hcju/proofwikips huggingface dataset:
Definition:Fermat Pseudoprime/Base 5
Let $q$ be a composite number such that $5^q \equiv 5 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $5$'''.
Definition:Null Sequence/Complex Numbers
Let $\sequence {z_n}$ be a sequence in $\C$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} z_n = 0$ Then $\sequence {z_n}$ is called a '''(complex) null sequence'''.
Definition:Null Sequence/Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0$ Then $\sequence {x_n}$ is called a '''(real) null sequence'''.
Definition:Null Sequence/Rational Numbers
Let $\sequence {x_n}$ be a sequence in $\Q$ which converges to a limit of $0$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0$ Then $\sequence {x_n}$ is called a '''(rational) null sequence'''.
Definition:Convergent Sequence
=== Topological Space === {{:Definition:Convergent Sequence/Topology}} === Metric Space === {{:Definition:Convergent Sequence/Metric Space}} For other equivalent definitions of a '''convergent sequence''' in a Metric Space see: Definition:Convergent Sequence in Metric Space === Normed Division Ring === {{:Definition:Convergent Sequence/Normed Division Ring}}
Definition:Convergent Sequence/Metric Space
Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space. Let $\sequence {x_k}$ be a sequence in $A$.
Definition:Convergent Sequence/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n} $ be a sequence in $R$.
Definition:Icosahedron/Truncated
A '''truncated icosahedron''' is an icosahedron whose $12$ vertices are truncated such that one third of each edge is cut off at each of both ends.
Definition:Floating Point Representation
'''Floating point''' representation is a technique whereby numbers are held in a computer in standard scientific notation.
Definition:Scientific Notation
'''Scientific notation''' is a technique for representing approximations to (usually large) numbers by presenting them in the form: :$n \approx m \times 10^e$ where: :$m$ is a rational number such that $1 \le m < 10$, expressed in decimal notation :$e$ is an integer.
Definition:Megaflop
A '''megaflop''' is a measure of calculation speed of a computer. :$1$ '''megaflop''' $= 1$ million floating point operations per second.
Definition:Gigaflop
A '''gigaflop''' is a measure of calculation speed of a computer. {{begin-eqn}} {{eqn | o = =       | r = 1        | c = '''gigaflop''' }} {{eqn | r = 1000       | c = megaflops }} {{eqn | r = 1 \times 10^9       | c = floating point operations per second }} {{end-eqn}}
Definition:Closed Disk
Let $V$ be a vectorspace and $x \in V$, $r \in \R$. Then the '''closed disk''' $\bar{D}$ centered at $x$ with radius $r$ is the set: {{begin-eqn}} $\bar{D}(x, r) := \{v \in V | \| x - v \| \le r\}$ {{end-eqn}}
Definition:Group Presentation/Relation
The standard form of a '''relation''' in a '''group presentation''' is: :$w = e$ where $w$ is a word in the group.
Definition:Topologically Equivalent Metrics/Definition 1
$d_1$ and $d_2$ are '''topologically equivalent''' : :For all metric spaces $\left({B, d}\right)$ and $\left({C, d'}\right)$: :For all mappings $f: B \to A$ and $g: A \to C$: :: $(1): \quad f$ is $\left({d, d_1}\right)$-continuous  $f$ is $\left({d, d_2}\right)$-continuous :: $(2): \quad g$ is $\left({d_1, d'}\right)$-continuous  $g$ is $\left({d_2, d'}\right)$-continuous.
Definition:Topologically Equivalent Metrics/Definition 2
$d_1$ and $d_2$ are '''topologically equivalent''' : :$U \subseteq A$ is $d_1$-open $\iff$ $U \subseteq A$ is $d_2$-open.
Definition:Lipschitz Equivalence/Terminology
Despite the close connection with the concept of Lipschitz continuity, this concept is rarely seen in mainstream mathematics, and appears not to have a well-established name. The name '''Lipschitz equivalence''' appears in {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland}}: :''There does not appear to be a standard name for this; the name we use is reasonably appropriate ...''
Definition:Equivalent Division Ring Norms
Let $R$ be a division ring. Let $\norm{\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm{\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$. Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ respectively. === Topologically Equivalent === {{:Definition:Equivalent Division Ring Norms/Topologically Equivalent}} === Convergently Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Convergently Equivalent|section=Definition}} === Null Sequence Equivalent === {{:Definition:Equivalent Division Ring Norms/Null Sequence Equivalent}} === Open Unit Ball Equivalent === {{:Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent}} === Norm is the Power of the Other Norm === {{:Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm}} === Cauchy Sequence Equivalent === {{Transclude:Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent|section=Definition}}
Definition:Equivalent Division Ring Norms/Topologically Equivalent
$\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are '''equivalent'''  $d_1$ and $d_2$ are topologically equivalent metrics.
Definition:Equivalent Division Ring Norms/Convergently Equivalent
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent'''  for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ converges to $l$ in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ converges to $l$ in $\norm {\,\cdot\,}_2$
Definition:Equivalent Division Ring Norms/Null Sequence Equivalent
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent'''  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'''  $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$
Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm
$\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are '''equivalent'''  $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$
Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent'''  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 $\norm{\,\cdot\,}_2$
Definition:International Standard Book Number
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  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'''.
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.
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.
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'''  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''' : :$\forall x, y \in S:\map \phi {x \circ y} = \map \phi y * \map \phi x$
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 $H_2$'''  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 $H_2$''' : :$(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ 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 $\sequence {H_n}$'''  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 $H_2$''' : :$(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ 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'''  it is divisible by $2$.
Definition:Even Integer/Definition 3
An integer $n \in \Z$ is '''even''' : :$x \equiv 0 \pmod 2$ where the notation denotes congruence modulo $2$.
Definition:Even Integer/Definition 2
An integer $n \in \Z$ is '''even'''  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''' : :$x \equiv 1 \pmod 2$ 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.