Definition:Equivalent Division Ring Norms/Null Sequence Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm {\,\cdot\,}_2$


Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$


Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm

$\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are '''equivalent''' {{iff}} $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$


Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ is a Cauchy sequence in $\norm {\,\cdot\,}_1 \iff \sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_2$


An '''international standard book number''' is an index number used to uniquely identify a published {{WP|Book|book}}. === ISBN-$10$ === {{:Definition:International Standard Book Number/ISBN-10}} === ISBN-$13$ === {{:Definition:International Standard Book Number/ISBN-13}}


Definition:International Standard Book Number/ISBN-10

The '''ISBN-$10$''' format consists of a string of $9$ digits followed by a check digit in the set $\set {0, 1, \ldots, 9, X}$. The check digit $d$ is calculated as follows. Let $c$ be calculated as: :$c = \displaystyle \sum_{k \mathop = 1}^9 k d_k \pmod {11}$ where $d_k$ denotes the $k$th digit in the '''ISBN-$10$'''. Then the check digit $d$ is defined as: :$d = \begin{cases} c & : c \le 9 \\ X &: c = 10 \end{cases}$


Definition:International Standard Book Number/ISBN-13

The '''ISBN-$13$''' format consists of a string of $12$ digits followed by a check digit. The check digit $d$ is calculated as: :$d = \displaystyle \sum_{k \mathop = 1}^{12} k d_k \pmod {10}$


Definition:International Standard Book Number/Linguistic Note

When referring to an '''international standard book number''' by its abbreviation '''ISBN''', beware not to vocalise it as '''"ISBN number"'''. The word '''number''' is already included in the abbreviation '''ISBN''', so does not need to be repeated.


Definition:Check Digit

A '''check digit''' is a digit or other symbol used as an aid to identifying whether a mistake has been made transmitting a codeword. It is calculated from the symbols in the codeword itself, and appended to the codeword as an extra symbol. If a mistake is made transmitting the codeword, the '''check digit''' will then be incompatible with the rest of the contents of the codeword, and reveal that a transmission error is present.


Definition:Error-Correcting Code

An '''error-correcting code''' is a technique whereby a transmitted codeword contains extra characters whose purpose is to detect whether the codeword has been corrupted during transmission.


Definition:Linear Code

Let $p$ be a prime number. Let $\Z_p$ be the set of residue classes modulo $p$. Let $\map V {n, p}$ denote the set of sequences of length $n$ of elements of $\Z_p$. A '''linear $\tuple {n, k}$-code''' is a $k$-dimensional subspace $C$ of $\map V {n, p}$ considered as a vector space over $\Z_p$ of $n$ dimensions.


Definition:Addition of Codewords in Linear Code

Let $\map V {n, p}$ denote the linear $\tuple {n, n}$-code modulo $p$. The operation of '''addition''' on $\map V {n, p}$ is defined as follows. Let $A$ and $B$ be elements of $\map V {n, p}$, that is, sequences of length $n$ of residue classes modulo $p$. Let $a_k$ and $b_k$ denote the $k$th term of $A$ and $B$ respectively. Then $C = A + B$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as: :$c_k : = a_k +_p b_k$ where $+_p$ denotes the operation of addition modulo $p$.


Definition:Linear Code/Master Code

$\map V {n, p}$ is itself a '''linear $\tuple {n, n}$-code''', and is referred to on {{ProofWiki}} as a '''master code''' (for a '''linear $\tuple {n, k}$-code''' over $\Z_p$).


Definition:Multiple of Codeword in Linear Code

Let $\map V {n, p}$ denote the linear $\tuple {n, n}$-code modulo $p$. The operation of '''multiplication''' on $\map V {n, p}$ is defined as follows. Let $A$ be an elements of $\map V {n, p}$, that is, a sequence of length $n$ of residue classes modulo $p$. Let $a_k$ denote the $k$th term of $A$. Let $c \in \Z_p$ be an element of the set $\Z_p$ of residue classes modulo $p$. Then $c A$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as: :$c_k : = c \times_p a_k$ where $\times_p$ denotes the operation of multiplication modulo $p$.


Definition:Codeword

=== Codeword of Linear Code === {{:Definition:Linear Code/Codeword}}


Definition:Linear Code/Codeword

The elements of $\map V {n, p}$ are referred to as '''codewords'''. A '''codeword''' is usually written without punctuation, so that, for example: :$\tuple {\eqclass 1 p, \eqclass 0 p, \eqclass 1 p}$ is presented as: :$101$


Definition:Weight of Linear Codeword

Let $C$ be a codeword of a linear code. The '''weight''' of $C$ is the number of non-zero terms of $C$.


Definition:Distance between Linear Codewords

Let $u$ and $v$ be two codewords of a linear code. The '''distance''' between $u$ and $v$ is the number of corresponding terms at which $u$ and $v$ are different.


Definition:Minimum Distance of Linear Code

Let $C$ be a linear code whose master code is $\map V {n, p}$. The '''minimum distance $d$ of $C$''' is defined as: :$\map d C := \displaystyle \min_{u, v \mathop \in C: u \mathop \ne v} \set {\map d {u, v} }$ where $\map d {u, v}$ denotes the distance between $u$ and $v$.


Definition:Standard Generator Matrix for Linear Code

Let $n, k \in \Z_{>0}$ be strictly positive integers such that $n > k$. Let $p$ be a prime number. Let $\Z_p$ denote the set of residue classes modulo $p$. A '''(standard) generator matrix''' $G$ over $\Z_p$ is a $k \times n$ matrix such that: :The elements of $G$ are elements of $\Z_p$ :The first $k$ columns form the $k \times k$ identity matrix.


Definition:Golay Ternary Code

The '''Golay ternary code''' is the linear $\tuple {11, 6}$ code over $\Z_3$ whose standard generator matrix $G$ is given by: :$G := \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1  \end{pmatrix}$


Definition:Transmission Error

Let $c$ be a transmitted codeword of a message sent over a transmission line $T$. Let $c'$ be the corresponding word received across $T$. Each term of $c'$ may or may not match the corresponding term of $c$. Each term of $c'$ which does ''not'' match the corresponding term of $c$ is referred to as an instance of a '''transmission error'''. Category:Definitions/Information Theory hx64fvv2g0bag2ytqohht7jnyphtbmj


Definition:Transmitted Codeword

Let $M$ be a message sent over a transmission line $T$. Let $M$ consist of a number of codewords. Each of the codewords comprising $M$, as it is before it has been transmitted across $T$, is referred to as a '''transmitted codeword'''.


Definition:Received Word

Let $M$ be a message sent over a transmission line $T$. Let $M$ consist of a number of codewords. Each of the codewords comprising $M$, as it is after it has been transmitted across $T$, is referred to as a '''received word'''. Note that if a '''received word''' $c'$ does not match its corresponding transmitted codeword $c$, then it is possible that $c'$ is not actually a codeword. Category:Definitions/Information Theory 6t0pb7pjlbka8zk4gzbsktw21q1p1eb


Definition:Trillion

=== Short Scale === {{:Definition:Trillion/Short Scale}} === Long Scale === {{:Definition:Trillion/Long Scale}}


Definition:Trillion/Short Scale

'''Trillion''' is a name for $10^{12}$ in the short scale system: :'''One trillion''' $= 10^{3 \times 3 + 3}$


Definition:Trillion/Long Scale

'''Trillion''' is a name for $10^{18}$ in the long scale system: :'''One trillion''' $= 10^{6 \times 3}$


Definition:Quadrillion

=== Short Scale === {{:Definition:Quadrillion/Short Scale}} === Long Scale === {{:Definition:Quadrillion/Long Scale}}


Definition:Quadrillion/Long Scale

'''Quadrillion''' is a name for $10^{24}$ in the long scale system: :'''One quadrillion''' $= 10^{6 \times 4}$


Definition:Quadrillion/Short Scale

'''Quadrillion''' is a name for $10^{15}$ in the short scale system: :'''One quadrillion''' $= 10^{3 \times 4 + 3}$


Definition:Quintillion/Short Scale

'''Quintillion''' is a name for $10^{18}$ in the short scale system: :'''One quintillion''' $= 10^{3 \times 5 + 3}$


Definition:Quintillion/Long Scale

'''Quintillion''' is a name for $10^{30}$ in the long scale system: :'''One quintillion''' $= 10^{6 \times 5}$


Definition:Milliard

'''Milliard''' is an archaic name for $10^9$.


Definition:Billiard

'''Billiard''' is an archaic name for $10^{15}$.


Definition:Zero Codeword

A '''zero codeword''' is a codeword whose terms are all zero. Category:Definitions/Information Theory eppxi3htxndryrm50ptdbnt07osvxvj


Definition:Coset Decoding Table

A '''coset decoding table''' is a technique for decoding a linear $\tuple {n, k}$ code. Let $C$ be a linear $\tuple {n, k}$ code whose master code is $\map V {n, p}$. Let $T$ be an array constructed as follows: :The first row consists of the codewords of $C$, starting with the zero codeword first. {{explain|It has not yet been rigorously proved that $C$ does actually contain the zero codeword.}} :Each subsequent row is a left coset of $C$. :The entries of the first column of $T$ are the coset representatives, now called '''coset leaders'''. :The $r$th coset leader is allocated by choosing any element of $\map V {n, p}$ of minimum weight which is not already included in the first $r - 1$ rows. Then $T$ is a '''coset decoding table'''. Note that it may not always be easy to find the $r$th coset leader.


Definition:Standard Parity Check Matrix

Let $C$ be a linear $\tuple {n, k}$ code whose master code is $\map V {n, p}$. Let $G$ be the $k \times n$ standard generator matrix of $C$: :$G = \paren {\begin{array} {c|c} \mathbf I_k & \mathbf A \end{array} }$ where: :$\mathbf I$ denotes the identity matrix of order $k$ :$\mathbf A$ denotes some $k \times \paren {n - k}$ matrix. The '''(standard) parity check matrix associated with $G$''' is the $\paren {n - k} \times n$ matrix: :$P = \paren {\begin{array} {c|c} -\mathbf A^\intercal & \mathbf I_{n - k} \end{array} }$ where: :$\mathbf A^\intercal$ denotes the transpose of $\mathbf A$ :$\mathbf I_{n - k}$ denotes the identity matrix of order $n - k$ :the $-$ sign before $\mathbf A^\intercal$ denotes that each of the elements of $\mathbf A$ is to be replaced with its inverse element in $\Z_p$, the additive group of integers modulo $p$.


Definition:Trivial Norm/Division Ring/Nontrivial

A norm $\norm {\, \cdot \,}$ on a division ring $R$ is '''nontrivial''' {{iff}} it is not trivial.


Definition:Antihomomorphism/Group Antihomomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be groups. Then $\phi: S \to T$ is a '''group antihomomorphism''' {{iff}}: :$\forall x, y \in S:\map \phi {x \circ y} = \map \phi y * \map \phi x$


Definition:Antihomomorphism/Field Antihomomorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be fields. Then a ring antihomomorphism $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is called a '''field antihomomorphism'''.


Definition:Absolute Galois Group/Definition 1

The '''absolute Galois group''' of $K$ is the Galois group $\Gal {K^{\operatorname{sep} } \mid K}$ of its separable closure.


Definition:Absolute Galois Group/Definition 2

The '''absolute Galois group''' of $K$ is the automorphism group $\Aut {\overline K \mid K}$ of its algebraic closure.


Definition:Acceleration/Dimension

'''Acceleration''' has dimension $\mathsf {L T}^{-2}$.


Definition:Acceleration/Units

=== SI === {{:Definition:Acceleration/Units/SI}} === CGS === {{:Definition:Acceleration/Units/CGS}} === FPS === {{:Definition:Acceleration/Units/FPS}}


Definition:Group Action/Left Group Action

A '''(left) group action''' is an operation $\phi: G \times X \to X$ such that: :$\forall \tuple {g, x} \in G \times X: g * x := \map \phi {g, x} \in X$ in such a way that the group action axioms are satisfied: {{:Definition:Group Action Axioms}}


Definition:Orbit (Group Theory)/Definition 1

The '''orbit''' of an element $x \in X$ is defined as: :$\Orb x := \set {y \in X: \exists g \in G: y = g * x}$ where $*$ denotes the group action. That is, $\Orb x = G * x$. Thus the '''orbit''' of an element is all its possible destinations under the group action.


Definition:Orbit (Group Theory)/Definition 2

Let $\mathcal R$ be the relation on $X$ defined as: :$\forall x, y \in X: x \mathrel {\mathcal R} y \iff \exists g \in G: y = g * x$ where $*$ denotes the group action. From Group Action Induces Equivalence Relation, $\mathcal R$ is an equivalence relation. The '''orbit''' of $x$, denoted $\Orb x$, is the equivalence class of $x$ under $\mathcal R$.


Definition:Gazillion

'''Gazillion''' is a rhetorical flourish to mean '''a very large number''', generally understood to be of the order of an unspecified power of $10$.


Definition:Internal Group Direct Product/Definition 1

The group $\struct {G, \circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}} the mapping: :$C: H_1 \times H_2 \to G: \map C {h_1, h_2} = h_1 \circ h_2$ is a group isomorphism from the (group) direct product $\struct {H_1, \circ {\restriction_{H_1} } } \times \struct {H_2, \circ {\restriction_{H_2} } }$ onto $\struct {G, \circ}$.


Definition:Internal Group Direct Product/Definition 2

The group $\struct {G, \circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}}: :$(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ and $\struct {H_2, \circ {\restriction_{H_2} } }$ are both normal subgroups of $\struct {G, \circ}$ :$(2): \quad$ every element of $G$ can be expressed uniquely in the form: ::::$g = h_1 \circ h_2$


Definition:Internal Group Direct Product/General Definition/Definition 1

The group $\struct {G, \circ}$ is the '''internal group direct product of $\sequence {H_n}$''' {{iff}} the mapping: :$\displaystyle C: \prod_{k \mathop = 1}^n H_k \to G: \map C {h_1, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$ is a group isomorphism from the group direct product $\struct {H_1, \circ {\restriction_{H_1} } } \times \cdots \times \struct {H_n, \circ {\restriction_{H_n} } }$ onto $\struct {G, \circ}$.


Definition:Internal Group Direct Product/General Definition/Definition 2

:$(1): \quad$ Each $H_1, H_2, \ldots, H_n$ is a normal subgroup of $G$ :$(2): \quad$ Each element $g$ of $G$ can be expressed uniquely in the form: ::::$g = h_1 \circ h_2 \circ \cdots \circ h_n$ :::where $h_i \in H_i$ for all $i \in \set {1, 2, \ldots, n}$.


Definition:Internal Group Direct Product/Definition 3

The group $\struct {G, \circ}$ is the '''internal group direct product of $H_1$ and $H_2$''' {{iff}}: :$(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ and $\struct {H_2, \circ {\restriction_{H_2} } }$ are both normal subgroups of $\struct {G, \circ}$ :$(2): \quad G$ is the subset product of $H_1$ and $H_2$, that is: $G = H_1 \circ H_2$ :$(3): \quad$ $H_1 \cap H_2 = \set e$ where $e$ is the identity element of $G$.


Definition:Even Integer/Definition 1

An integer $n \in \Z$ is '''even''' {{iff}} it is divisible by $2$.


Definition:Even Integer/Definition 3

An integer $n \in \Z$ is '''even''' {{iff}}: :$x \equiv 0 \pmod 2$ where the notation denotes congruence modulo $2$.


Definition:Even Integer/Definition 2

An integer $n \in \Z$ is '''even''' {{iff}} it is of the form: :$n = 2 r$ where $r \in \Z$ is an integer.


Definition:Odd Integer/Definition 3

An integer $n \in \Z$ is '''odd''' {{iff}}: :$x \equiv 1 \pmod 2$ where the notation denotes congruence modulo $2$.


Definition:Balanced Ternary Representation

Let $n \in \Z$ be an integer. '''Balanced ternary representation''' is the unique representation of $n$ in the form: :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}$ such that: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j 3^j$ where: :$m \in \Z_{>0}$ is a strictly positive integer such that $3^m \le \size n < 3^{m + 1}$ :all the $r_j$ are such that $r_j \in \set {\underline 1, 0, 1}$, where $\underline 1 := -1$.


Definition:Integral Ideal

Let $J \subseteq \Z$ be a non-empty subset of the set of integers. Let $J$ fulfil the following conditions: :$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$ :$(2): \quad n \in J, r \in \Z \implies r n \in J$ Then $J$ is an '''integral ideal'''.


Definition:Greatest Common Divisor/Integers/Definition 1

The '''greatest common divisor of $a$ and $b$''' is defined as: :the largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$


Definition:Greatest Common Divisor/Integers/Definition 2

The '''greatest common divisor of $a$ and $b$''' is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that: :$(1): \quad d \divides a \land d \divides b$ :$(2): \quad c \divides a \land c \divides b \implies c \divides d$


Definition:Lowest Common Multiple/Also known as

The '''lowest common multiple''' is also known as the '''least common multiple'''. It is usually abbreviated '''LCM''', '''lcm''' or '''l.c.m.''' The notation $\lcm \set {a, b}$ can be found written as $\sqbrk {a, b}$. This usage is not recommended as it can cause confusion.


Definition:Open Ball/Normed Division Ring

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$''' is defined as: :$\map {B_\epsilon} a = \set {x \in R: \norm{x - a} < \epsilon}$ If it is necessary to show the norm itself, then the notation $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ can be used.


Definition:Open Ball/Normed Division Ring/Radius

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the open $\epsilon$-ball.


Definition:Open Ball/Normed Division Ring/Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\epsilon$-ball.


Definition:Closed Ball/Normed Division Ring

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''closed $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$''' is defined as: :$\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a} \le \epsilon}$ If it is necessary to show the norm itself, then the notation $\map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ can be used.


Definition:Closed Ball/Normed Division Ring/Radius

In $\map { {B_\epsilon}^-} a$, the value $\epsilon$ is referred to as the '''radius''' of the closed $\epsilon$-ball.


Definition:Closed Ball/Normed Division Ring/Center

In $\map { {B_\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\epsilon$-ball.


Definition:Lattice Point

Let $\CC$ be a cartesian coordinate system of $n$ dimensions. Let $P = \tuple {a_1, a_2, \ldots, a_n}$ be a point in $\CC$. Then $P$ is a '''lattice point''' of $\CC$ {{iff}} $a_1, a_2, \ldots, a_n$ are all integers. === Rational Lattice Point === {{:Definition:Lattice Point/Rational}}


Definition:Intercept

Let $\mathcal C$ be a curve embedded in a cartesian plane. An '''intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects one of the coordinate axes. In particular: === $x$-Intercept === {{:Definition:Intercept/X-Intercept}} === $y$-Intercept === {{:Definition:Intercept/Y-Intercept}}


Definition:Intercept/X-Intercept

An '''$x$-intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects the $x$-axis.


Definition:Intercept/Y-Intercept

A '''$y$-intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects the $y$-axis.


Definition:Perpendicular Distance

Let $\mathcal L$ be a straight line. Let $P$ be a point not on $\mathcal L$. Let a perpendicular be dropped from $P$ to $\mathcal L$, intersecting $\mathcal L$ at $A$. The '''perpendicular distance''' from $P$ to $\mathcal L$ is defined as being the length of the line $AP$. :300px


Definition:Lowest Common Multiple/Integers/General Definition

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\displaystyle \prod_{a \mathop \in S} a = 0$ (that is, all elements of $S$ are non-zero). Then the '''lowest common multiple''' of $S$: :$\lcm \paren S = \lcm \set {a_1, a_2, \ldots, a_n}$ is defined as the smallest $m \in \Z_{>0}$ such that: :$\forall x \in S: x \divides m$


Definition:Sphere/Metric Space

Let $M = \struct{A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $M$''' is defined as: :$\map {S_\epsilon} a = \set {x \in A: \map d {x, a} = \epsilon}$


Definition:Sphere/Metric Space/Center

In $S_\epsilon \paren{a}$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.


Definition:Sphere/Metric Space/Radius

In $S_\epsilon \paren{a}$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.


Definition:Sphere/Normed Division Ring

Let $\struct{R, \norm{\,\cdot\,}}$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $\struct{R, \norm{\,\cdot\,}}$''' is defined as: :$S_\epsilon \paren{a} = \set {x \in R: \norm{x - a} = \epsilon}$


Definition:Sphere/Normed Division Ring/Radius

In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.


Definition:Sphere/Normed Division Ring/Center

In $\map {S_\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.


Definition:Stellation/Polygon/Regular

A '''regular stellated polygon''' is a stellated polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Triskaidecagon

A '''triskaidecagon''' is a polygon with exactly $13$ sides. :300px === Regular Triskaidecagon === {{:Definition:Triskaidecagon/Regular}}


Definition:Topology Induced by Division Ring Norm

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring whose norm is $\norm {\,\cdot\,}$. The '''topology on $\struct {R, \norm {\,\cdot\,} }$ induced by $\norm {\,\cdot\,}$''' is defined as the topology $\tau$ generated by the basis consisting of the set of all open balls of $\struct {R, \norm {\,\cdot\,} }$. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. By the definition of the metric induced by a norm on division ring it follows that the '''topology on the normed division ring $\struct{R, \norm{\,\cdot\,}}$ induced by (the norm) $\norm{\,\cdot\,}$''' is the topology induced by the metric d.


Definition:Complete Residue System

Let $m \in \Z_{\ne 0}$ be a non-zero integer. Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a set of integers with the properties that: :$(1): \quad i \ne j \implies r_i \not \equiv r_j \pmod m$ :$(2): \quad \forall n \in \Z: \exists r_i \in S: n \equiv r_i \pmod m$ Then $S$ is a '''complete residue system modulo $m$'''.


Definition:Deck of Cards

A '''deck of cards''' is a set of pieces of (traditionally) cardboard with a design on one side uniquely identifying the card, and on the other side a single design which is exactly the same for all cards in the '''deck'''. === Card === {{:Definition:Deck of Cards/Card}} === Suit === {{:Definition:Deck of Cards/Suit}}


Definition:Deck of Cards/Card

Each of the elements of a deck of cards is a '''card'''.


Definition:Deck of Cards/Suit

The cards in a deck of cards are partitioned into a number of '''suits'''. The traditional '''deck of cards''' that is usually discussed in the context of probability theory is divided into $4$ '''suits''' of $13$ cards each. There may also be a small number (usually between $1$ and $3$) of extra cards, usually called '''jokers''', which do not belong to any '''suit'''.


Definition:Shuffle

A '''shuffle''' is an operation on a deck of cards which changes the order of the cards in the deck. Category:Definitions/Cards qljox4ecvgily80liwz5ry5pm6nkhr4


Definition:Order of Cards

Let $D$ be a deck of cards. The cards in the deck can be considered to be stacked uniformly, usually all in the same orientation, with a card on the top and a card on the bottom. The '''order''' of the cards is the total ordering $\prec$ defined as: :$a \prec b$ {{iff}} $a$ is somewhere underneath $b$ Category:Definitions/Cards kn84y5p05gjai83ol0n83qrxiewm53o


Definition:Modified Perfect Faro Shuffle

A '''modified perfect Faro shuffle''' is a shuffle performed on a deck of cards $D$ of size $2 n$. Cut $D$ into $2$ piles $D_1$ (the top half) and $D_2$ (the bottom half) of exactly $n$ cards each. The shuffled deck $D'$ is assembled by alternating the cards in $D$: :The cards in positions $2, 4, 6, \ldots, 2 n$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_1$ in the same order :The cards in positions $1, 3, 5, \ldots, 2 n - 1$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_2$ in the same order. Thus a card in position $k$ in $D$ ends up in position $2 k \bmod \paren {2 n + 1}$ in $D'$.


Definition:Fifth Root/Real

Let $x \in \R_{\ge 0}$ be a positive real number. The '''fifth root of $x$''' is the real number defined as: :$x^{\paren {1 / 5} } := \set {y \in \R: y^5 = x}$ where $x^{\paren {1 / 5} }$ is the $5$th root of $x$. The notation: :$y = \sqrt [5] x$ is usually encountered.


Definition:Affine Space/Weyl's Axioms

Let $K$ be a field. Let $\struct{V, +_V, \circ}$ be a vector space over $K$. Let $\EE$ be a set on which a mapping is defined: :$- : \EE \times \EE \to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \text W 1         | q = \forall p \in \EE: \forall v \in V: \exists ! q \in \EE         | m = v = q - p }} {{axiom | n = \text W 2         | q = \forall p, q, r \in \EE         | m = \paren{r - q} +_V \paren{q - p} = r - p }} {{end-axiom}} Then the ordered pair $\tuple {\EE, -}$ is an '''affine space'''.


Definition:Affine Space/Group Action

Let $K$ be a field. Let $\left({V, +_V, \circ}\right)$ be a vector space over $K$. Let $\mathcal E$ be a set. Let $\phi: \mathcal E \times V \to \mathcal E$ be a free and transitive group action of $\struct{V, +_V}$ on $\mathcal E$. Then the ordered pair $\tuple{\mathcal E, \phi}$ is an '''affine space'''.


Definition:Affine Space/Associativity Axioms

Let $K$ be a field. Let $\struct {V, +_V, \circ}$ be a vector space over $K$. Let $\EE$ be a set on which two mappings are defined: :$+ : \EE \times V \to \EE$ :$- : \EE \times \EE \to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \text A 1         | q = \forall p, q \in \EE         | m = p + \paren {q - p} = q }} {{axiom | n = \text A 2         | q = \forall p \in \EE: \forall u, v \in V         | m = \paren {p + u} + v = p + \paren {u +_V v} }} {{axiom | n = \text A 3         | q = \forall p, q \in \EE: \forall u \in V         | m = \paren {p - q} +_V u = \paren {p + u} - q }} {{end-axiom}} Then the ordered triple $\struct {\EE, +, -}$ is an '''affine space'''.


Definition:Tangent Space (Affine Geometry)

Let $\tuple {\EE, +, -}$ be an affine space. Let $V$ be the vector space that is the codomain of $-$. Then $V$ is the '''tangent space''' of $\EE$.


Definition:Vector (Affine Geometry)

Let $\EE$ be an affine space. Let $V$ be the tangent space of $\EE$. Any element $v$ of $V$ is called a '''vector'''.


Definition:Subtraction (Affine Geometry)

Let $\tuple {\EE, +, -}$ be an affine space. Then the mapping $-$ is called '''affine subtraction'''.


Definition:Addition (Affine Geometry)

Let $\tuple {\EE, +, -}$ be an affine space. Then the mapping $+$ is called '''affine addition'''.


Definition:Addition/Sum

Let $a + b$ denote the operation of addition on two objects $a$ and $b$. Then the result $a + b$ is referred to as the '''sum''' of $a$ and $b$.


Definition:Multiplication/Notation

There are several variants of the notation for '''multiplication''': :$n \times m$ This is usually used when numbers are under consideration, for example: $3 \times 5 = 15$. However, it can be used in the context of algebra where extra clarity is needed. :$n m$ This is most common in algebra, but not with numbers, as it is difficult to make it obvious where one number ends and the next number begins. :$\paren n \paren m$ This form can be used for either symbols denoting variables or numbers, for example: $\paren 3 \paren 4 = 12$. :$n \cdot m$ or $n . m$ These have their uses in algebra, but the dot has the danger of being confused with the decimal point when used for numbers. :$n * m$ This notation specifically evolved in the field of computer science, but can occasionally be seen encroaching into mathematics. Its use is not recommended, as it can be confused with other operations that use the same or similar notation, for example convolution.


Definition:Multiplication/Multiplicand

Let $a \times b$ denote the operation of multiplication on two objects. The object $b$ is known as the '''multiplicand of $a$'''. That is, it is the object which is to be multiplied by the multiplier.


Definition:Multiplication/Multiplier

Let $a \times b$ denote the operation of multiplication on two objects. The object $a$ is known as the '''multiplier of $b$'''. That is, it is the object which is to multiply the multiplicand.


Definition:Multiplication/Product

Let $a \times b$ denote the operation of multiplication on two objects $a$ and $b$. Then the result $a \times b$ is referred to as the '''product''' of $a$ and $b$.


Definition:Real Number/Real Number Line/Origin

The point representing the number $0$ (zero) is referred to as the '''origin'''.


Definition:Variable/Real

A '''real variable''' is a symbol which can stand for any one of a set of real numbers.


Definition:Complex Number/Definition 2/Scalar Product

Let $\tuple {x, y}$ be a complex numbers. Let $m \in \R$ be a real number. Then $m \tuple {x, y}$ is defined as: :$m \tuple {x, y} := \tuple {m x, m y}$


Definition:Complex Conjugate Coordinates

Let $P$ be a point in the complex plane. $P$ may be located using '''complex conjugate coordinates''' $\tuple {z, \overline z}$ based on: {{begin-eqn}} {{eqn | l = x       | r = \dfrac {z + \overline z} 2       | c = Sum of Complex Number with Conjugate }} {{eqn | l = y       | r = \dfrac {z - \overline z} {2 i}       | c = Difference of Complex Number with Conjugate }} {{end-eqn}} where $P = \tuple {x, y}$ is expressed in Cartesian coordinates.


Definition:Connected Set (Complex Analysis)/Definition 1

$D$ is '''connected''' {{iff}} every pair of points in $D$ can be joined by a staircase contour.


Definition:Connected Set (Complex Analysis)/Definition 2

$D$ is '''connected''' {{iff}} every pair of points in $D$ can be joined by a polygonal path all points of which are in $D$.


Definition:Polygonal Path

Let $\C$ denote the complex plane. A '''polygonal path''' in $\C$ is a contour in $\C$ which is composed of a finite number of straight line segments.


Definition:Valuation Ring Induced by Non-Archimedean Norm

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ring induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the set: :$\mathcal O = \set {x \in R: \norm x \le 1}$ That is, the '''valuation ring induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the closed ball ${B_1}^- \paren {0_R}$.


Definition:Valuation Ideal Induced by Non-Archimedean Norm

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ideal induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the set: :$\mathcal P = \set{x \in R: \norm{x} \lt 1}$ That is, the '''valuation ideal induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the open ball $B_1 \paren {0_R}$.


Definition:Residue Division Ring Induced by Non-Archimedean Norm

Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$. Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$. The '''residue division ring induced by the norm''' $\norm {\,\cdot\,}$ is the quotient ring $\OO / \PP$. If $R$ is a field then the quotient ring $\OO / \PP$ is called the '''residue field induced by the norm''' $\norm {\,\cdot\,}$.


Definition:P-adic Integers

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the $p$-adic numbers $\Q_p$ for some prime $p$. The set of '''$p$-adic integers''', denoted $\Z_p$, is the valuation ring induced by $\norm {\,\cdot\,}_p$, that is: :$\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$


Definition:Coincident Straight Lines

Two straight lines are '''coincident''' {{iff}} they lie on top of each other. That is, two straight lines $L_1$ and $L_2$ are '''coincident''' {{iff}}: :$L_1 \cap L_2 = L_1 = L_2$. Category:Definitions/Geometry qmrzpxwjm4xris3ziy02ig6puschszu


Definition:Equilibrant

Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space. The '''equilibrant''' of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ is defined as the force which is needed to prevent $B$ from moving.


Definition:Lemniscate of Bernoulli/Major Axis

The line $P_1 P_2$ is the '''major axis''' of the lemniscate.


Definition:Lemniscate of Bernoulli/Major Semiaxis

Each of the lines $O P_1$ and $O P_2$ is a '''major semiaxis''' of the lemniscate.


Definition:Division Ring/Definition 2

:Every non-zero element of $R$ is a unit.


Definition:Ring of Quaternions

The '''ring of quaternions''' $\struct {\H, +, \times}$ is the set of quaternions under the two operations of quaternion addition and quaternion multiplication.


Definition:Subfield/Field

Let $\struct {F, +, \circ}$ be a field. Let $K$ be a subset of $F$ such that $\struct {K, +, \circ}$ is also a field. Then $\struct {K, +, \circ}$ is a '''subfield''' of $\struct {F, +, \circ}$.


Definition:Subfield/Ring

Let $\struct {R, +, \circ}$ be a ring with unity. Let $K$ be a subset of $R$ such that $\struct {K, +, \circ}$ is a field. Then $\struct {K, +, \circ}$ is a '''subfield''' of $\struct {R, +, \circ}$.


Definition:Subfield/Proper Subfield

Let $\struct {K, +, \circ}$ be a subfield of $\struct {F, +, \circ}$. Then $\struct {K, +, \circ}$ is a '''proper subfield''' of $\struct {F, +, \circ}$ {{iff}} $K \ne F$. That is, $\struct {K, +, \circ}$ is a '''proper subfield''' of $\struct {F, +, \circ}$ {{iff}}: :$(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$ :$(2): \quad K$ is a proper subset of $F$.


Definition:Periodic Function/Periodic Element

Let $L \in X_{\ne 0}$. Then $L$ is a '''periodic element''' of $f$ {{iff}}: :$\forall x \in X: \map f x = \map f {x + L}$


Definition:Trivial Factorization/Non-Trivial Factorization

A factorization in $\struct {D, +, \circ}$ of the form $x = z \circ y$, where neither $y$ nor $z$ is a unit of $D$, is called a '''non-trivial factorization'''.


Definition:Associate/Integral Domain/Definition 3

$x$ and $y$ are '''associates (in $D$)''' {{iff}} there exists a unit $u$ of $\struct {D, +, \circ}$ such that: :$y = u \circ x$ and consequently: :$x = u^{-1} \circ y$ That is, {{iff}} $x$ and $y$ are unit multiples of each other.


Definition:Cauchy 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 '''Cauchy distribution''' if it has probability density function: :$\map {f_X} x = \dfrac 1 {\pi \gamma \paren {1 + \paren {\frac {x - x_0} \gamma}^2} }$ for some $\gamma > 0$.  This is written:  :$X \sim \Cauchy {x_0} \gamma$


Definition:Antiperiodic Function

=== Antiperiodic Real Function === {{:Definition:Antiperiodic Function/Real}} === Antiperiodic Complex Function === {{:Definition:Antiperiodic Function/Complex}}


Definition:Antiperiodic Function/Antiperiodic Element

Let $L \in X_{\ne 0}$. Then $L$ is an '''anti-periodic element''' of $f$ {{iff}}: : $\forall x \in X: - \map f x = \map f {x + L}$


Definition:Antiperiodic Function/Antiperiod

The '''antiperiod''' of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that: : $\forall x \in X: - \map f x = \map f {x + L}$ where $\cmod L$ is the modulus of $L$.


Definition:Skewness

'''Skewness''' is a measure of the asymmetry of a probability distribution about its mean. Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the '''skewness''' of $X$, usually denoted $\gamma_1$, is defined as:  :$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$ where $\expect X$ denotes the expectation of $X$.


Definition:Uniform Distribution

=== Discrete Uniform Distribution === {{:Definition:Discrete Uniform Distribution}} === Continuous Uniform Distribution === {{:Definition:Continuous Uniform Distribution}}


Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 1

A '''prime ideal''' of $R$ is a proper ideal $P$ such that: :$\forall a, b \in R : a \circ b \in P \implies a \in P$ or $b \in P$


Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 2

A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :$I \circ J \subseteq P \implies I \subseteq P \text { or } J \subseteq P$ for all ideals $I$ and $J$ of $R$.


Definition:Prime Ideal of Ring/Commutative and Unitary Ring

=== Definition 1 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 1}} === Definition 2 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 2}} === Definition 3 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 3}}


Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 3

A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :the complement $R \setminus P$ of $P$ in $R$ is closed under the ring product $\circ$.


Definition:Moment (Probability Theory)

Let $X$ be a random variable on some probability space.  Let $a$ be a real number. Then the '''$n$th moment of $X$ about $a$''', usually denoted $\map {\mu_n} a$, is defined as:  :$\map {\mu_n} a = \expect {\paren {X - a}^n}$ where $\expect X$ denotes the expectation of $X$.


Definition:Raw Moment

Let $X$ be a random variable on some probability space.  Then the '''$n$th raw moment of $X$''', usually denoted $\mu'_n$, is defined as:  :$\mu'_n = \expect {X^n}$ where $\expect X$ denotes the expectation of $X$. That is, the $n$th raw moment of $X$ is its $n$th moment about $0$.  The first raw moment of $X$ is the mean of $X$, and is instead usually denoted $\mu$.


Definition:Central Moment

Let $X$ be a random variable on some probability space with mean $\mu$.  Then the '''$n$th central moment of $X$''', usually denoted $\mu_n$, is defined as:  :$\mu_n = \expect {\paren {X - \mu}^n}$ where $\expect X$ denotes the expectation of $X$. That is, the $n$th central moment of $X$ is its $n$th moment about $\mu$.  The second central moment of $X$ is the variance of $X$, and is instead usually denoted $\sigma^2$.


Definition:Norm Axioms

=== Multiplicative Norm Axioms === Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a multiplicative norm on $R$. The '''multiplicative norm axioms''' are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a multiplicative norm: {{:Definition:Multiplicative Norm Axioms}} When the concept of '''norm axioms''' is raised without qualification, it is usually the case that '''multiplicative norm axioms''' are under discussion. === Submultiplicative Norm Axioms === Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a submultiplicative norm on $R$. The '''submultiplicative norm axioms''' are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to b a submultiplicative norm: {{:Definition:Submultiplicative Norm Axioms}} === Norm Axioms (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 $\mathbf 0_V$. Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$. The '''norm axioms''' are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm: {{:Definition:Norm Axioms (Vector Space)}} Category:Definitions/Norm Theory gyi8v5yi0lz255ux3g3vc1fwvcw6qr1


Definition:Submultiplicative Norm Axioms

{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Submultiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= \le         | mr= \norm x \times \norm y }} {{axiom | n = \text N 3         | lc= Triangle Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}


Definition:Multiplicative Function on Ring

Let $\struct {R, +, \circ}$ be a ring. Let $f: R \to \R$ be a (real-valued) function on $R$. $f$ is a '''multiplicative function on $R$''' {{iff}}: :$\forall x, y \in R: \map f {x \circ y} = \map f x \times \map f y$


Definition:Standardized Moment

Let $X$ be a random variable on some probability space with standard deviation $\sigma$.  Then the '''$n$th standardized moment of $X$''', usually denoted $\alpha_n$, is defined as:  :$\alpha_n = \dfrac {\mu_n} {\sigma^n}$ where $\mu_n$ is the $n$th central moment of $X$. The third standardized moment of $X$ is the skewness of $X$, and is instead usually denoted $\gamma_1$.  The fourth standardized moment of $X$ is the kurtosis of $X$.


Definition:Cyclotomic Ring

Let $\Z \sqbrk {i \sqrt n}$ be the set $\set {a + i b \sqrt n: a, b \in \Z}$. The algebraic structure $\struct {\Z \sqbrk {i \sqrt n}, +, \times}$ is the '''$n$th cyclotomic ring'''.


Definition:Kurtosis

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the '''kurtosis''' of $X$, usually denoted $\alpha_4$, is defined as:  :$\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$ where $\expect X$ denotes the expectation of $X$. That is, the '''kurtosis''' of $X$ is the fourth standardized moment of $X$.


Definition:Constant Term of Polynomial

Let $R$ be a commutative ring with unity. Let $P \in R \sqbrk X$ be a nonzero polynomial over $R$: :$\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ where $n$ is the degree of $P$. The '''constant term''' of $P$ is the coefficient of $a_0$.


Definition:Ring of Eisenstein Integers

The '''ring of Eisenstein integers''' $\struct {\Z \sqbrk \omega, +, \times}$ is the algebraic structure formed from: :the set of Eisenstein integers $\Z \sqbrk \omega$, where $\omega = e^{2 \pi i / 3}$ :the operation of complex addition :the operation of complex multiplication.


Definition:Ring of Square Matrices

Let $R$ be a ring. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map {\mathcal M_R} n$ denote the $n \times n$ matrix space over $R$. Let $+$ denote the operation of matrix entrywise addition. Let $\times$ be (temporarily) used to denote the operation of conventional matrix multiplication.  The algebraic structure: :$\struct {\map {\mathcal M_R} n, +, \times}$ is known as the '''ring of square matrices of order $n$ over $R$'''


Definition:Additive Group

=== Additive Group of Integers === {{:Definition:Additive Group of Integers}} === Additive Group of Integer Multiples === {{:Definition:Additive Group of Integer Multiples}} === Additive Group of Integers Modulo $m$ === {{:Definition:Additive Group of Integers Modulo m}} === Additive Group of Rational Numbers === {{:Definition:Additive Group of Rational Numbers}} === Additive Group of Real Numbers === {{:Definition:Additive Group of Real Numbers}} === Additive Group of Complex Numbers === {{:Definition:Additive Group of Complex Numbers}}


Definition:Excess Kurtosis

'''Excess kurtosis''' is defined as the difference between the kurtosis of a particular probability distribution and that of the Gaussian distribution. Let $X$ be a random variable with kurtosis $\alpha_4$.  By Kurtosis of Gaussian Distribution, the kurtosis of a Gaussian distribution is equal to $3$.  So the '''excess kurtosis''' of $X$, usually denoted $\gamma_2$, is given by:  :$\gamma_2 = \alpha_4 - 3$


Definition:Die/Historical Note

Dice have been around for thousands of years. An early reference to the design of a die can be found in {{AuthorRef|W.R. Paton}}'s $1918$ translation of {{BookLink|The Greek Anthology Book XIV|W.R. Paton}}: :''The numbers on a die run so: six one, five two, three four.''


Definition:Ordered Integral Domain/Definition 1

An '''ordered integral domain''' is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$: {{:Definition:Strict Positivity Property}}


Definition:Ordered Integral Domain/Definition 2

An '''ordered integral domain''' is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain. That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$: {{begin-axiom}} {{axiom | n = \text {OID} 1         | lc= $\le$ is compatible with ring addition:         | q = \forall a, b, c \in D         | ml= a \le b         | mo= \implies         | mr= \paren {a + c} \le \paren {b + c} }} {{axiom | n = \text {OID} 2         | lc= Strict positivity is closed under ring product:         | q = \forall a, b \in D         | ml= 0_D \le a, 0_D \le b         | mo= \implies         | mr= 0_D \le a \times b }} {{end-axiom}}


Definition:Positivity Property

{{begin-axiom}} {{axiom | n = \text P 1         | m = P + P \subseteq P }} {{axiom | n = \text P 2         | m = P \cap \paren {-P} = \set {0_R} }} {{axiom | n = \text P 3         | m = P \circ P \subseteq P }} {{end-axiom}}


Definition:Total Ordering Induced by Strict Positivity Property

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $P: D \to \set {\mathrm T, \mathrm F}$ denote the strict positivity property: {{:Definition:Strict Positivity Property}} Then the total ordering $\le$ compatible with the ring structure of $D$ is called the '''(total) ordering induced by (the strict positivity property) $P$'''.


Definition:Dottie Number

The '''Dottie number''' $\d$ is defined as the unique fixed point of the cosine function: :$\cos \rd = \d$


Definition:Strict Ordering on Integers

=== Definition 1 === {{:Definition:Strict Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Strict Ordering on Integers/Definition 2}}


Definition:Well-Ordered Integral Domain/Definition 1

$\struct {D, +, \times \le}$ is a '''well-ordered integral domain''' {{iff}} the ordering $\le$ is a well-ordering on the set $P$ of (strictly) positive elements of $D$.


Definition:Well-Ordered Integral Domain/Definition 2

$\struct {D, +, \times \le}$ is a '''well-ordered integral domain''' {{iff}} every subset $S$ of the set $P$ of (strictly) positive elements of $D$ has a minimal element: :$\forall S \subseteq D_{\ge 0_D}: \exists x \in S: \forall a \in S: x \le a$ where $D_{\ge 0_D}$ denotes all the elements $d \in D$ such that $\map P d$.


Definition:P-adic Number/P-adic Norm Completion of Rational Numbers

The '''p-adic numbers''', denoted $\struct {\Q_p, \norm {\,\cdot\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$. Each element of $\Q_p$ is called a '''$p$-adic number'''.


Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm

Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$. Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by: :$\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$ By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring, then $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field. Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a '''$p$-adic number'''. The '''p-adic numbers''' is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.


Definition:Multiple/Integral Domain

Let $\struct {D, +, \circ}$ be an integral domain. Let $x, y \in D$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$.


Definition:Multiple/Integer

Let $\Z$ denote the set of integers. Let $x, y \in \Z$. Let $x$ be a divisor of $y$. Then $y$ is a '''multiple''' of $x$.


Definition:Unit of Ring/Definition 1

An element $x \in R$ is a '''unit of $\struct {R, +, \circ}$''' {{iff}} $x$ is invertible under $\circ$. That is, a '''unit''' of $R$ is an element of $R$ which has an inverse. :$\exists y \in R: x \circ y = 1_R = y \circ x$


Definition:Unit of Ring/Definition 2

An element $x \in R$ is a '''unit of $\struct {R, +, \circ}$''' {{iff}} $x$ is divisor of $1_R$.


Definition:Proper Divisor/Integer

Let $\struct {\Z, +, \times}$ be the ring of integers. Let $x, y \in \Z$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z: y = t \times x$ Then $x$ is a '''proper divisor''' of $y$ {{iff}}: :$(1): \quad x \divides y$ :$(2): \quad \size x \ne \size y$ :$(3): \quad x \ne \pm 1$ That is: :$(1): \quad x$ is a divisor of $y$ :$(2): \quad x$ and $y$ are not equal in absolute value :$(3): \quad x$ is not equal to either $1$ or $-1$.


Definition:Content of Polynomial/Integer

Let $f \in \Z \sqbrk X$ be a polynomial with integer coefficients. Then the '''content''' of $f$, denoted $\cont f$, is the greatest common divisor of the coefficients of $f$.


Definition:Content of Polynomial/Rational

Let $f \in \Q \sqbrk X$ be a polynomial with rational coefficients. The '''content''' of $f$ is defined as: :$\cont f := \dfrac {\cont {n f} } n$ where $n \in \N$ is such that $n f \in \Z \sqbrk X$.


Definition:Content of Polynomial/GCD Domain

Let $D$ be a GCD domain. Let $K$ be the field of quotients of $D$. Let $f \in K \sqbrk X$ be a polynomial. Let $a \in D$ be such that $a f \in D \sqbrk X$. Let $d$ be the greatest common divisor of the coefficients of $a f$. Then we define the '''content''' of $f$ to be: :$\cont f := \dfrac d a$


Definition:Z-Module Associated with Abelian Group/Definition 1

The '''$\Z$-module associated with $G$''' is the $\Z$-module $\struct {G, *, \circ}$ with ring action: :$\circ: \Z \times G \to G$: :$\tuple {n, x} \mapsto *^n x$ where $*^n x$ is the $n$th power of $x$.


Definition:Z-Module Associated with Abelian Group/Definition 2

The '''$\Z$-module associated with $G$''' is the $\Z$-module on $G$ with ring representation $\Z \to \map {\operatorname {End} } G$ equal to the initial homomorphism.


Definition:Linear Ring Action/Left

A '''(left) linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\circ : R \times M \to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \forall \lambda \in R: \forall m, n \in M         | ml= \lambda \circ \paren {m + n}         | mo= =         | mr= \paren {\lambda \circ m} + \paren {\lambda \circ n} }} {{axiom | n = 2         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= \paren {\lambda + \mu} \circ m         | mo= =         | mr= \paren {\lambda \circ m} + \paren {\mu \circ m} }} {{axiom | n = 3         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= \paren {\lambda \mu} \circ m         | mo= =         | mr= \lambda \circ \paren {\mu \circ m} }} {{end-axiom}}


Definition:Linear Ring Action/Right

A '''right linear ring action''' of $R$ on $M$ is a mapping from the cartesian product $\circ : M \times R \to M$ such that: {{begin-axiom}} {{axiom | n = 1         | q = \forall \lambda \in R: \forall m, n \in M         | ml= \paren {m + n} \circ \lambda         | mo= =         | mr= \paren {m \circ \lambda} + \paren {n \circ \lambda} }} {{axiom | n = 2         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= m \circ \paren {\lambda + \mu}         | mo= =         | mr= \paren {m \circ \lambda} + \paren {m \circ \mu} }} {{axiom | n = 3         | q = \forall \lambda, \mu \in R: \forall m \in M         | ml= m \circ \paren {\lambda\mu}         | mo= =         | mr= \paren {m \circ \lambda} \circ \mu }} {{end-axiom}}


Definition:Closed Statement

Let $P$ be a statement. $P$ is a '''closed statement''' {{iff}} $P$ contains only bound occurrences of any variables that may appear in it. That is, such that it contains no free occurrences of variables.


Definition:Open Statement

Let $P$ be a statement. $P$ is an '''open statement''' {{iff}} $P$ contains at least one free occurrence of a variables that appears in it.


Definition:Generated Subspace

=== Definition 1 === {{:Definition:Generated Subspace/Definition 1}} === Definition 2 === {{:Definition:Generated Subspace/Definition 2}}


Definition:Generated Subspace/Definition 1

The '''subspace generated by $S$''' is the intersection of all subspaces of $\mathbf V$ containing $S$.


Definition:Generated Subspace/Definition 2

The '''subspace generated by $S$''' is the set of all linear combinations of elements of $S$.


Definition:Dimension of Vector Space/Definition 2

The '''dimension of $V$''' is the maximum cardinality of a linearly independent subset of $V$.


Definition:Dimension of Vector Space/Definition 1

The '''dimension of $V$''' is the number of vectors in a basis for $V$.


Definition:Sheldon Prime

A '''Sheldon prime''' is a prime number which satisfies both the product property and the mirror property.


Definition:Product Property of Primes

Let $p_n$ denote the $n$th prime number. $p_n$ satisfies the '''product property''' if the product of the digits of its base $10$ representation equals $n$.


Definition:Mirror Property of Primes

Let $p_n$ denote the $n$th prime number. Let $\map {\operatorname {rev} } {p_n}$ denote the reversal of $p_n$. $p_n$ satisfies the '''mirror property''' {{iff}}: :$(1): \quad \map {\operatorname {rev} } {p_n}$ is also a prime number :$(2): \quad$ the product of the digits of the base $10$ representation of $\map {\operatorname {rev} } {p_n}$ equals $\map {\operatorname {rev} } n$.


Let $F$ be a field. Let $\theta \in \C$ be a complex number. Let $S$ be the intersection of all fields such that: :$S \subseteq F$ :$\theta \in F$ Then $S$ is denoted $\map F \theta$ and referred to as the '''smallest field containing $F$ and $\theta$'''. {{help|I encounter this definition in {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham}} without having much idea of the context. The element $\theta$ is specifically defined as being a complex number. At the moment I have not got a clue as to why the specific nature of $\theta$, but I will go along with it as defined in Clapham. If the context needs to be expanded, that can happen in due course.}} === General Definition === {{:Definition:Smallest Field containing Subfield and Complex Number/General Definition}}


Definition:Smallest Field containing Subfield and Complex Number/General Definition

Let $F$ be a field. Let $\theta_1, \theta_2, \ldots, \theta_n \in \C$ be complex numbers. Let $S$ be the intersection of all fields such that: :$S \subseteq F$ :$\theta_1, \theta_2, \ldots, \theta_n \in F$ Then $S$ is denoted $\map F {\theta_1, \theta_2, \ldots, \theta_n}$ and referred to as the '''smallest field containing $F$ and $\theta_1, \theta_2, \ldots, \theta_n$'''.


Definition:Algebraic Number over Field

Let $F$ be a field. Let $z$ be a complex number. $z$ is '''algebraic over $F$''' {{iff}} $z$ is a root of a polynomial with coefficients in $F$.


Definition:Generated Field Extension/Definition 1

The '''field extension $F \sqbrk S$ generated by $S$''' is the smallest subfield extension of $E$ containing $S$, that is, the intersection of all subfields of $E$ containing $S$ and $F$. Thus $S$ is a '''generator''' of $F \sqbrk S$ {{iff}} $F \sqbrk S$ has no proper subfield extension containing $S$.


Definition:Simple Algebraic Field Extension

Let $F$ be a field. Let $z$ be a complex number which is algebraic over $F$. Then the smallest field $\map F z$ containing $F$ and $z$ is known as a '''simple algebraic (field) extension''' of $F$.


Definition:Algebraic Number over Field/Degree

Let $F$ be a field. Let $z \in \C$ be algebraic over $F$. The '''degree''' of $\alpha$ is the degree of the minimal polynomial $\map m x$ whose coefficients are all in $F$.


Definition:Algebraic Element of Field Extension/Definition 1

$\alpha$ is '''algebraic over $F$''' {{iff}} it is a root of some nonzero polynomial over $F$: :$\exists f \in F \sqbrk X \setminus \set 0: \map f \alpha = 0$ where $F \sqbrk X$ denotes the ring of polynomial forms in $X$.


Definition:Algebraic Element of Field Extension/Definition 2

$\alpha$ is '''algebraic over $F$''' {{iff}} the evaluation homomorphism $F \sqbrk X \to K$ at $\alpha$ is not injective.


Definition:Polynomial Evaluation Homomorphism/Single Indeterminate

Let $\struct {R \sqbrk X, \iota, X}$ be a polynomial ring in one variable over $R$. Let $s\in S$. A ring homomorphism $h : R \sqbrk X \to S$ is called an '''evaluation in $s$''' {{iff}}: :$\map h X = s$ :$h \circ \iota = \kappa$ where $\circ$ denotes composition of mappings.


Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates

Let $\family {s_i}_{i \mathop \in I}$ be an indexed family of elements of $S$. Let $R \sqbrk {\set {X_i: i \mathop \in I} }$ be a polynomial ring over $R$. A ring homomorphism $g: R \sqbrk {\set {X_i: i \mathop \in I} } \to S$ is called an '''evaluation at $\family {s_i}_{i \mathop \in I}$''' {{iff}}: :$\forall r \in R : \map g r = \map \kappa r$ :$\forall j \in J : \map g {X_j} = s_j$


Definition:Constructible Point in Plane

Let $\mathcal C$ be a Cartesian coordinate plane. Let $S$ be a set of points in $\mathcal C$. Let $P$ be a point in $\mathcal C$. Let there exist a compass and straightedge construction for $P$ from a line segment $AB$, where $A, B \in S$ Then $P$ is defined as '''constructible from $S$'''.


Definition:Coherent Sequence

Let $p$ be a prime number. Let $\sequence {\alpha_n}$ be an integer sequence such that: :$(1): \quad \forall n \in \N: 0 \le \alpha_n \le p^{n + 1} - 1$ :$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$ The sequence $\sequence {\alpha_n}$ is said to be a '''coherent sequence'''. If it is necessary to emphasize the choice of prime $p$ then the sequence $\sequence {\alpha_n}$ is said to be a '''$p$-adically coherent sequence'''.


Definition:P-adic Norm/P-adic Numbers/P-adic Metric

The '''$p$-adic metric''' on $\Q_p$ is the metric induced by $\norm{\cdot}_p$: :$\forall x, y \in \Q_p: \map d {x, y} = \norm{x - y}_p$


Definition:P-adic Norm/P-adic Numbers

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the '''$p$-adic norm''' on $\Q_p$.


Definition:Irreducible Element of Ring/Definition 1

$x$ is defined as '''irreducible''' {{iff}} it has no non-trivial factorization in $D$. That is, {{iff}} $x$ cannot be written as a product of two non-units.


Definition:Irreducible Element of Ring/Definition 2

$x$ is defined as '''irreducible''' {{iff}} the only divisors of $x$ are its associates and the units of $D$. That is, {{iff}} $x$ has no proper divisors.


Definition:Laplace Transform/Restriction to Reals

Although the definition of the '''Laplace transform''' has $s$ be a complex variable, sometimes the restriction of $\map {\laptrans f} s$ to wholly real $s$ is sufficient to solve a particular differential equation.  Therefore, elementary textbooks introducing the Laplace transform will often write something like the following: :''... where we assume at present that the parameter $s$ is real. Later it will be found useful to consider $s$ complex.'' ::::-- {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel}}: Chapter $1$: The Laplace Transform: Definition of the Laplace Transform :''A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.'' ::::-- {{BookReference|Fourier Analysis and its Applications|2003|Anders Vretblad}}: $\S 3.1$


Definition:Laplace Transform/Graphical Interpretation

Define $\gamma$ as the integrand of $\displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ as a function of $\map f t$, $s$, and $t$: :$\map \gamma {\map f t, t; s} = \map f t \, e^{-s t}$ For any particular function $f$, holding $s$ fixed, the integrand of the Laplace Transform $\kappa$ can be interpreted as a contour. That is, for a given function $f$ and a particular complex number $s_0$ held constant: :$\map \kappa t: \R_{\ge 0} \to \C$ :$\map \kappa t = \map \gamma {\map f t, t; s_0}$ is a parameterization of a contour.


Definition:Laplace Transform/Discontinuity at Zero

Let $f: \R_{> 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$. Let $f$ be discontinuous or not defined at $t = 0$. Then the '''Laplace transform''' of $f$  is defined as: :$\displaystyle \laptrans {\map f t} = \map F s = \int_{0^+}^{\to +\infty} e^{-s t} \map f t \rd t = \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} e^{-s t} \map f t \rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.


Definition:Cauchy Principal Value/Real Integral

Let $f: \R \to \R$ be a real function which is piecewise continuous everywhere. Then the '''Cauchy principal value of $\displaystyle \int f$''' is defined as: :$\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$ where $\displaystyle \int_{-R}^R \map f t \rd t$ is a Riemann integral.


Definition:Jump Discontinuity/Jump

Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$ such that $c$ is a '''jump discontinuity''' of $f$. The '''jump''' at $c$ is defined as: :$\displaystyle \lim_{x \mathop \to c^+} \map f x - \lim_{x \mathop \to c^-} \map f x$


Definition:Bessel Function/Order

The parameter $n$ is known as the '''order''' of the Bessel function.


Definition:Modified Bessel Function

The '''modified Bessel functions''' are solutions to Bessel's modified equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} - \paren {x^2 + n^2} y = 0$ These solutions have two main classes: :the modified Bessel functions of the first kind $I_n$ and: :the modified Bessel functions of the second kind $K_n$.


Definition:Modified Bessel Function/Order

The parameter $n$ is known as the '''order''' of the modified Bessel function.


Definition:Modified Bessel Function/First Kind

A '''modified Bessel function of the first kind of order $n$''' is a modified Bessel function which is non-singular at the origin. It is usually denoted $\map {I_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\map {I_n} x$ forms a solution.


Definition:Modified Bessel Function/Second Kind

A '''modified Bessel function of the second kind of order $n$''' is a modified Bessel function which is singular at the origin. It is usually denoted $\map {K_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's modified equation''' to which $\map {K_n} x$ forms a solution.


Definition:Error Function

The '''error function''' is the following improper integral, considered as a real function $\erf : \R \to \R$: :$\map {\erf} x = \displaystyle \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t$ where $\exp$ is the real exponential function.


Definition:Complementary Error Function

The '''complementary error function''' is the real function $\erfc: \R \to \R$: {{begin-eqn}} {{eqn | l = \map {\erfc} x       | r = 1 - \map \erf x       | c = where $\erf$ denotes the Error Function }} {{eqn | r = 1 - \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t       | c = where $\exp$ denotes the Real Exponential Function }} {{eqn | r = \dfrac 2 {\sqrt \pi} \int_x^\infty \map \exp {-t^2} \rd t       | c =  }} {{end-eqn}}


Definition:Exponential Integral Function

The '''exponential integral function''' is the real function $\Ei: \R_{> 0} \to \R$ defined as: :$\map \Ei x = \displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$


Definition:Null Function

Let $\mathcal N: \R \to \R$ be a real function such that: :$\forall x \in \R_{>0}: \displaystyle \int_0^x \map {\mathcal N} t \rd t = 0$ Then $\mathcal N$ is a '''null function'''.


The half wave rectified sine curve is the real function $f: \R \to \R$ defined as: :$\forall t \in \R: \map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$ for all integers $n$. === Graph of Half Wave Rectified Sine Curve === {{:Definition:Half Wave Rectified Sine Curve/Graph}}


Definition:Convolution Integral

Let $f$ and $g$ be real functions which are integrable. The '''convolution integral''' of $f$ and $g$ is defined as: :$\displaystyle \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$


Definition:Convergent Sequence/Real Numbers/Graphical Illustration

The following diagram illustrates the first few terms of a convergent real sequence. :510px For the value of  $\epsilon$ given, a suitable value of $N$ such that: :$n > N \implies \size {x_n - l} < \epsilon$ is $6$.


Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity

$\sequence {x_n}$ '''diverges to $+\infty$''' {{iff}}: :$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n > H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will exceed $H$. We write: :$x_n \to +\infty$ as $n \to \infty$ or: :$\displaystyle \lim_{n \mathop \to \infty} x_n \to +\infty$


Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity

$\sequence {x_n}$ '''diverges to $-\infty$''' {{iff}}: :$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n < -H$ That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will be less than $-H$. We write: :$x_n \to -\infty$ as $n \to \infty$ or: :$\displaystyle \lim_{n \mathop \to \infty} x_n \to -\infty$


Definition:Unbounded Divergent Sequence/Real Sequence/Infinity

$\sequence {x_n}$ '''diverges to $\infty$''' {{iff}}: :$\forall H > 0: \exists N: \forall n > N: \size {x_n} > H$


Definition:Local Basis/Local Basis for Open Sets

A '''local basis''' at $x$ is a set $\BB$ of open neighborhoods of $x$ such that: :$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$ That is, such that every open neighborhood of $x$ also contains some set in $\BB$.


Definition:Local Basis/Neighborhood Basis of Open Sets

A '''local basis''' at $x$ is a set $\BB$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\BB$. That is, a '''local basis''' at $x$ is a neighborhood basis of $x$ consisting of open sets.


Definition:Limit Superior/Definition 1

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum. This maximum is called the '''limit superior'''. It can be denoted: :$\displaystyle \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$


Definition:Limit Superior/Definition 2

The '''limit superior of $\sequence {x_n}$''' is defined and denoted as: :$\displaystyle \map {\limsup_{n \mathop \to \infty} } {x_n} = \inf \set {\sup_{m \mathop \ge n} x_m: n \in \N}$


Definition:Limit Inferior/Definition 1

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a minimum. This minimum is called the '''limit inferior'''. It can be denoted: :$\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$


Definition:Limit Inferior/Definition 2

The '''limit inferior of $\sequence {x_n}$''' is defined and denoted as: :$\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \sup \set {\inf_{m \mathop \ge n} x_m: n \in \N}$


Definition:Chi-Squared Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi-squared distribution''' with $r$ degrees of freedom {{iff}} it has probability density function: :$\displaystyle \map {f_X} x = \dfrac 1 {2^{r / 2} \map \Gamma {r / 2} } x^{\paren {r / 2} - 1} e^{- x / 2}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \chi^2_r$


Definition:Operations Research

'''Operations research''' is the branch of applied mathematics which studies the management of human organizations.


Definition:Chi Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $r$ be a strictly positive integer. $X$ is said to have a '''chi distribution''' with $r$ degrees of freedom {{iff}} it has probability density function: :$\displaystyle \map {f_X} x = \dfrac 1 {2^{\paren {r / 2} - 1} \map \Gamma {r / 2} } x^{r - 1} e^{- x^2 / 2}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \chi_r$


Definition:F-Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $n, m$ be strictly positive integers. $X$ is said to have an '''F-distribution''' with $\tuple {n, m}$ degrees of freedom {{iff}} it has probability density function: :$\displaystyle \map {f_X} x = \frac {m^{m / 2} n^{n / 2} x^{\paren {n / 2} - 1} } {\paren {m + n x}^{\paren {n + m} / 2} \map \Beta {n / 2, m / 2} }$ where $\Beta$ denotes the beta function.  This is written:  :$X \sim F_{n, m}$


Definition:Constant/Linguistic Note

The word '''constant''' can be used either as a {{WP|Noun|noun}}: :''Let $c$ be a constant between $0$ and $1$'' or as an {{WP|Adjective|adjective}}: :''Let $c$ be a constant real number between $0$ and $1$'' Which is intended can usually be deduced from the context.


Definition:Chess/Chessboard

A '''chessboard''' is an array (usually square) of alternating dark and light squares (conventionally referred to as '''black''' and '''white''', even though they may well not be those actual colours). {{ChessDiagram| | |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   | }} This may be referred to as an $m \times n$ '''chessboard''', where: :$m$ is the number of ranks :$n$ is the number of files. For the conventional '''chessboard''', as depicted above, $m = n = 8$. === Rank === {{:Definition:Chess/Chessboard/Rank}} === File === {{:Definition:Chess/Chessboard/File}} === Square === {{:Definition:Chess/Chessboard/Square}}


Definition:Hypothesis (Inductive Statistics)

A '''hypothesis''' is a statement about a population parameter.


Definition:Chess/Chessboard/Rank

A '''row''' of a '''chessboard''' is known as a '''rank'''. It is identified by '''number''': :$1$ is the '''rank''' nearest white :The '''ranks''' are numbered sequentially to the '''rank''' nearest black.


Definition:Chess/Player

The two players in a game of chess are identified by the colour of the pieces they are using. The sets of pieces are made from materials of two contrasting colours, conventionally referred to as '''black''' and '''white'''. Most everyday chess sets actually ''are'' '''black''' and '''white'''. === White === {{:Definition:Chess/Player/White}} === Black === {{:Definition:Chess/Player/Black}} A diagram of a chessboard is conventionally positioned on a page such that '''white''' is at the bottom, playing upwards towards '''black''', and '''black''' is at the top, playing downwards towards '''white'''.


Definition:Chess/Player/White

The '''chess player''' who is using the lighter-coloured pieces is conventionally referred to as '''white'''.


Definition:Chess/Player/Black

The '''chess player''' who is using the darker-coloured pieces is conventionally referred to as '''black'''.


Definition:Chess/Chessboard/File

A '''column''' of a '''chessboard''' is known as a '''file'''. It is identified by '''letter''' :$\text a$ is the '''file''' on the edge of the board to the left of white (that is, to the right of black) :The '''files''' are lettered sequentially up to the '''file''' on the edge of the board to the left of black (that is, to the right of white).


Definition:Chess/Chessboard/Square

The individual spaces of the chessboard are referred to as its '''squares'''. They are conventionally identified by a letter-number pair, where: :the letter identifies its file :the number identifies its rank.


Definition:Chess/Piece/King

The '''king''' File:Chess klt45.svg can be moved one square in any direction, including diagonally. It cannot be moved into a position where it is in check.


Definition:Chess/Rules/Castling

'''Castling''' is a special move where, in one move, the king moves two squares towards a rook, and the rook jumps over the king and lands on the next square the other side of the king. This can only be done if: :$(1): \quad$ The king and rook are both in their initial positions. :$(2): \quad$ The squares in between the king and rook are empty. :$(3): \quad$ The king is not in check, and neither of the $2$ squares it moves through are under attack.


Definition:Chess/Piece/Queen

The '''queen''' File:Chess qlt45.svg can be moved any number of squares in any of $8$ directions: forwards, backwards, left, right, and the $4$ diagonals. It cannot be moved past a square occupied by another piece.


Definition:Chess/Draw/Insufficient Pieces

If neither player has sufficient pieces to force checkmate, then the game of chess is '''drawn'''.


Definition:Chess/Rules/Under Attack

If a chess piece $A$ may legally move to a square which is occupied by a piece $B$ of the opposite colour, $B$ is said to be '''under attack''' by $A$. In the same way, any square to which $A$ may legally move may also be said to be '''under attack'''.


Definition:Chess/Rules/In Check

Uniquely among the chess pieces the king may not legally move into a square which is under attack by a piece of the opposite colour. That is, a king may not be moved so as to be placed under attack. If player $a$ moves a piece so as to place the king of player $b$ under attack, then player $b$ is said to be '''in check'''. Player $a$ is then obliged to say '''"Check."''' Player $b$ must make a move so that his or her king is no longer under attack. Such a move by player $b$ is known as '''getting out of check'''.


Definition:Chess/Rules/Checkmate

If player $a$ has put player $b$'s king in check, but player $b$ has no legal move that will get him or her out of check, then player $a$ has won. This situation is known as '''checkmate'''. Player $a$ is said to have '''checkmated''' player $b$.


Definition:Chess/Rules

With the exception of the knight, no chess piece may move past a square which is occupied by another piece. === Under Attack === {{:Definition:Chess/Rules/Under Attack}} === Capture === {{:Definition:Chess/Rules/Capture}} === Pawn === {{:Definition:Chess/Rules/Capture/Pawn}} === Castling === {{:Definition:Chess/Rules/Castling}} === In Check === {{:Definition:Chess/Rules/In Check}} === Checkmate === {{:Definition:Chess/Rules/Checkmate}}


Definition:Chess/Piece/Bishop

The '''bishop''' File:Chess blt45.svg can be moved any number of squares in any of the $4$ diagonal directions. It cannot be moved past a square occupied by another piece.


Definition:Chess/Piece/Rook

The '''rook''' File:Chess rlt45.svg can be moved any number of squares in any of the $4$ orthogonal directions: up, down, left or right. It cannot be moved past a square occupied by another piece.


Definition:Chess/Piece/Knight

The '''knight''' File:Chess nlt45.svg can be moved two squares in any orthogonal direction (up, down, left or right), and then one square at right angles to that direction. It passes over intervening pieces and does not stop until reaching the end of its move. It cannot be moved onto a square occupied by another piece of the same colour.


Definition:Chess/Pawn/Promotion

If a pawn moves onto the square on the final rank, it is then exchanged for any other type of piece which replaces the pawn on that square. This is called '''promotion'''. The player of that pawn decides which piece to replace it with, which may be neither another pawn nor a king. If it is promoted to a queen (which is usual, the queen being the most powerful piece), this operation is colloquially referred to as '''queening'''.


Definition:Chess/Pawn

A '''pawn''' File:Chess plt45.svg may move in the following modes. $(1): \quad$ It may move one square towards the opposing player, {{iff}} that square is empty. $(2): \quad$ If it has not yet been moved, and {{iff}} both of the squares is empty, it may move $2$ squares forward instead of $1$. Some authorities do not classify the pawn as a piece. === Promotion === {{:Definition:Chess/Pawn/Promotion}}


Definition:Chess/Rules/Capture/Pawn/Normal

If one of the $2$ diagonally adjacent squares towards the opposing player is occupied by an opposing piece, the pawn may move into that square and capture that piece.


Definition:Chess/Rules/Capture/Pawn/En Passant

Let pawn $a$ be on the player's $5$th rank. Let pawn $b$ be of the opposite colour to pawn $a$. Let pawn $b$ be on its starting position, on one of the files adjacent to the one occupied by pawn $a$. Let pawn $b$ move forward $2$ spaces, in the process crossing over one of the squares which is under attack from pawn $a$. Then pawn $a$, '''on its next move only''', may move into that square crossed over by pawn $b$, and capture pawn $b$ "while it is passing". This mode of capture is known as '''capture ''en passant'''''.


Definition:Chess/Rules/Capture/Pawn

A capture in chess by a pawn works completely differently from that by any other piece. It may happen in one of $2$ circumstances. ==== Normal Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/Normal}} ==== En Passant Pawn Capture ==== {{:Definition:Chess/Rules/Capture/Pawn/En Passant}}


Definition:Chess/Piece

There are a number of different kinds of '''chess piece'''. They are distinguished by their modes of movement, as follows: === King === {{:Definition:Chess/Piece/King}} === Queen === {{:Definition:Chess/Piece/Queen}} === Bishop === {{:Definition:Chess/Piece/Bishop}} === Knight === {{:Definition:Chess/Piece/Knight}} === Rook === {{:Definition:Chess/Piece/Rook}} === Pawn === {{:Definition:Chess/Pawn}}


Definition:Chess/Starting Position

The '''starting position''' of a game of chess is: {{ChessDiagram| | |rd|nd|bd|qd|kd|bd|nd|rd |pd|pd|pd|pd|pd|pd|pd|pd |  |  |  |  |  |  |  |    |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |  |  |  |  |  |  |  |   |pl|pl|pl|pl|pl|pl|pl|pl |rl|nl|bl|ql|kl|bl|nl|rl | }} Note the following: :$(1): \quad$ The top left square is white :$(2): \quad$ The queens start on squares of their own colours :$(3): \quad$ White is always first to move.


Definition:Chess/Move

Each turn in a game of chess consists of transferring a piece from one square to another, according to the rules. Such a transfer is known as a '''move'''. There is one exception: the operation of '''castling''' involves the moving of $2$ pieces.


Definition:Retrograde Analysis

'''Retrograde analysis''' is a technique used to determine the sequence of chess moves which were played that led to a given position.


Definition:Hypothesis Test

A '''hypothesis test''' is a rule that specifies, for a null hypothesis $H_0$ and alternative hypothesis $H_1$:  * For which sample values the decision is made to accept $H_0$. * For which sample values $H_0$ is rejected and $H_1$ is accepted.


Definition:Erlang Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \hointr 0 \infty$.  Let $k$ be a strictly positive integer. Let $\lambda$ be a strictly positive real number. $X$ is said to have an '''Erlang distribution''' with parameters $k$ and $\lambda$ {{iff}} it has probability density function: :$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$ where $\Gamma$ denotes the gamma function. This is written:  :$X \sim \map {\operatorname {Erlang} } {k, \lambda}$


Definition:Laplace Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \R$.  Let $\mu$ be a real number.  Let $b$ be a strictly positive real number. $X$ is said to have a '''Laplace distribution''' with parameters $\mu$ and $b$ {{iff}} it has probability density function: :$\map {f_X} x = \dfrac 1 {2 b} \map \exp {- \dfrac {\size {x - \mu} } b}$ This is written:  :$X \sim \map {\operatorname {Laplace} } {\mu, b}$


Definition:Bijection/Technical Note

{{LatexFor|for = f: S \leftrightarrow T}} {{LatexFor|for = f: S \cong T}} {{LatexFor|for = S \stackrel f \cong T}}


Definition:Simple Hypothesis

Let $\theta$ be a population parameter of some population.  Let the parameter space of $\theta$ be $\Omega$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$.  Consider the hypotheses:  :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ We call $H_i$, for $i \in \set {0, 1}$, a '''simple hypothesis''' if $\Omega_i$ contains only a single element.


Definition:Composite Hypothesis

Let $\theta$ be a population parameter of some population.  Let the parameter space of $\theta$ be $\Omega$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$.  Consider the hypotheses:  :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ We call $H_i$, for $i \in \set {0, 1}$, a '''composite hypothesis''' if $\Omega_i$ contains more than a single element.


Definition:Critical Region

Let $\theta$ be a population parameter of some population $P$.  Let $\Omega$ be the parameter space of $\theta$.  Let $\mathbf X$ be a random sample from $P$.  Let $T = \map f {\mathbf X}$ be a sample statistic. Let $\delta$ be a test procedure of the form:  :reject $H_0$ if $T \in C$ for some null hypothesis $H_0$ and some $C \subset \Omega$. We refer to $C$ as the '''critical region''' of $\delta$.


Definition:Test Statistic

Let $\theta$ be a population parameter of some population $P$.  Let $\Omega$ be the parameter space of $\theta$.  Let $\mathbf X$ be a random sample from $P$.  Let $T = \map f {\mathbf X}$ be a sample statistic. Let $\delta$ be a test procedure of the form:  :reject $H_0$ if $T \in C$ for some null hypothesis $H_0$ and some $C \subset \Omega$. We refer to $T$ as the '''test statistic''' of $\delta$.


Definition:Power Function

Let $\theta$ be a population parameter of some population. Let $\Omega$ be the parameter space of $\theta$.    Let $\delta$ be a test procedure for hypotheses about the value of $\theta$.  Let $C$ be the critical region of $\delta$.  Let $T$ be the test statistic of $\delta$.  The '''power function''' of $\delta$, written $\map \pi {\theta_0}$, is defined by: :$\map \pi {\theta_0} = \map \Pr {T \in C \mid \theta = \theta_0 }$ for all $\theta_0 \in \Omega$.


Definition:Parameter Space

Let $\theta$ be a population parameter of some population. The '''parameter space''' of $\theta$, typically denoted $\Omega$, is the domain of $\theta$. That is, it is the set of all possible values that $\theta$ can take.


Definition:Type I Error

A '''type I error''' is the rejection of a true null hypothesis as the result of a test procedure.


Definition:Type II Error

A '''type II error''' is the failure to reject a false null hypothesis as the result of a test procedure.


Definition:Size (Inductive Statistics)

Let $\theta$ be a population parameter of some population. Let $\Omega$ be the parameter space of $\theta$.  Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$. Let $\delta$ be a test procedure of the hypotheses: :$H_0: \theta \in \Omega_0$ :$H_1: \theta \in \Omega_1$ Let $\pi$ be the power function of $\delta$.  The '''size''' of $\delta$, usually denoted $\alpha$, is defined as:  :$\displaystyle \alpha = \sup_{\theta \in \Omega_0} \map \pi \theta$


Definition:Covariance

Let $X$ and $Y$ be random variables.  Let $\mu_X = \expect X$ and $\mu_Y = \expect Y$, the expectations of $X$ and $Y$ respectively, exist and be finite. Then the '''covariance''' of $X$ and $Y$ is defined by:  :$\cov {X, Y} = \expect {\paren {X - \mu_X} \paren {Y - \mu_Y} }$ where this expectation exists.


Definition:Pearson Correlation Coefficient

Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite. Then the '''Pearson correlation coefficient''' of $X$ and $Y$, typically denoted $\map \rho {X, Y}$, is defined by:  :$\map \rho {X, Y} = \dfrac {\cov {X, Y} } {\sqrt {\var X \, \var Y} }$ where $\cov {X, Y}$ is the covariance of $X$ and $Y$.


Definition:Transversality Conditions

Let $\map {\mathbf y} x$ be a differentiable vector-valued function. Let $J \sqbrk {\mathbf y}$ be a functional of the following form: :$\ds J \sqbrk {\mathbf y} = \int_{P_1}^{P_2} \map F {x, \mathbf y, \mathbf y', \ldots} \rd x$ where $P_1$, $P_2$ are points on given differentiable manifolds $M_1$ and $M_2$. Suppose we are looking for $\mathbf y$ extremizing $J$. The system of equations to be solved consists of differential Euler equations and algebraic equations at both endpoints. Then the set of all algebraic equations at both endpoints are called '''transversality conditions'''.


Definition:Unbiased Estimator

Let $\theta$ be a population parameter of some statistical model. Let $\delta$ be an estimator of $\theta$. We call $\delta$ an '''unbiased estimator''' if its bias is equal to $0$ regardless of the true value of $\theta$.


Definition:Hardy-Littlewood Maximal Function

{{MissingLinks|throughout}} {{explain|presumably the "Hardy-Littlewood Maximal Function" is that function which is returned by the Hardy-Littlewood maximal operator? More precision needed here.}} The Hardy-Littlewood maximal operator takes a locally integrable function $f: \R^d \to \R$  and returns another function $M f$ that, at each point $x \in \R^d$ gives the maximum average value that $f$ can have on balls centered at that point.  More precisely, :$\displaystyle \map {M f} x := \sup_{r \mathop > 0} \frac 1 {\size {\map B {x, r} } } \int_{\map B {x, r} } \size {\map f y} \rd y$ where: :$\map B {x, r}$  is the ball of radius $r$ centered at $x$ :$\size E$ denotes the Lebesgue measure of $E \subset \R^d$.


Definition:Precisely One Function

Let $p_1, p_2, \ldots, p_n$ be statements. The '''precisely one function''' is the propositional function $\map P {p_1, p_2, \ldots, p_n}$ defined as: :$\map P {p_1, p_2, \ldots, p_n}$ is true {{iff}} '''precisely one''' of $p_1, p_2, \ldots, p_n$ is true.


Definition:Differential Equation/Total

A '''total differential equation''' is a '''differential equation''' which contains: :more than one dependent variable :one independent variable which may or may not appear explicitly in that differential equation.


Definition:Differential Equation/Degree

Let $f$ be a '''differential equation''' which can be expressed as a polynomial in all the derivatives involved. The '''degree''' of $f$ is defined as being the power to which the derivative of the highest order is raised. By default, if not specifically mentioned, the '''degree''' of a '''differential equation''' is assumed to be $1$.


Definition:Alexandrian School

The '''Alexandrian School''' was active in the city of {{WP|Alexandria|Alexandria}} for the approximate millennium leading up to the city's destruction in $641$ C.E. It stood as the cultural hub of the Greek sphere of influence. The {{WP|Musaeum|Alexandrian Musaeum}} stood at the centre of this hub, and has been argued as being the archetypal forerunner of the modern University. It had a library which, at its peak, is believed to have contained over $700 \, 000$ volumes. After the fall of {{WP|Alexandria|Alexandria}}, most of its scholars migrated to {{WP|Constantinople|Constantinople}}, preserving the ancient Greek mathematical learning, where it remained for some $800$ more years until the {{WP|Renaissance|Renaissance}}.


Definition:Convolution of Probability Distributions

Let $X$ and $Y$ be independent random variables.  Let $Z = X + Y$.  The probability distribution of $Z$ is called the '''convolution''' of the probability distributions of $X$ and $Y$.


Definition:Convergence in Probability

Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''converges in probability''' to $X$ if:  :$\displaystyle \lim_{n \to \infty} \map \Pr {\size {X_n - X} < \varepsilon} = 1$ for all real $\varepsilon > 0$. This is written:  :$X_n \xrightarrow p X$


Definition:Convergence in Distribution

Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''converges in distribution''' to $X$ if:  :$\displaystyle \lim_{n \mathop \to \infty} \map \Pr {X_n \le x} = \map \Pr {X \le x}$ for all $x$ for which the map $x \mapsto \map \Pr {X \le x}$ is continuous. This is written:  :$X_n \xrightarrow d X$


Definition:Random Sample

Let $X_i$ be a random variable with $\Img {X_i} = \Omega$, for all $1 \le i \le n$. Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \le i \le n$.  We say that $X_1, X_2, \ldots, X_n$ form a '''random sample''' of size $n$ if:  :$X_i$ and $X_j$ are independent if $i \ne j$ :$\map {F_1} x = \map {F_i} x$ for all $x \in \Omega$ for all $1 \le i, j \le n$. If $X_1, X_2, \ldots, X_n$ form a '''random sample''', they are said to be '''independent and identically distributed''', commonly abbreviated '''i.i.d'''.


Definition:Almost Sure Convergence

Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables. Let $X$ be a random variable.  We say that $\sequence {X_n}$ '''almost surely converges''' to $X$ if:  :$\displaystyle \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \varepsilon} = 1$ for all real $\varepsilon > 0$. This is written: :$X_n \xrightarrow {\text {a.s.}} X$


Definition:Path Component/Equivalence Class

Let $\sim$ be the equivalence relation on $T$ defined as: :$x \sim y \iff x$ and $y$ are path-connected. The equivalence classes of $\sim$ are called the '''path components of $T$'''. If $x \in T$, then the '''path component of $T$''' containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname{PC}_x} T$.


Definition:Path Component/Union of Path-Connected Sets

The '''path component of $T$ containing $x$''' is defined as: : $\displaystyle \map {\operatorname{PC}_x} T = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is path-connected $\left.\right\}$


Definition:Path Component/Maximal Path-Connected Set

The '''path component of $T$ containing $x$''' is defined as: :the maximal path-connected set of $T$ that contains $x$.


Definition:Sufficient Statistic

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population indexed by a parameter $\theta$. Let $T$ be a sample statistic. Let $I = \Img {\map T {X_1, X_2, \ldots, X_n} }$. Let $D$ be the conditional joint distribution of $X_1, X_2, \ldots, X_n$ given $T = t$ and $\theta$.   We call $T$ a '''sufficient statistic''' for $\theta$ if $D$ is independent of the value of $\theta$ for all $t \in I$.


Definition:Error/Absolute

The '''absolute error''' $\varepsilon$ is the difference between $x$ and a $X$, and can be defined in one of three ways: {{begin-eqn}} {{eqn | n = 1       | l = \varepsilon       | o = :=       | r = X - x }} {{eqn | n = 2       | l = \varepsilon       | o = :=       | r = x - X }} {{eqn | n = 3       | l = \varepsilon       | o = :=       | r = \size {X - x} }} {{end-eqn}} where $\size {X - x}$ denotes the absolute value of $X - x$. Different sources use different conventions.


Definition:Error/Relative

The '''relative error''' in $x$ is defined as: :$\dfrac {\size {X - x} } X$ :where $\size {X - x}$ denotes the absolute value of $X - x$.


Definition:Abstraction

'''Abstraction''' is the process of making a general statement from a collection of particular statements so as to summarise a property that applies to the subject of those particular statements.


Definition:Imperial/Area/Square Mile

One '''square mile''' is equal to a square whose side measures $1$ mile. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square mile''' }} {{eqn | r = 640       | c = acres }} {{end-eqn}}


Definition:Hectare

One '''hectare''' is equal to a square whose side measures $100$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hectare''' }} {{eqn | r = 100       | c = ares }} {{eqn | r = 10 \, 000       | c = square metres }} {{eqn | o = \approx       | r = 0.4       | c = acres }} {{end-eqn}}


Definition:Angle/Adjacent/Also defined as

Some sources give that '''adjacent angles''' are the angles on a straight line at the intersection of that line and another. Under such a definition, $\angle \alpha$ and $\angle \beta$ are '''adjacent''' in the diagram below: :250px Such '''adjacent angles''' are seen to be supplementary.


Definition:The Algebra of Sets

Let $E$ be a universal set. The power set $\powerset E$, together with: :the binary operations union $\cup$ and intersection $\cap$ :the unary operation complement $\complement$ is referred to as '''the algebra of sets''' on $E$.


Definition:Absolute Geometry

'''Absolute geometry''' is the study of '''Euclidean geometry''' without the parallel postulate.


Definition:Normal Real Number

A real number $r$ is '''normal''' with respect to a number base $b$ {{iff}} its basis expansion in number base $b$ is such that: :no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$.


Definition:Disjunction/Disjunct

The substatements $p$ and $q$ are known as the '''disjuncts''', or the '''members of the disjunction'''.


Definition:Dirichlet Beta Function

The '''Dirichlet beta function''' $\beta$ is the complex function defined on the half-plane $\map \Re s > 0$ by the series:  :$\displaystyle \map \beta s = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}$


Definition:Convergent Sequence/Metric Space/Definition 4

$\sequence {x_k}$ '''converges to the limit $l \in A$''' {{iff}}: :for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.


Definition:Alexandroff Extension of Real Number Line

The '''Alexandroff extension of the real number line''' $\R^*$ is defined as: :$\R^* := \R \cup \set \infty$ that is, the set of real numbers together with an element $\infty$ which is not in $\R$.


Definition:Möbius Transformation/Real Numbers

A '''Möbius transformation''' is a mapping $f: \R^* \to \R^*$ of the form: :$\map f x = \dfrac {a x + b} {c x + d}$ where: :$\R^*$ denotes the Alexandroff extension of the real number line :$a, b, c, d \in \R$ such that $a d - b c \ne 0$ We define: :$\map f {-\dfrac d c} = \infty$ if $c \ne 0$, and: :$\map f \infty = \begin{cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$


Definition:P-adic Expansion

Let $p$ be a prime number. A '''$p$-adic expansion''' is a power series in the rational numbers $\Q$ of the form: :$\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ where: :$m \in \Z_{\le 0}$ :$\forall n \in \Z_{\ge m}: d_n \in \N \mathop {\text{ and } } 0 \le d_n < p$ :$m < 0 \implies d_m \ne 0$


Definition:Composition of Mappings/Definition 2

The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \tuple {\map {f_1} x, z} \in f_2}$


Definition:Composition of Mappings/Definition 1

The '''composite mapping''' $f_2 \circ f_1$ is defined as: :$\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$


Definition:Generated Ring Extension

Let $S$ be a commutative rings with unity. Let $R$ be a subring of $S$ with unity such that the unity of $R$ is the unity of $S$. That is, $S$ is a ring extension of $R$. Let $T \subseteq S$ be a subset of $S$.


Definition:Generated Ring Extension/Smallest Subring

The '''ring extension $R \sqbrk T$ generated by $T$''' is the smallest subring of $S$ containing $T$ and $R$, that is, the intersection of all subrings of $S$ containing $T$ and $R$. Thus $T$ is a '''generator''' of $R \sqbrk T$ {{iff}} $R \sqbrk T$ has no proper subring containing $T$ and $R$.


Definition:Generated Ring Extension/Evaluation of Polynomial Ring

Let $R \sqbrk {\set {X_t} }$ be the polynomial ring in $T$ variables $X_t$. Let $\operatorname {ev} : R \sqbrk {\set {X_t} } \to S$ be the evaluation homomorphism associated with the inclusion $T \hookrightarrow S$. The '''ring extension $R \sqbrk T$ generated by $T$''' is $\map {\operatorname {Img}} {\operatorname {ev}}$, the image  of $\operatorname {ev}$. $T$ is said to be a '''generator''' of $R \sqbrk T$.


Definition:Composition of Mappings/Commutative Diagram

The concept of '''composition of mappings''' can be illustrated by means of a commutative diagram. This diagram illustrates the specific example of $f_2 \circ f_1$: ::$\begin{xy}\xymatrix@+1em{  S_1   \ar[r]^*+{f_1}   \ar@{-->}[rd]_*[l]+{f_2 \mathop \circ f_1} &  S_2   \ar[d]^*+{f_2} \\ &  S_3 }\end{xy}$


Definition:Composition of Mappings/Binary Operation

Let $\sqbrk {S \to S}$ be the set of all mappings from a set $S$ to itself. Then the concept of '''composite mapping''' defines a binary operation on $\sqbrk {S \to S}$: :$\forall f, g \in \sqbrk {S \to S}: g \circ f = \set {\tuple {s, t}: s \in S, \tuple {f \paren s, t} \in g} \in \sqbrk {S \to S}$ Thus, for every pair $\tuple {f, g}$ of mappings in $\sqbrk {S \to S}$, the composition $g \circ f$ is another element of $\sqbrk {S \to S}$.