parameterization of Contour}}


Definition:Directed Smooth Curve/Endpoints

Let $C$ be parameterized by a smooth path $\rho: \left[{a \,.\,.\, b}\right] \to \C$. Then: : $\rho \left({a}\right)$ is the '''start point''' of $C$ : $\rho \left({b}\right)$ is the '''end point''' of $C$. Collectively, $\rho \left({a}\right)$ and $\rho \left({b}\right)$ are known as the '''endpoints''' of $\rho$.


Definition:Bias of Estimator

Let $\theta$ be a population parameter of some statistical model.  Let $\mathbf X$ be a random sample from this population. Let $\delta$ be an estimator of $\theta$.  The '''bias''' of $\delta$ is defined as:  :$\map {\operatorname{bias} } \delta = \expect {\map \delta {\mathbf X} } - \theta$


Definition:Contour/Parameterization

Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\rho: \left[{a_1 \,.\,.\, c_n}\right] \to \R^n$ with: :$\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \rho_i \left({t}\right)$ where: : $\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$ : $\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\rho$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.


Definition:Contour/Closed

Let $C$ be the '''contour''' in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. $C$ is a '''closed contour''' {{iff}} the start point of $C$ is equal to the end point of $C$: :$\rho_1 \left({a_1}\right) = \rho_n \left({b_n}\right)$


Definition:Contour/Simple

Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour in $\R^n$ defined by the finite sequence $C_1, \ldots, C_n$. $C$ is a '''simple contour''' {{iff}}: :$(1): \quad$ For all $i, j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\rho_i \left({t_1}\right) \ne \rho_j \left({t_2}\right)$ :$(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\rho_k \left({t}\right) \ne \rho_n \left({b_n}\right)$.


Definition:Contour/Length

Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''length''' of $C$ is defined as: :$\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\rho_i' \left({t}\right) }\right\vert \rd t$


Definition:Contour/Image

Let $C$ be a '''contour''' in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''image of $C$''' is defined as: :$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\rho_i}\right)$ where $\operatorname{Im} \left({\rho_i}\right)$ denotes the image of $\rho_i$.


Definition:Contour/Endpoints

Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''start point''' of $C$ is $\rho_1 \left({a_1}\right)$. The '''end point''' of $C$ is $\rho_n \left({b_n}\right)$. Collectively, $\rho_1 \left({a_1}\right)$ and $\rho_n \left({b_n}\right)$ are referred to as the '''endpoints''' of $C$.


Definition:Concatenation of Contours/Complex Plane

Let $C$ and $D$ be contours in the complex plane $\C$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\sigma_i: \closedint {c_i} {d_i} \to \C$ for all $i \in \set {1, \ldots, m}$ Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence: :$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$


Definition:Pointwise Operation/Real-Valued Functions/Multiary

For ease of notation, write $\left[{S \to \R}\right]$ for $\R^S$. Let $I$ be some index set. Let $\oplus^I: \R^I \to \R$ be an $I$-ary operation on $\R$. Then $\oplus^I: \left[{S \to \R}\right]^I \to \left[{S \to \R}\right]$, referred to as '''pointwise $\oplus^I$''', is defined as: :$\forall \left({f_i}\right)_{i \mathop \in I} \in \left[{S \to \R}\right]^I: \forall s \in S: \left({\oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I \left({f_i \left({s}\right) }\right)_{i \in I}$


Definition:Confidence Interval

Let $\theta$ be a population parameter of some population. Let $\mathbf X$ be a random sample from this population. Let $I = \openint {\map f {\mathbf X} } {\map g {\mathbf X} }$ for some real-valued functions $f$, $g$. $I$ is said to be a '''$100\gamma \%$ confidence interval''' for $\theta$ if:  :$\map \Pr {\theta \in I} = \gamma$ where $0 < \gamma < 1$.


Definition:Random Vector

Let $X_1, X_2, \ldots, X_n$ be random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Then the vector $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)$ is referred to as a '''random vector'''.


Definition:Mean Squared Error of Estimator

Let $\theta$ be a population parameter of some population. Let $\mathbf X$ be a random sample from this population. Let $\hat \theta$ be an estimator of $\theta$.  The '''mean squared error''' of $\hat \theta$ is defined by: :$\map {\operatorname {MSE} } {\hat \theta} = \expect {\paren {\map {\hat \theta} {\mathbf X} - \theta}^2} $


Definition:Density (Physics)/Area

The '''area density''' of a two-dimensional body is its mass per unit area.


Definition:Density (Physics)/Linear

The '''linear density''' of a one-dimensional body is its mass per unit length.


Definition:Expectation of Random Vector

Let $X_1, X_2, \ldots, X_n$ be random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Let $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)$ be a random vector.  Then the expected value of $\mathbf X$, $\mathbb E \left[{\mathbf X}\right]$, is defined by:  :$\mathbb E \left[{\mathbf X}\right] = \left({\mathbb E \left[{X_1}\right], \mathbb E \left[{X_2}\right], \ldots, \mathbb E \left[{X_n}\right]}\right)$


Definition:Field of Rational Functions

Let $K$ be a field. Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\map K x$ be the set of rational functions on $K$: :$\map K x := \set {\forall f \in K \sqbrk x, g \in K \sqbrk x^*: \dfrac {\map f x} {\map g x} }$ where $K \sqbrk x^* = K \sqbrk x \setminus \set {\text {the null polynomial} }$. Then $\map K x$ is the '''field of rational functions''' on $K$.


Definition:Sylow p-Subgroup/Definition 3

Let $n$ be the largest integer such that: :$p^n \divides \order G$ where $\divides$ denotes divisibility. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements.


Definition:Cross-Covariance Matrix

Let $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)^T$ and $\mathbf Y = \left({Y_1, Y_2, \ldots, Y_m}\right)^T$ be random vectors.   Then the '''cross-covariance matrix''' of $\mathbf X$ and $\mathbf Y$ is defined by:  :$\operatorname{cov} \left({\mathbf X, \mathbf Y}\right) = \mathbb E \left[{\left({\mathbf X - \mathbb E \left[{\mathbf X}\right]}\right) \left({\mathbf Y - \mathbb E \left[{\mathbf Y}\right]}\right)^T}\right]$ where this expectation exists.


Definition:Doubleton/Class Theory

Let $a$ and $b$ be sets. The class $\set {a, b}$ is a '''doubleton (class)'''. It is defined as the class of all $x$ such that $x = a$ or $x = b$: :$\set {a, b} = \set {x: x = a \lor x = b: a \ne b}$


Definition:Variance of Random Vector

Let $\mathbf X$ be a random vector. The '''variance''' of $\mathbf X$ is defined by:  :$\operatorname{var} \left({\mathbf X}\right) = \operatorname{cov} \left({\mathbf X, \mathbf X}\right)$ where $\operatorname{cov}$ is the cross-covariance matrix.


Definition:Even Permutation

$\rho$ is an '''even permutation''' {{iff}}: :$\map \sgn \rho = 1$


Definition:Odd Permutation

$\rho$ is an '''odd permutation''' {{iff}}: :$\map \sgn \rho = -1$


Definition:Zero Residue Class

Let $m \in \Z$. Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$: :$\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ Let $\eqclass 0 m$ be the residue class of $0$ (modulo $m$): :$\eqclass 0 m = \set {x \in \Z: \exists k \in \Z: x = k m}$ Then $\eqclass 0 m$ is known as the '''zero residue class (modulo $m$)'''.


Definition:Binomial Coefficient/Complex Numbers

Let $z, w \in \C$. Then $\dbinom z w$ is defined as: :$\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$ where $\Gamma$ denotes the Gamma function. When $z$ is a negative integer and $w$ is not an integer, $\dbinom z w$ is infinite.


Definition:Power (Algebra)/Real Number/Definition 3

First let $x > 1$. Let $r$ be expressed by its decimal expansion: :$r = n \cdotp d_1 d_2 d_3 \ldots$ For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {10^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as: :$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \left({\dfrac 1 x}\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$


Definition:Power (Algebra)/Real Number/Definition 3/Binary Expansion

First let $x > 1$. Let $r$ be expressed in binary notation: :$r = n \cdotp d_1 d_2 d_3 \ldots$ where $d_1, d_2, d_3 \ldots$ are in $\left\{ {0, 1}\right\}$. For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {2^k} = n + \frac {d_1} 2 + \cdots + \frac {d_k} {2^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {2^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as: :$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \left({\dfrac 1 x}\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$


Definition:General Logarithm/Common/Historical Note

'''Common logarithms''' were developed by {{AuthorRef|Henry Briggs}}, as a direct offshoot of the work of {{AuthorRef|John Napier}}. After seeing the tables that {{AuthorRef|John Napier|Napier}} published, {{AuthorRef|Henry Briggs|Briggs}} consulted {{AuthorRef|John Napier|Napier}}, and suggested defining them differently, using base $10$. In $1617$, {{AuthorRef|Henry Briggs|Briggs}} published a set of tables of logarithms of the first $1000$ positive integers. In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places. Before the advent of cheap means of electronic calculation, '''common logarithms''' were widely used as a technique for performing multiplication.


Definition:Bit

A '''bit''' is a '''binary digit'''. Category:Definitions/Computer Science 7exngkynvfq4qe904b9xsblx6x5yu1k


Definition:Product Notation (Algebra)/Propositional Function/Iverson's Convention

Let $\displaystyle \prod_{R \left({j}\right)} a_j$ be the product over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\displaystyle \prod_{j \mathop \in \Z} a_j^{\left[{R \left({j}\right)}\right]}$ where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.


Definition:Uncountable Sum

Let $X$ be an uncountable set. Let $f: X \to \left[{0 \,.\,.\, +\infty}\right]$ be an extended real-valued function. The '''uncountable sum''' of $f$ over $X$ is defined to be the supremum of the finite sums: :$\displaystyle \sum_{x \mathop \in X} f \left({x}\right) := \sup \left\{ {\sum_{x \mathop \in F} f \left({x}\right): F \subseteq X, F \text{ finite} }\right\}$


Definition:Supremum of Set/Real Numbers/Propositional Function

Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$. Let $\map R j$ be a propositional function of $j \in I$. Then we can define the '''supremum of $\family {a_j}_{j \mathop \in I}$''' as: :$\displaystyle \sup_{\map R j} a_j := \text{ the supremum of all $a_j$ such that $\map R j$ holds}$ If more than one propositional function is written under the supremum sign, they must ''all'' hold.


Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range

Let the fiber of truth of $\map R j$ be finite. Then the '''supremum of $\family {a_j}_{j \mathop \in I}$''' can be expressed as: :$\displaystyle \max_{\map R j} a_j = \text{ the maxmum of all $a_j$ such that $\map R j$ holds}$ and can be referred to as the '''maximum''' of $\family {a_j}_{j \mathop \in I}$.


Definition:Supremum of Set/Real Numbers/Propositional Function/Vacuous Supremum

Take the indexed supremum: :$\displaystyle \sup _{\Phi \left({j}\right)} a_j$ where $\Phi \left({j}\right)$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\Phi \left({j}\right)$ is true. Then $\displaystyle \sup_{\Phi \left({j}\right)} a_j$ is defined as being $-\infty$. This supremum is called a '''vacuous supremum'''. This is because: :$\forall a \in \R: \sup \left\{ {a, -\infty}\right\} = a$ Hence for all $j$ for which $\Phi \left({j}\right)$ is false, the supremum is unaffected. In this context $-\infty$ is considered as '''minus infinity''', the hypothetical quantity that has the property: : $\forall n \in \Z: -\infty< n$


Definition:Rounding/Integer

Let $y \in \R$ such that: :$y = \floor {x + \dfrac 1 2}$ Then $y$ is defined as '''$x$ rounded to the nearest integer'''.


Definition:Rounding/Treatment of Half

Consider the situation when $\dfrac x {10^n} + \dfrac 1 2$ is an integer. That is, $\dfrac x {10^n}$ is exactly midway between the two integers $\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$. Recall that the general philosophy of the process of rounding is to find the closest approximation to $x$ to a given power of $10$. Thus there are two equally valid such approximations: :$\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$ between which $\dfrac x {10^n}$ is exactly midway. There are a number of conventions which determine which is to be used.


Definition:General Dirichlet Series

Let $a_n$ be a sequence in $\C$. Let $\sequence {\lambda_n}$ be a strictly increasing sequence of non-negative real numbers whose limit is infinity. A '''general Dirichlet series of type $\lambda_n$''' is a complex function $f: \C \to \C$ defined by the series: :$\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$


Definition:Sufficiently Small

Let $P$ be a property of real numbers. Then: : '''$P \left({x}\right)$ holds for all sufficiently small $x$''' {{iff}}: :$\exists \epsilon \in \R: \forall x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$ That is, {{iff}}: :''There exists a real number $\epsilon$ such that for every (real) number not more than $\epsilon$ in in absolute value, 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 $\epsilon$ actually needs to be known, merely that such a value can be demonstrated to exist.


Definition:Extended Real Sequence

An '''extended real sequence''' is a sequence (usually infinite) whose codomain is the extended real number line.


Definition:Limit Superior of Extended Real Sequence

Let $\left \langle {x_n} \right \rangle$ be an extended real sequence. The '''limit superior''' of $\left \langle {x_n} \right \rangle$ is defined as: :$\displaystyle \limsup x_n : = \inf_{k \mathop \ge 1} \left({\sup_{n \mathop \ge k} x_n}\right)$


Definition:Limit Inferior of Extended Real Sequence

Let $\left \langle {x_n} \right \rangle$ be an extended real sequence. The '''limit inferior''' of $\left \langle {x_n} \right \rangle$ is defined as: :$\displaystyle \liminf x_n : = \sup_{k \mathop \ge 1} \left({\inf_{n \mathop \ge k} x_n}\right)$


Definition:N-Cube (Euclidean Space)

Let $R, c \in \R$ be real numbers with $R > 0$. Let $\struct {\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. An '''$n$-cube''' is a subset of $\struct {\R^n, d}$ defined as the cartesian product of closed real intervals of the form: :$\displaystyle \prod_{i \mathop = 1}^n \closedint {c - R} {c + R}_i$ where $\closedint {c - R} {c + R}_i$ is an interval in the $i$th factor of $\R^n$. The $n$-cube can be concisely expressed as: :$\closedint {c - R} {c + R}^n$ in contexts where the indices of the product are unimportant.


Definition:Diameter of Subset of Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $S \subseteq A$ be subset of $A$. Then the '''diameter''' of $S$ is the extended real number defined by: : $\operatorname {diam} \left({S}\right) := \begin{cases} \sup \left\{ {d \left({x, y}\right): x, y \in S }\right\}, \ \text{if this quantity is finite} \\ +\infty, \ \text{otherwise} \end{cases}$ In the finite case, this is, by the definition of the supremum, the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.


Definition:Arbitrarily Small

Let $P$ be a property of real numbers. We say that '''$P \left({x}\right)$ holds for arbitrarily small $\epsilon$''' (or '''there exist arbitrarily small $x$ such that $P \left({x}\right)$ holds''') {{iff}}: :$\forall \epsilon \in \R_{> 0}: \exists x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$ That is: :''For any real number $a$, there exists a (real) number not more than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However small a number you can think of, there will be an even smaller one for which $P$ still holds.''


Definition:Half-Plane of Convergence

Let $f \left({s}\right)$ be a (general) Dirichlet series. The '''half-plane of convergence of $f$''' is the set: :$\left\{ {s: s \in \C, \operatorname{Re} \left({s}\right) > \sigma_0}\right\}$ where $\sigma_0$ is abscissa of convergence of $f$.


Definition:Stirling Numbers of the First Kind/Unsigned/Complex Numbers

{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''unsigned Stirling numbers of the first kind''' $\displaystyle {r \brack r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.


Definition:Stirling Numbers of the Second Kind/Complex Numbers

{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''Stirling numbers of the second kind''' $\displaystyle {r \brace r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.


Definition:Minimal Uncountable Well-Ordered Set

Let $\Omega$ be an uncountable well-ordered set. Then $\Omega$ is the '''minimal uncountable well-ordered set''' if every initial segment in $\Omega$ is countable.


Definition:Combinatorial Number System

The '''combinatorial number system''' is a system for representing a positive integer $m$ by a sequence of digits which are the upper coefficient of a sequence of $n$ binomial coefficients for some $n \in \Z_{>0}$: :$m := k_1 k_2 k_3 \ldots k_n$ where: :$m = \dbinom {k_1} 1 + \dbinom {k_2} 2 + \dbinom {k_3} 3 + \cdots + \dbinom {k_n} n$ :$0 \le k_1 < k_2 < \cdots < k_n$


Definition:Combination with Repetition

Let $S$ be a (finite) set with $n$ elements. A '''$k$ combination of $S$ with repetition''' is a multiset with $k$ elements selected from $S$.


Definition:Bernoulli Polynomial

The '''Bernoulli polynomials''' $B_n \left({x}\right)$ are the terms of the sequence of polynomials defined as:  :$\displaystyle B_n \left({x}\right) = \sum_{k \mathop = 0}^n \binom n k B_{n - k} x^k$ where $B_k$ are the Bernoulli numbers.


Definition:Midline of Triangle

Let $\triangle ABC$ be a triangle. Let $D$ and $E$ be the midpoints of any two of the sides of $\triangle ABC$. The line $DE$ is a '''midline of $\triangle ABC$'''. 400px


Definition:Second Order Fibonacci Number

The '''second order Fibonacci numbers''' are a sequence $\left \langle {\mathcal F_n}\right \rangle$ of integers which is formally defined recursively for all $n \in \Z_{\ge 0}$ as: :$\mathcal F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ \mathcal F_{n - 1} + \mathcal F_{n - 2} + F_{n - 2} & : \text{otherwise} \end{cases}$ where $F_{n - 2}$ denotes the $n - 2$th Fibonacci number.


Definition:Fibonomial Coefficient

Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''Fibonomial coefficient''' $\dbinom n k$ is defined as: :$\dbinom n k_\mathcal F = \begin{cases} 0 & : n < 0, n > k \\ 1 & : n \ge 0, k = 0 \\ \dfrac {F_n F_{n - 1} \cdots F_{n - k + 1} } {F_k F_{k - 1} \cdots F_1} = \displaystyle \prod_{j \mathop = 1}^k \dfrac {F_{n - k + j} } {F_j} & : \text{otherwise} \end{cases}$ where $F_n$ denotes the $n$th Fibonacci number.


Definition:Tower in Set

Let $X$ be a set. Let $T$ be any non-empty subset of $X$ Let $c$ be a fixed choice function on the non-empty subsets $T$ of $X$. Let $\preccurlyeq$ be a well-ordering on $T$. The well-ordered set $\left({T,\preccurlyeq}\right)$ is a '''tower in $X$''' {{iff}}, for all $t \in T$: :$t = c \left({X \setminus S_t \left({T}\right)}\right)$ where $S_t \left({T}\right)$ is the initial segment of $T$ determined by $t$.


Definition:Zeckendorf Representation

'''Zeckendorf representation''' is a system for representing a positive integer $m$ by a sequence of digits which are the indices of a sequence of $r$ Fibonacci numbers: :$n := k_1 k_2 k_3 \ldots k_r$ where: :$n = F_{k_1} + F_{k_2} + F_{k_3} + \cdots + F_{k_r}$ :$k_1 \gg k_2 \gg k_3 \gg \cdots \gg k_r \gg 0$ where $n \gg k$ denotes that $n \ge k + 2$.


Definition:Golden Mean Number System

The '''golden mean number system''' is a system for representing a non-negative real number $x$ by a sequence of zeroes and  ones using the golden mean $\phi$ as a number base.


Definition:Proper Subtower in Set

Let $X$ be a set. Let $\left({T, \preccurlyeq}\right)$ be a tower in $X$. Then $\left({T, \preccurlyeq}\right)$ is a '''proper subtower in $X$''' {{iff}}: : $T$ is a proper subset of some set $T' \subseteq X$ and: : $\left({T', \preccurlyeq}\right)$ is a tower in $X$.


Definition:Golden Mean Number System/Simplification

Let $x \in \R_{\ge 0}$ have a representation which includes the string $011$, say: :$x = p011q$ where $p$ and $q$ are strings in $\left\{ {0, 1}\right\}$. From 100 in Golden Mean Number System is Equivalent to 011, $x$ can also be written as: :$x = p100q$ The expression $p100q$ is a '''simplification''' of $p011q$.


Definition:String/Infinite

A '''string''' $S$ in $\mathcal A$ is an '''infinite string''' {{iff}} the sequence of symbols of which it is composed is infinite.


Definition:String/Finite

A '''string''' $S$ in $\mathcal A$ is a '''finite string''' {{iff}} the sequence of symbols of which it is composed is finite.


Definition:Fibonacci String

Consider the alphabet $\left\{ {\text {a}, \text {b} }\right\}$. For all $n \in \Z_{>0}$, let $S_n$ be the (finite) string formed as: :$S_n = \begin{cases} \text {a} & : n = 1 \\ \text {b} & : n = 2 \\ S_{n - 1} S_{n - 2} & : n > 2 \end{cases}$ where $S_{n - 1} S_{n - 2}$ denotes that $S_{n - 1}$ and $S_{n - 2}$ are concatenated. The terms of the sequence $\left\langle{S_n}\right\rangle$ are '''Fibonacci strings'''.


Definition:Fibonacci Nim

'''Fibonacci nim''' is a two-person game whose rules are as follows: :$(1): \quad$ The game starts with one pile of $n$ counters. :$(2): \quad$ The first player removes a number of counters $c_1$ such that $1 \le c_1 < n$. :$(3): \quad$ Each player takes it in turns to remove $c_n$ counters such that $1 \le c_n \le 2 c_{n - 1}$ where $c_{n - 1}$ is the number of counters taken in the other player's prevous move. :$(3): \quad$ The person who removes the last counter (or counters) is the winner.


Definition:Generating Function/Parameter/Also denoted as

The symbol used for the '''parameter''' varies. $x$ is often used instead. In the field of probability theory $s$ tends to be the symbol of choice. Some authors use $\zeta$.


Definition:Linearly Recurrent Sequence

A '''linearly recurrent sequence''' is a sequence which can be defined by a recurrence relation of the form: :$a_n = c_1 a_{n - 1} + c_2 a_{n - 2} + \cdots + c_m a_{n - m}$ with appropriate initial values for $a_1, a_2, \ldots, a_{n - 1}$.


Definition:Generating Function/Extraction of Coefficient

Let $\map G z$ be the generating function for the sequence $S = \sequence {a_n}$. The '''coefficient of $z^n$ extracted from $\map G z$''' is the $n$th term of $S$, and can be denoted: :$\sqbrk {z^n} \map G z := a_n$


Definition:Elementary Symmetric Function

Let $a, b \in \Z$ be integers such that $b \ge a$. Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$. Let $m \in \Z_{>0}$ be a (strictly) positive integer. An '''elementary symmetric function of degree $m$''' is a polynomial which can be defined by the formula: {{begin-eqn}} {{eqn | l = \map {e_m} U       | r = \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \paren {\prod_{i \mathop = 1}^m x_{j_i} }       | c =  }} {{eqn | r = \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}       | c =  }} {{end-eqn}} That is, it is the sum of all products of $m$ distinct elements of $\set {x_a, x_{a + 1}, \dotsc, x_b}$.


Definition:Power Sum

Let $S$ be a finite set of numbers. Let $p \in \R$ be a real number. Then the '''$p$th power sum''' of $S$ is defined as: :$S_p = \displaystyle \sum_{x \mathop \in S} x^p$ That is, it is the summation of the $p$th powers of all the elements of $S$.


Definition:Sequence/Doubly Subscripted

A '''doubly subscripted sequence''' is a mapping whose domain is a subset of the cartesian product $\N \times \N$ of the set of natural numbers $\N$ with itself. It can be seen that a '''doubly subscripted sequence''' is an instance of a family of elements indexed by $\N^2$. A '''doubly subscripted sequence''' can be denoted $\left\langle{a_{m n} }\right\rangle_{m, \, n \mathop \ge 0}$


Definition:Generating Function/Doubly Subscripted Sequence

Let $A = \left \langle {a_{m, n} }\right \rangle$ be a doubly subscripted sequence in $\R$ for $m, n \in \Z_{\ge 0}$. Then $\displaystyle G_A \left({w, z}\right) = \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ is called the '''generating function''' for the sequence $A$.


Definition:Complex Arithmetic

'''Complex arithmetic''' is the branch of arithmetic which concerns the manipulation of complex numbers


Definition:Conjugate Pair

Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Then $z$ and $\overline z$ are a '''conjugate pair'''. Category:Definitions/Complex Conjugates 6kg0eyhxfetdrg3qh1bqlt7grgic968


Definition:Complex Algebra

'''Complex algebra''' is the branch of algebra which specifically involves complex arithmetic. Thus it is the branch of mathematics which studies the techniques of manipulation of expressions in the domain of complex numbers.


Definition:Half-Plane

Let $P$ denote the plane. Let $L$ denote an infinite straight line in $P$. Then each of the areas of $P$ on either side of $L$ is a '''half-plane'''.


Definition:Circle of Apollonius

Let $A, B$ be distinct points in the plane. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $X$ be the locus of points in the plane such that: :$X A = \map \lambda {X B}$ :400px Then $X$ is a '''circle of Apollonius'''.


Definition:Root of Unity/Complex/Order

Let $z \in U_n$. The '''order''' of $z$ is the smallest $p \in \Z_{> 0}$ such that: :$z^p = 1$


Definition:Cauchy Sequence/Normed Vector Space

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $V$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$


Definition:Cauchy Sequence/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} } $ be a normed division ring Let $\sequence {x_n}$ be a sequence in $R$. Then $\sequence {x_n} $ is a '''Cauchy sequence in the norm $\norm {\, \cdot \,}$''' {{iff}}: :$\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$ That is: : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$


Definition:Sextic Equation

Let $f \paren x = a x^6 + b x^5 + c x^4 + d x^3 + e x^2 + f x + g$ be a polynomial function over a field $\mathbb k$ of degree $6$. Then the equation $f \paren x = 0$ is the general '''sextic equation''' over $\mathbb k$.


=== Definition 1 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 2}}


Definition:Convergent Sequence/Complex Numbers/Definition 2

Let $\sequence {z_k} = \sequence {x_k + i y_k}$ be a sequence in $\C$. $\sequence {z_k}$ '''converges to the limit $c = a + i b$''' {{iff}} both: :$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$ where $\size {x_n - a}$ denotes the absolute value of $x_n - a$.


Definition:Series/Complex

A '''complex series''' $S_n$ is the limit to infinity of the sequence of partial sums of a complex sequence $\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \cdots       | c =  }} {{end-eqn}}


Definition:Bounded Mapping/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $f: S \to R$ be a mapping from $S$ into $R$. Then $f$ is '''bounded''' {{iff}} the real-valued function $\norm {\,\cdot\,} \circ f: S \to \R$ is bounded, where $\norm {\,\cdot\,} \circ f$ is the composite of $\norm {\,\cdot\,}$ and $f$. That is, $f$ is '''bounded''' if there is a constant $K \in \R_{\ge 0}$ such that $\norm{f \paren {s}} \le K$ for all $s \in S$.


Definition:Bounded Sequence/Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Then $\sequence {x_n}$ is '''bounded''' {{iff}}: :$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$


Definition:Bounded Sequence/Normed Division Ring/Unbounded

$\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded.


Definition:Bounded Sequence/Metric Space

Let $M$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Then $\sequence {x_n}$ is a '''bounded sequence''' {{iff}} $\sequence {x_n}$ is bounded in $M$. That is: :$\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$


Definition:Bounded Sequence/Metric Space/Unbounded


Definition:Metric Induced by Norm on Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Then the '''induced metric''' or the '''metric induced by $\norm {\,\cdot\,}$''' is the map $d: R \times R \to \R_{\ge 0}$ defined as: :$d \paren {x, y} = \norm {x - y}$


Definition:Series/Real

A '''real series''' $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \cdots       | c =  }} {{end-eqn}}


=== Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}}


Definition:Non-Archimedean/Norm (Division Ring)/Archimedean

A norm $\norm {\, \cdot \,}$ on a division ring $R$ is '''Archimedean''' {{iff}} it is not non-Archimedean.


Definition:Bell Number/Sequence

The sequence of Bell numbers begins: :$(1), 1, 2, 5, 15, 52, 203, 877, 4140, 21 \, 147, \ldots$ where the first $(1)$ is the zeroth '''Bell number''' $B_0$.


Definition:Divisor (Algebra)/Also known as

A '''divisor''' can also be referred to as a '''factor'''.


Definition:Greatest Common Divisor/Also known as

The '''greatest common divisor''' is often seen abbreviated as '''GCD''', '''gcd''' or '''g.c.d.''' Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs. The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable. The '''greatest common divisor''' is also known as the '''highest common factor''', or '''greatest common factor'''. '''Highest common factor''' when it occurs, is usually abbreviated as '''HCF''', '''hcf''' or '''h.c.f.''' It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$. The archaic term '''greatest common measure''' can also be found, mainly in such as {{ElementsLink}}.


Definition:Ring of Sequences

Let $\struct {R, +, \circ}$ be a ring. Given the natural numbers $\N$, the '''ring of sequences over $R$''' is the ring of mappings $\struct {R^\N, +', \circ'}$ where: :$(1): \quad R^\N$ is the set of sequences in $R$ :$(2): \quad +'$ and $\circ'$ are the (pointwise) operations induced by $+$ and $\circ$.


Definition:Ring of Cauchy Sequences

Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. Let $\CC$ be the set of Cauchy sequences on $R$. The '''ring of Cauchy sequences over $R$''' is the subring $\struct {\CC, +', \circ'}$ of $R^\N$ with unity. The (pointwise) ring operations on the '''ring of Cauchy sequences over $R$''' are defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$ :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$ The zero of the '''ring of Cauchy sequences''' is the sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero in $R$. The unity of the '''ring of Cauchy sequences''' is the sequence $\tuple {1, 1, 1, \dots}$, where $1$ is the unity of $R$. By Corollary to Cauchy Sequences form Ring with Unity, if $R$ is a valued field then the '''ring of Cauchy sequences over $R$''' is a commutative ring with unity.


Definition:Image/Also denoted as

The notation $\Img f$ to denote the image of a mapping $f$ is specific to {{ProofWiki}}. The usual notation is $\map {\mathrm {Im} } f$ or a variant, but this is too easily confused with $\map \Im z$, the imaginary part of a complex number. Hence the non-standard usage $\Img f$. Some sources use $f \sqbrk S$, where $S$ is the domain of $f$. Others just use $\map f S$, but that notation is deprecated on {{ProofWiki}} so as not to confuse it with the notation for the image of an element.


Definition:Inverse Image Mapping/Relation/Definition 1

The '''inverse image mapping''' of $\mathcal R$ is the mapping $\mathcal R^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $\map {\mathcal R^{-1} } Y$ under $\mathcal R$: :$\forall Y \in \powerset T: \map {\mathcal R^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} } & : \Img {\mathcal R} \cap Y \ne \O \\ \O & : \Img {\mathcal R} \cap Y = \O \end {cases}$


Definition:Inverse Image Mapping/Relation/Definition 2

The '''inverse image mapping''' of $\mathcal R$ is the direct image mapping of the inverse $\mathcal R^{-1}$ of $\mathcal R$: :$\mathcal R^\gets = \paren {\mathcal R^{-1} }^\to: \powerset T \to \powerset S$ That is: :$\forall Y \in \powerset T: \map {\mathcal R^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} }$


Definition:Inverse Image Mapping/Mapping/Definition 1

The '''inverse image mapping''' of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$


Definition:Inverse Image Mapping/Mapping/Definition 2

The '''inverse image mapping''' of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$: :$f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$: That is: :$\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$


Definition:Mapping/Also known as

Words which are often used to mean the same thing as '''mapping''' include: :'''transformation''' (particularly in the context of self-maps) :'''operator''' or '''operation''' :'''function''' (usually in the context of numbers) :'''map''' (but this term is discouraged, as the term is also used by some writers for '''planar graph'''). Some sources introduce the concept with informal words such as '''rule''' or '''idea''' or '''mathematical notion'''. Sources which define a '''mapping (function)''' to be only a many-to-one relation refer to a '''mapping (function)''' as a '''total mapping (total function)'''. Some use the term '''single-valued relation'''. Sources which go into analysis of multifunctions often refer to a conventional '''mapping''' as: :a '''single-valued mapping''' or '''single-valued function''' :a '''many-to-one mapping''', '''many-to-one transformation''', or '''many-to-one correspondence''', and so on. The wording can vary, for example: '''many-one''' can be seen for '''many-to-one'''. A '''mapping $f$ from $S$ to $T$''' is also described as a '''mapping on $S$ into $T$'''.


Definition:Image (Set Theory)/Mapping/Element/Also known as

The image of an element $s$ under a mapping $f$ is also called the '''functional value''', or '''value''', of $f$ at $s$. The terminology: :'''$f$ maps $s$ to $\map f s$''' :'''$f$ assigns the value $\map f s$ to $s$''' :'''$f$ carries $s$ into $\map f s$ can be found. The modifier '''by $f$''' can also be used for '''under $f$'''. Thus, for example, the '''image of $s$ by $f$''' means the same as the '''image of $s$ under $f$'''. In the context of computability theory, the following terms are frequently found: If $\tuple {x, y} \in f$, then $y$ is often called the '''output''' of $f$ for '''input''' $x$, or simply, the '''output of $f$ at $x$'''.


Definition:Surjection/Also known as

The phrase '''$f$ is surjective''' is often used for '''$f$ is a surjection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''an onto mapping''' for '''a surjection''', and '''onto''' for '''surjective'''. A mapping which is '''not surjective''' is thence described as '''into'''. A surjection $f$ from $S$ to $T$ is sometimes denoted: :$f: S \twoheadrightarrow T$ to emphasize surjectivity. {{LatexFor|for = f: S \twoheadrightarrow T}}


Definition:Constructed Semantics/Instance 2

Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_2$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A1)$ === Structures === Define the structures of $\mathscr C_2$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 2 & : \text{if $v \left({\phi}\right) = 2$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 0 \\ \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_2$ by declaring: :$\models_{\mathscr C_2} \phi$ {{iff}} $v \left({\phi}\right) = 0$


Definition:Constructed Semantics/Instance 3

Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_3$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A2)$ === Structures === Define the structures of $\mathscr C_3$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 2 & : \text{if $v \left({\phi}\right) = 0$} \\ 1 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 2 \\ \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_3$ by declaring: :$\models_{\mathscr C_3} \phi$ {{iff}} $v \left({\phi}\right) \in \set{ 0, 1 }$


Definition:Constructed Semantics/Instance 4

Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_4$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A3)$ === Structures === Define the structures of $\mathscr C_4$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2, 3 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 3$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}cccc} \phi \lor \psi & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 3 & 3 \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_4$ by declaring: :$\models_{\mathscr C_4} \phi$ {{iff}} $v \left({\phi}\right) = 0$


Definition:Cauchy Functional Equation

Let $f: \R \to \R$ be a real function. Then the '''Cauchy functional equation''' is: :$\forall x, y \in \R: f \paren {x + y} = f \paren x + f \paren y$


Definition:Constructed Semantics/Instance 5

Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_5$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A4)$ === Structures === Define the structures of $\mathscr C_5$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2, 3 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 3$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}cccc} \phi \lor \psi & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 0 & 3 \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_5$ by declaring: :$\models_{\mathscr C_5} \phi$ {{iff}} $v \left({\phi}\right) = 0$


Definition:Subtraction/Real Numbers

Let $\struct {\R, +, \times}$ be the field of real numbers. The operation of '''subtraction''' is defined on $\R$ as: :$\forall a, b \in \R: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\R$.


Definition:Subtraction/Rational Numbers

Let $\struct {\Q, +, \times}$ be the field of rational numbers. The operation of '''subtraction''' is defined on $\Q$ as: :$\forall a, b \in \Q: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\Q$.


Definition:Self-Inverse Element/Definition 1

$x$ is a '''self-inverse element of $\struct {S, \circ}$''' {{iff}} $x \circ x = e$.


Definition:Self-Inverse Element/Definition 2

$x$ is a '''self-inverse element of $\struct {S, \circ}$''' {{iff}}: :$x$ is invertible and: :$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.


Definition:Symmetric Group/n Letters

Let $S_n$ denote the set of permutations on $n$ letters. Let $\struct {S_n, \circ}$ denote the symmetric group on $S_n$. Then $\struct {S_n, \circ}$ is referred to as the '''symmetric group on $n$ letters'''.


Definition:Symmetric Group/Notation

In order not to make notation for operations on a '''symmetric group''' overly cumbersome, '''product notation''' is usually used for mapping composition. Thus $\pi \circ \rho$ is written $\pi \rho$. Also, for the same reason, rather than using $I_{S_n}$ for the identity mapping, the symbol $e$ is usually used.


Definition:Symmetric Group/Also known as

In view of the isomorphism between symmetric groups on sets of the same cardinality, the terminology '''symmetric group of degree $n$''' is often used when the nature of the underlying set is immaterial. Some sources use the term '''$n$th symmetric group'''. These terms will sometimes be used on {{ProofWiki}}. Some sources refer to the '''symmetric group''' on a set as the '''full symmetric group (on $S$)'''. Others use the term '''complete symmetric group'''. Similarly, the '''symmetric group on $n$ letters''' can be found referred to as the '''full symmetric group on $n$ letters'''. The term '''(full) symmetric group on $n$ objects''' can be found for both the general '''symmetric group''' and the '''symmetric group on $n$ letters''' Some sources use the notation $S \paren A$ to denote the set of permutations on a given set $A$, and thence $S \paren A$ to denote the '''symmetric group''' on $A$. In line with this, the notation $S \paren n$ is often used for $S_n$ to denote the '''symmetric group on $n$ letters'''. Others use $\mathcal S_n$ or some such variant. The notation $\operatorname {Sym} \paren n$ for $S_n$ can also be found. Some older sources denote the '''symmetric group on $A$''' as $\mathfrak S_A$. Such sources consequently denote the '''symmetric group on $n$ letters''' as $\mathfrak S_n$. However, this ''fraktur'' font is rarely used nowadays as it is cumbersome to reproduce and awkward to read. Be careful not to refer to $\struct {\Gamma \paren S, \circ}$ for $\card S = n$ or $S_n$ as the '''symmetric group of order $n$''', as the order of these groups is not $n$ but $n!$, from Order of Symmetric Group. === Isomorphism between Symmetric Groups === {{:Definition:Symmetric Group/Isomorphism}}


Definition:Rotation (Geometry)

A '''rotation''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows. A '''rotation''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Rotation in the Plane === {{:Definition:Rotation (Geometry)/Plane}} === Rotation in Space === {{:Definition:Rotation (Geometry)/Space}}


Definition:Rotation (Geometry)/Plane

A '''rotation''' $r_\alpha$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows. Let $O$ be a distinguished point in $\Gamma$, which has the property that: :$\map {r_\alpha} O = O$ That is, $O$ maps to itself. Let $P \in \Gamma$ such that $P \ne O$. Let $OP$ be joined by a straight line. Let a straight line $OP'$ be constructed such that: :$(1): \quad OP' = OP$ :$(2): \angle POP' = \alpha$ such that $OP \to OP'$ is in the anticlockwise direction: :300px Then: :$\map {r_\alpha} P = P'$ Thus $r_\alpha$ is a '''rotation (in the plane) of (angle) $\alpha$ about (the axis) $O$'''.


Definition:Rotation (Geometry)/Space

A '''rotation''' $r_\theta$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows. Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that: :$\forall P \in AB: \map {r_\theta} P = P$ That is, all points on $AB$ map to themselves. Let $P \in \Gamma$ such that $P \notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let a straight line $OP'$ be constructed perpendicular to $AB$ such that: :$(1): \quad OP' = OP$ :$(2): \quad \angle POP' = \theta$ such that $OP \to OP'$ is in the anticlockwise direction: :400px Then: :$\map {r_\theta} P = P'$ Thus $r_\theta$ is a '''rotation (in space) of (angle) $\theta$ about (the axis) $O$'''.


Definition:Rotation (Geometry)/Axis

Let $r_\theta$ be a rotation in the Euclidean Space $\Gamma = \R^n$. The set $A$ of points in $\Gamma$ such that: :$\forall P \in A: \map {r_\theta} P = P$ is called the '''axis of rotation''' of $r_\theta$.


Definition:Reflection (Geometry)

A '''reflection''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows. A '''reflection''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Reflection in the Plane === {{:Definition:Reflection (Geometry)/Plane}} === Reflection in Space === {{:Definition:Reflection (Geometry)/Space}} === Point Reflection in Space === {{:Definition:Reflection (Geometry)/Point}}


Definition:Reflection (Geometry)/Plane

A '''reflection''' $\phi_{AB}$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows. Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that: :$\forall P \in AB: \map {\phi_{AB} } P = P$ That is, every point on $AB$ maps to itself. Let $P \in \Gamma$ such that $P \notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let $PO$ be produced to $P'$ such that $OP = OP'$. :400px Then: :$\map {\phi_{AB} } P = P'$ Thus $\phi_{AB}$ is a '''reflection (in the plane) in (the axis of reflection) $AB$'''.


Definition:Translation Mapping/Abelian Group

Let $\struct {G, +}$ be an abelian group. Let $g \in G$. Then '''translation by $g$''' is the mapping $\tau_g: G \to G$ defined by: :$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$ where $-g$ is the inverse of $g$ with respect to $+$ in $G$.


Definition:Translation Mapping/Euclidean Space

A '''translation''' $\tau_\mathbf x$ is an isometry on the Euclidean Space $\Gamma = \R^n$ defined as: :$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$ where $\mathbf x$ is a vector in $\R^n$.


Definition:Aleph Number

The '''aleph numbers''' are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. The cardinality of the natural numbers is $\aleph_0$. The next larger cardinality is $\aleph_1$. Then $\aleph_2$ and so on.


Definition:Order of Group Element/Also known as

Some sources refer to the '''order''' of an element of a group as its '''period'''.


Definition:Order of Group Element/Also denoted as

The '''order''' of an element $x$ in a group is sometimes seen as $\map o x$. Some sources render it as $\map {\operatorname {Ord} } x$.


Definition:Cyclic Group/Notation

A '''cyclic group''' with $n$ elements is often denoted $C_n$. Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the '''cyclic group''' generated by $g$. From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a '''cyclic group'''. Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a '''cyclic group''', and the notation $\Z_m$ is used. This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$. In certain contexts $\Z_m$ is particularly useful, as it allows results about '''cyclic groups''' to be demonstrated using number theoretical techniques.


Definition:Cyclic Group/Definition 1

The group $G$ is '''cyclic''' {{iff}} every element of $G$ can be expressed as the power of one element of $G$: :$\exists g \in G: \forall h \in G: h = g^n$ for some $n \in \Z$.


Definition:Cyclic Group/Definition 2

The group $G$ is '''cyclic''' {{iff}} it is generated by one element $g \in G$: :$G = \gen g$


Definition:Generator of Group

Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. Then '''$S$ is a generator of $G$''', denoted $G = \gen S$, {{iff}} $G$ is the subgroup generated by $S$.


Definition:Cyclic Group/Generator

Let $a \in G$ be an element of $G$ such that $\gen a = G$. Then $a$ is '''a generator of $G$'''.


Definition:Multiplicative Group of Reduced Residues

Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ denote the reduced residue system modulo $m$. Consider the algebraic structure: :$\struct {\Z'_m, \times_m}$ where $\times_m$ denotes multiplication modulo $m$. Then $\struct {\Z'_m, \times_m}$ is referred to as '''the multiplicative group of reduced residues modulo $m$'''.


Definition:Completion (Normed Division Ring)

Let $\struct {R_1, \norm {\, \cdot \,}_1}$ and $\struct {R_2, \norm {\, \cdot \,}_2}$ be normed division rings. Let $M_1 = \struct {R_1, d_1}$ and $M_2 = \struct {R_2, d_2}$ be the metric spaces where $d_1: R_1 \times R_1 \to \R_{\ge 0}$ and $d_2: R_2 \times R_2 \to \R_{\ge 0}$ are the metrics induced by $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively. Then '''$\struct {R_2, \norm {\, \cdot \,}_2}$ is a completion of $\struct {R_1, \norm {\, \cdot \,}_1}$''' {{iff}}: :$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$ :$(2): \quad M_2$ is a metric completion of $\map \phi {M_1}$. That is, '''$\struct {R_2, \norm{\,\cdot\,}_2}$ is a completion of $\struct {R_1,\norm{\,\cdot\,}_1}$''' {{iff}}: :$(a): \quad M_2$ is a complete metric space :$(b): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$ :$(c): \quad \map \phi {R_1}$ is a dense subspace in $M_2$.


Definition:Existential Quantifier/Unique/Also denoted as

The symbol $\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\exists_n$. Some sources, for example {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes}}, use $\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\exists$ for the general existential quantifier.


Definition:Order of Group Element/Infinite/Definition 1

$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$: :$\order x = \infty$


Definition:Order of Group Element/Infinite/Definition 2

$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} the powers $x, x^2, x^3, \ldots$ of $x$ are all distinct: :$\order x = \infty$


Definition:Order of Group Element/Finite/Also known as

An '''element of finite order''' of $G$ is also known as a '''torsion element''' of $G$.


Definition:Order of Group Element/Finite/Definition 1

$x$ '''is of finite order''', or '''has finite order''' {{iff}} there exists $k \in \Z_{> 0}$ such that $x^k = e_G$.


Definition:Order of Group Element/Finite/Definition 2

$x$ '''is of finite order''', or '''has finite order''' {{iff}} there exist $m, n \in \Z_{> 0}$ such that $m \ne n$ but $x^m = x^n$.


Definition:Complete Normed Division Ring

A normed division ring $\struct {R, \norm {\, \cdot \,} }$ is '''complete''' {{iff}} the metric space $\struct {R, d}$ is a complete metric space where $d$ is the metric induced by the norm $\norm {\, \cdot \,}$. That is, a normed division ring $\struct {R, \norm {\, \cdot \,} }$ is '''complete''' {{iff}} every Cauchy sequence is convergent.


Definition:Order of Group Element/Infinite/Definition 3

$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} the group $\gen x$ generated by $x$ is of infinite order. :$\order x = \infty \iff \order {\gen x} = \infty$


Definition:Cardinality/Also known as

Some authors prefer the term '''order''' instead of '''cardinality''', particularly in the context of finite sets. {{AuthorRef|Georg Cantor}} used the term '''power''' and equated it with the term '''cardinal number''', using the notation $\overline {\overline M}$ for the '''cardinality''' of $M$. Some sources cut through all the complicated language and call it the '''size'''. Some sources use $\map {\#} S$ (or a variant) to denote '''set cardinality'''. This notation has its advantages in certain contexts, and is used on occasion on this website. Others use $\map C S$, but this is easy to confuse with other uses of the same or similar notation. A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$. {{BookReference|Introductory Real Analysis|1968|A.N. Kolmogorov|author2 = S.V. Fomin}} use $\map m A$ for the '''power''' of the set $A$. Further notations are $\map n A$ and $\overline A$.


Definition:Cardinality/Infinite

Let $S$ be an infinite set. The '''cardinality''' $\card S$ of $S$ can be indicated as: :$\card S = \infty$ However, it needs to be noted that this just means that the cardinality of $S$ cannot be assigned a number $n \in \N$. It means that $\card S$ is ''at least'' $\aleph_0$ (aleph null).


Definition:Field of Integers Modulo Prime

Let $p \in \Bbb P$ be a prime number. Let $\Z_p$ be the set of integers modulo $p$. Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively. The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is '''the field of integers modulo $p$'''.


Definition:Conjugate (Group Theory)/Element/Also known as

Some sources refer to the '''conjugate''' of $x$ as the '''transform''' of $x$. Some sources refer to '''conjugacy''' as '''conjugation'''.


Definition:Conjugate (Group Theory)/Element/Also defined as

Some sources define the '''conjugate of $x$ by $a$ in $G$''' as: :$x \sim y \iff \exists a \in G: x \circ a = a \circ y$ or: :$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$


Definition:Conjugate (Group Theory)/Element/Definition 1

The  '''conjugacy relation''' $\sim$ is defined on $G$ as: :$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$


Definition:Conjugate (Group Theory)/Element/Definition 2

The  '''conjugacy relation''' $\sim$ is defined on $G$ as: :$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$


Definition:Ring (Abstract Algebra)/Historical Note

According to {{AuthorRef|Ian Stewart}}, in his {{BookLink|Galois Theory|Ian Stewart|ed = 3rd|edpage = Third Edition}} of $2004$, the ring axioms were first formulated by {{AuthorRef|Heinrich Martin Weber}} in $1893$.


Definition:Norm/Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is denoted $0_R$. A '''(submultiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Submultiplicative Norm on Ring}}


Definition:Norm/Division Ring

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$. A '''(multiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Multiplicative Norm on Ring}}


Definition:Norm/Vector Space

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$. Let $V$ be a vector space over $R$, with zero $0_V$. A '''norm''' on $V$ is a map from $V$ to the nonnegative reals: :$\norm{\,\cdot\,}: V \to \R_{\ge 0}$ satisfying the '''(vector space) norm axioms''': {{Definition:Norm Axioms (Vector Space)}}


Definition:Norm/Algebra

Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. Let $A$ be an algebra over $R$. A '''norm''' on $A$ is a vector space norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ on $A$ as a vector space such that: :for all $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$


Definition:Norm/Unital Algebra

Let $A$ be an unital algebra over $R$ with unit $e$. A '''norm''' on $A$ is an algebra norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ such that: :$\norm e = 1$


Definition:Norm/Bounded Linear Transformation

Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation. Then the '''norm''' of $A$, denoted $\norm A$, is the real number defined by: :$(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$ :$(2): \quad \norm A = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne \mathbf 0_H}$ :$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$ :$(4): \quad \norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$ These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Transformation.


Definition:Norm/Bounded Linear Functional/Definition 1

The '''norm''' of $L$ is the real number defined as the supremum: :$\norm L = \sup \set {\size {L h}: \norm h_H \le 1}$


Definition:Norm/Bounded Linear Functional/Definition 2

The '''norm''' of $L$ is the supremum: :$\norm{L} = \sup \set{\left|{Lh}\right|: \norm{h}_H = 1}$


Definition:Norm/Bounded Linear Functional/Definition 3

The '''norm''' of $L$ is the supremum: :$\norm L = \displaystyle \sup \set {\frac {\size {L h} } {\norm h _H}: h \in H, h \ne \bszero_H}$


Definition:Norm/Bounded Linear Functional/Definition 4

The '''norm''' of $L$ is the infimum: :$\norm{L} = \inf \set{c > 0: \forall h \in H: \left|{Lh}\right| \le c \norm{h}_H}$


Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$. === Definition 1 === {{:Definition:Norm/Bounded Linear Functional/Definition 1}} === Definition 2 === {{:Definition:Norm/Bounded Linear Functional/Definition 2}} === Definition 3 === {{:Definition:Norm/Bounded Linear Functional/Definition 3}} === Definition 4 === {{:Definition:Norm/Bounded Linear Functional/Definition 4}} {{refactor|These ought to go into their own pages, appropriately linked}} As a consequence of definition $(4)$, we have for all $h \in H$ that $\size {L h} \le \norm L \norm h$. As $L$ is bounded, it is assured that $\norm L < \infty$.


Definition:Group Direct Product/Finite Product

Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be groups. Let $\displaystyle G = \prod_{k \mathop = 1}^n G_k$ be their cartesian product. Let $\circ$ be the operation defined on $G$ as: :$\circ := \tuple {g_1, g_2, \ldots, g_n} \circ \tuple {h_1, h_2, \ldots, h_n} = \tuple {g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}$ for all ordered $n$-tuples in $G$. The group $\struct {G, \circ}$ is called the '''(external) direct product''' of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.


Definition:Bilateral Symmetry/Axis

Let $F$ be a plane geometric figure. Let $\map R F$ be a reflection in an axis of reflection $AB$ which is a symmetry. Then $AB$ is referred to as an '''axis of symmetry''' of $F$.


Definition:Multiplicative Group of Complex Roots of Unity

Let $n \in \Z$ be an integer such that $n > 0$. Let $U_n := \set {z \in \C: z^n = 1}$ denote the set of complex $n$th roots of unity. Let $\struct {U_n, \times}$ be the algebraic structure formed by $U_n$ under complex multiplication. Then $\struct {U_n, \times}$ is the '''multiplicative group of complex $n$th roots of unity'''.


Definition:Isolated Zero

Let $f: \C \to \C$ be a complex function. Let $\map f {z_0} = 0$ for some $z_0 \in \C$.  We say $z_0$ is an '''isolated zero''' of $f$ if there exists an open ball $B$ containing $z_0$, such that $\map f w \ne 0$ for all $w \in B \setminus \set {z_0}$.


Definition:Conjugacy Action/Subsets

Let $\powerset G$ be the power set of $G$. The '''(left) conjugacy action on subsets''' is the group action $* : G \times \powerset G \to \powerset G$: :$g * S = g \circ S \circ g^{-1}$ The '''right conjugacy action on subsets''' is the group action $* : \powerset G \times G \to \powerset G$: :$S * g = g^{-1} \circ S \circ g$


Definition:Subgroup Action

Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the operation defined as: :$\forall h \in H, g \in G: h * g := h \circ g$ This is the '''subgroup action''' of $H$ on $G$.


Definition:Norm/Ring/Multiplicative

:$\norm {\,\cdot\,}: R \to \R_{\ge 0}$ satisfying the '''(ring) multiplicative norm axioms''': {{:Definition:Multiplicative Norm Axioms}}


Definition:Norm/Ring/Submultiplicative

:$\norm {\,\cdot\,}: R \to \R_{\ge 0}$ satisfying the '''(ring) submultiplicative norm axioms''': {{:Definition:Submultiplicative Norm Axioms}}


Definition:Isometric Isomorphism

Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively. Let $\phi:R \to S$ be a bijection such that: :$(1): \quad \phi: \struct {R, d_R} \to \struct {S, d_S}$ is an isometry :$(2): \quad \phi: R \to S$ is a ring isomorphism. Then $\phi$ is called an '''isometric isomorphism'''. The normed division rings $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ are said to be '''isometrically isomorphic'''. Category:Definitions/Normed Division Rings 64o9alx30idd000xrp4c8v9wwl428ll


Definition:Cartesian Product/Cartesian Space/Two Dimensions

The '''cartesian $2$nd power of $S$''' is: :$S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$ The set $S^2$ called a '''cartesian space of $2$ dimensions'''.


Definition:Bijection/Also known as

The terms :'''biunique correspondence''' :'''bijective correspondence''' are sometimes seen for '''bijection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''one-one and onto mapping''' for '''bijection'''. If a '''bijection''' exists between two sets $S$ and $T$, then $S$ and $T$ are said to be '''in one-to-one correspondence'''. Occasionally you will see the term '''set isomorphism''', but the term '''isomorphism''' is usually reserved for mathematical structures of greater complexity than a set. Some authors, developing the concept of '''inverse mapping''' independently from that of the '''bijection''', call such a mapping '''invertible'''. The symbol $f: S \leftrightarrow T$ is sometimes seen to denote that $f$ is a '''bijection''' from $S$ to $T$. Also seen sometimes is the notation $f: S \cong T$ or $S \stackrel f \cong T$ but this is cumbersome and the symbol has already got several uses. == Technical Note == {{:Definition:Bijection/Technical Note}}


Definition:Normed Division Subring

Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. A '''normed division subring''' of $\struct {R, +, \circ, \norm {\, \cdot \,} }$ is a subset $S$ of $R$ such that $\struct{S, +_S, \circ_S, \norm{\, \cdot \,}_S}$ is a normed division ring where: :$(1) \quad +_S$ is the binary operation $+$ restricted to $S \times S$ :$(2) \quad \circ_S$ is the binary operation $\circ$ restricted to $S \times S$ :$(3) \quad \norm {\, \cdot \,}_S$ is the norm $\norm {\, \cdot \,}$ restricted to $S$. Category:Definitions/Normed Division Rings qkji2ugsu8opfzad3kzqdsibjpr37iy


Definition:Square/Mapping

Let $\struct {S, \circ}$ be an algebraic structure. Let $f: S \to S$ be the mapping from $S$ to $S$ defined as: :$\forall x \in S: \map f x := x \circ x$ This is usually denoted $x^2$: :$x^2 := x \circ x$


Definition:Square/Mapping/Element

A '''square (element of $S$)''' is an element $y$ of $S$ for which: :$\exists x \in S: y = x^2$ Such a $y = x^2$ is referred to as '''the square of $x$'''.


Definition:Square/Function/Definition 1

The '''square (function) on $\F$''' is the mapping $f: \F \to \F$ defined as: :$\forall x \in \F: \map f x = x \times x$ where $\times$ denotes multiplication.


Definition:Square/Function

Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$. === Definition 1 === {{:Definition:Square/Function/Definition 1}} === Definition 2 === {{:Definition:Square/Function/Definition 2}}


Definition:Square/Function/Definition 2

The '''square (function) on $\F$''' is the mapping $f: \F \to \F$ defined as: :$\forall x \in \F: \map f x = x^2$ where $x^2$ denotes the $2$nd power of $x$.


Definition:Square/Function/Integer

The '''(integer) square function''' is the integer function $f: \Z \to \Z$ defined as: :$\forall x \in \Z: \map f x = x^2$


Definition:Ultrametric Space

An '''ultrametric space''' is a metric space $\struct {X, d}$ where $d$ is non-Archimedean.


Definition:Unique up to Isomorphism

Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to isomorphism''' {{iff}} for all objects $s,t \in S$ there is a isomorphism from $s$ to $t$.


Definition:Power of Element/Also defined as

Some treatments do not define the '''power of an element''' for a magma whose operation is non-associative.


Definition:Power Associativity

Let $\struct {S, \circ}$ be a magma. $\struct {S, \circ}$ is '''power associative''' {{iff}}: :$\forall a \in S: a \circ \paren {a \circ a} = \paren {a \circ a} \circ a$ Category:Definitions/Powers (Abstract Algebra) dyk1d393oyc3d4lh49p5uynq81ugw4t


Definition:Finitely Generated Group

Let $G$ be a group. $G$ is '''finitely generated''' {{iff}} $G$ has a generator which is finite.


Definition:Infinite Cyclic Group/Definition 1

An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\forall n \in \Z_{> 0}: n > 0 \implies \nexists a \in G, a \ne e: a^n = e$


Definition:Infinite Cyclic Group/Definition 2

An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\forall a \in G, a \ne e: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$ where $e$ is the identity element of $G$. That is, such that all the powers of $a$ are distinct.


Definition:Subset Product Action/Left

The '''(left) subset product action''' of $G$ is the group action $*: G \times \powerset G \to \powerset G$: :$\forall g \in G, S \in \powerset G: g * S = g \circ S$


Definition:Subset Product Action

Let $\struct {G, \circ}$ be a group. Let $\HH$ be the set of subgroups of $G$. === Left Subset Product Action === {{:Definition:Subset Product Action/Left}} === Right Subset Product Action === {{:Definition:Subset Product Action/Right}}


Definition:Subset Product Action/Right

The '''(right) subset product action''' of $G$ is the group action $*: G \times \powerset G \to \powerset G$: :$\forall g \in G, S \in \powerset G: g * S = S \circ g$


Definition:Generated Subgroup/Definition 2

The '''subgroup generated by $S$''' is the intersection of all subgroups of $G$ containing $S$.


Definition:Group Action on Coset Space

Let $G$ be a group. Let $H$ be a subgroup of $G$. The mapping $*: G \times G / H \to G / H$ by the rule: :$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$ is the '''group action on the (left) coset space''' $G / H$.


Definition:Trivial Quotient Group

Let $G$ be a group. The quotient group: :$G / \set {e_G} \cong G$ where: :$\cong$ denotes group isomorphism :$e_G$ denotes the identity element of $G$. is known as the '''trivial quotient group'''.


Definition:Inner Automorphism Group

Let $G$ be a group. The '''inner automorphism group''' of $G$ is the algebraic structure: :$\struct {\Inn G, \circ}$ where: :$\Inn G$ is the '''set of inner automorphisms''' of $G$ :$\circ$ denotes composition of mappings.


Definition:Zeraoulia Function

The '''Zeraoulia function''' is a new special function see <ref>E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969 </ref> , <ref> E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969. View at Google Scholar · View at MathSciNet</ref>   proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia Rafik,Alvaro Humberto Salas  Davide L.Ocampo  and published in  International Journal of Mathematics and Mathematical Sciences , <ref> Zeraoulia Rafik, Alvaro H. Salas, and David L. Ocampo, “A New Special Function and Its Application in Probability,” International Journal of Mathematics and Mathematical Sciences, vol. 2018, Article ID 5146794, 12 pages, 2018. https://doi.org/10.1155/2018/5146794</ref>,  it behave like more than error function and it is defined as : :$\displaystyle \map T a = \int_0^a \paren {e^{-x^2} }^{\map \erf x} \rd x$ Mathematica gives for the first $100$ digits: :$\map T \infty = 0 \cdotp 9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$ with $\map T 0 = 0$


Definition:Generator of Subgroup/Definition by Predicate

A generator of a subgroup can be defined by a '''predicate'''. For example: :$\gen {x \in G: x^2 = e}$ defines the subgroup of $G$ generated by the elements of $G$ of order $2$.


Definition:Generated Normal Subgroup/Definition 3

The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$,  is the smallest normal subgroup of $G$ containing $S$: :$\gen {S^G} = \gen {x G x^{-1}: x \in G}$


Definition:Nu Function/Sequence

The sequence of values of '''$\nu$ function''': $\map \nu n$ for $n = 1, 2, 3, \ldots$ begins: :$1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, \ldots$


Definition:Extension Problem

Let $H$ and $K$ be groups. The '''extension problem''' is to determine all groups $G$ such that: :$K \trianglelefteq G$ :$G / K \cong H$ where: :$K \trianglelefteq G$ denotes that $K$ is a normal subgroup of $G$ :$G / K$ denotes the quotient group of $G$ by $K$ :$\cong$ denotes group isomorphism.


Definition:Involution (Mapping)/Definition 1

$f: A \to A$ is an '''involution''' precisely when: :$\forall x \in A: \map f {\map f x} = x$ That is: :$f \circ f = I_A$ where $I_A$ denotes the identity mapping on $A$.


Definition:Involution (Mapping)/Definition 3

Let $f: A \to A$ be a mapping on $A$. Then $f$ is an involution {{iff}} $f$ is both a bijection and a symmetric relation. That is, {{iff}} $f$ is a bijection such that $f = f^{-1}$.


Definition:Self-Inverse Element/Also known as

The definition of a '''self-inverse element''' is usually made in the context of a group. Some sources refer to such an element as an '''involution'''.


Definition:Characterization Theorem

Let $E$ be a non-abelian finite simple group. Let $u \in E$ be a self-inverse element of $E$. Let $H = \map {C_E} u$ be the centralizer of $u$ in $E$. Let $G$ be a finite simple group with a self-inverse element $t$ such that $H \cong \map {C_G} t$. A '''characterization theorem''' is a theorem that proves there is only one such group type $G$. That is, that $G \cong E$ necessarily.


Definition:Group of Units/Ring

Let $\struct {R, +, \circ}$ be a ring with unity. Then the set $U_R$ of units of $\struct {R, +, \circ}$ is called the '''group of units''' of $\struct {R, +, \circ}$. This can be denoted explicitly as $\struct {U_R, \circ}$.


Definition:Chess

'''Chess''' is a game for two players. It is played using '''pieces''' placed on a '''chessboard''', where the players take turns to move a piece. A piece is '''captured''' if an opposing piece moves onto the same space. The opposing piece is '''attacking'''. The captured piece is then removed from the board, unless it is a king. It is forbidden to capture your own piece. If a piece could be captured in one move by any piece, it is '''threatened'''. A piece cannot move over another piece, unless it is a knight, or it is castling. If a king is captured, the player who captured it wins immediately. A king is put in '''check''' if it is threatened by any piece. This is '''checkmate''' if the player who owns the king cannot prevent the king from being captured using one move. The only other way to win is if a player indicates that he resigns, traditionally by knocking over his own king.


Definition:Odd Integer/Odd-Times Odd/Sequence

The sequence of '''odd-times odd integers''' begins: :$9, 15, 21, 25, 27, \ldots$


Definition:Completely Metrizable Topology

Let $\struct {S, \tau}$ be a topological space. The space $\struct {S, \tau}$ is said to be '''completely metrizable''' {{iff}} there exists a metric $d$ such that: :$\struct {S, d}$ is a complete metric space and: :$\tau$ is the topological space induced by the metric $d$. == Also see == {{refactor|Trivial, but the following statement needs to be justified in its own page}} All completely metrizable topologies are in particular metrizable topologies.


Definition:Limit of Sequence/Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n} $ be a sequence in $R$. <section notitle="true" name="LimitDefinition"> Let $\sequence {x_n}$ converge to $x \in R$. Then $x$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle x = \lim_{n \mathop \to \infty} x_n$ </section>


Definition:Null Sequence/Analysis

=== Complex Numbers === {{:Definition:Null Sequence/Complex Numbers}} === Real Numbers === {{:Definition:Null Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Null Sequence/Rational Numbers}}


Definition:Null Sequence/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0_R$. Let $\sequence {x_n}$ be a sequence in $R$ which converges to the limit $0_R$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0_R$ Then $\sequence {x_n}$ is called a '''null sequence'''.


Definition:Null Sequence

=== Normed Division Ring === {{:Definition:Null Sequence/Normed Division Ring}}


Definition:Fermat Pseudoprime/Base 4

Let $q$ be a composite number such that $4^q \equiv 4 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $4$'''.


Definition:Fermat Pseudoprime/Base 5

Let $q$ be a composite number such that $5^q \equiv 5 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $5$'''.


Definition:Null Sequence/Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} z_n = 0$ Then $\sequence {z_n}$ is called a '''(complex) null sequence'''.


Definition:Null Sequence/Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0$ Then $\sequence {x_n}$ is called a '''(real) null sequence'''.


Definition:Null Sequence/Rational Numbers

Let $\sequence {x_n}$ be a sequence in $\Q$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0$ Then $\sequence {x_n}$ is called a '''(rational) null sequence'''.


Definition:Convergent Sequence

=== Topological Space === {{:Definition:Convergent Sequence/Topology}} === Metric Space === {{:Definition:Convergent Sequence/Metric Space}} For other equivalent definitions of a '''convergent sequence''' in a Metric Space see: Definition:Convergent Sequence in Metric Space === Normed Division Ring === {{:Definition:Convergent Sequence/Normed Division Ring}}


Definition:Convergent Sequence/Metric Space

Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space. Let $\sequence {x_k}$ be a sequence in $A$.


Definition:Convergent Sequence/Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n} $ be a sequence in $R$.


Definition:Icosahedron/Truncated

A '''truncated icosahedron''' is an icosahedron whose $12$ vertices are truncated such that one third of each edge is cut off at each of both ends.


Definition:Floating Point Representation

'''Floating point''' representation is a technique whereby numbers are held in a computer in standard scientific notation.


Definition:Scientific Notation

'''Scientific notation''' is a technique for representing approximations to (usually large) numbers by presenting them in the form: :$n \approx m \times 10^e$ where: :$m$ is a rational number such that $1 \le m < 10$, expressed in decimal notation :$e$ is an integer.


Definition:Megaflop

A '''megaflop''' is a measure of calculation speed of a computer. :$1$ '''megaflop''' $= 1$ million floating point operations per second.


Definition:Gigaflop

A '''gigaflop''' is a measure of calculation speed of a computer. {{begin-eqn}} {{eqn | o = =       | r = 1        | c = '''gigaflop''' }} {{eqn | r = 1000       | c = megaflops }} {{eqn | r = 1 \times 10^9       | c = floating point operations per second }} {{end-eqn}}


Definition:Closed Disk

Let $V$ be a vectorspace and $x \in V$, $r \in \R$. Then the '''closed disk''' $\bar{D}$ centered at $x$ with radius $r$ is the set: {{begin-eqn}} $\bar{D}(x, r) := \{v \in V | \| x - v \| \le r\}$ {{end-eqn}}


Definition:Group Presentation/Relation

The standard form of a '''relation''' in a '''group presentation''' is: :$w = e$ where $w$ is a word in the group.


Definition:Topologically Equivalent Metrics/Definition 1

$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}: :For all metric spaces $\left({B, d}\right)$ and $\left({C, d'}\right)$: :For all mappings $f: B \to A$ and $g: A \to C$: :: $(1): \quad f$ is $\left({d, d_1}\right)$-continuous {{iff}} $f$ is $\left({d, d_2}\right)$-continuous :: $(2): \quad g$ is $\left({d_1, d'}\right)$-continuous {{iff}} $g$ is $\left({d_2, d'}\right)$-continuous.


Definition:Topologically Equivalent Metrics/Definition 2

$d_1$ and $d_2$ are '''topologically equivalent''' {{iff}}: :$U \subseteq A$ is $d_1$-open $\iff$ $U \subseteq A$ is $d_2$-open.


Definition:Lipschitz Equivalence/Terminology

Despite the close connection with the concept of Lipschitz continuity, this concept is rarely seen in mainstream mathematics, and appears not to have a well-established name. The name '''Lipschitz equivalence''' appears in {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland}}: :''There does not appear to be a standard name for this; the name we use is reasonably appropriate ...''


Let $R$ be a division ring. Let $\norm{\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm{\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$. Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ respectively. === Topologically Equivalent === {{:Definition:Equivalent Division Ring Norms/Topologically Equivalent}} === Convergently Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Convergently Equivalent|section=Definition}} === Null Sequence Equivalent === {{:Definition:Equivalent Division Ring Norms/Null Sequence Equivalent}} === Open Unit Ball Equivalent === {{:Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent}} === Norm is the Power of the Other Norm === {{:Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm}} === Cauchy Sequence Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent|section=Definition}}


Definition:Equivalent Division Ring Norms/Topologically Equivalent

$\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are '''equivalent''' {{iff}} $d_1$ and $d_2$ are topologically equivalent metrics.


Definition:Equivalent Division Ring Norms/Convergently Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ converges to $l$ in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ converges to $l$ in $\norm {\,\cdot\,}_2$