Definition:Floor Function/Notation

Before around $1970$, the usual symbol for the '''floor function''' was $\sqbrk x$. The notation $\floor x$ for the '''floor function''' is a relatively recent development. Compare the notation for the corresponding ceiling function, $\ceiling x$, which in the context of discrete mathematics is used almost as much. Some sources use $\map {\mathrm {fl} } x$ for the '''floor function''' of $x$. However, this notation is clumsy, and will not be used on {{ProofWiki}}.


Definition:Floor Function/Definition 3

The '''floor function of $x$''' is the unique integer $\floor x$ such that: :$\floor x \le x < \floor x + 1$


Definition:Ceiling Function/Notation

The notation $\ceiling x$ for the '''ceiling function''' is a relatively recent development. Compare the notation $\floor x$ for the corresponding  floor function.


Definition:Language of Predicate Logic/Alphabet

The alphabet $\AA$ of the language of predicate logic $\LL_1$ is defined as follows: <section notitle="true" name="tc"> === Letters === The letters of $\LL_1$ are separated in three classes: * variables; * predicate or relation symbols; * function symbols. Each of these three classes is handled differently by the formal grammar of predicate logic. ==== Variables ==== The variables constitute an infinite set $\mathrm{VAR}$ of arbitrary symbols, for example: :$\mathrm{VAR} = \set {x, y, z, x_0, y_0, z_0, x_1, y_1, z_1, \ldots}$ ==== Predicate Symbols ==== The predicate symbols are a collection of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$). We agree to write $\PP$ for the set of predicate symbols, grouped by their arity: :$\PP = \set {\PP_0, \PP_1, \PP_2, \ldots, \PP_k, \ldots}$ The symbols in $\PP_0$ are inherited from the language of propositional logic. For example, if $P \in \PP_5$ then $P$ is a quinternary predicate symbol. ==== Function Symbols ==== The function symbols are a collection (possibly empty) of arbitrary symbols. Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$). We agree to write $\FF$ for the set of function symbols, grouped by their arity: :$\FF = \set {\FF_0, \FF_1, \ldots, \FF_k, \ldots}$ The symbols in $\FF_0$ are often called '''parameters''' or '''constants'''. Some sources write $\KK$ for the collection of '''parameters'''. === Signs === The signs of $\LL_1$ are an extension of the signs of propositional logic. They split in three classes: * connectives; * quantifiers; * punctuation. ==== Connectives ==== The '''connectives''' of $\LL_1$ comprise: {{begin-axiom}} {{axiom | ml= \land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \neg         | mo= :         | t = the negation sign }} {{axiom | ml= \top         | mo= :         | t = the top sign }} {{axiom | ml= \bot         | mo= :         | t = the bottom sign }} {{end-axiom}} The symbols $\land, \lor, \implies$ and $\iff$ are called the '''binary connectives'''. The symbols $\neg$ is called a '''unary connective'''. The symbols $\top$ and $\bot$ are called the '''nullary connectives'''. ==== Quantifiers ==== The '''quantifiers''' of $\LL_1$ are: {{begin-axiom}} {{axiom | ml= \exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Punctuation ==== The punctuation symbols used in $\LL_1$ are: {{begin-axiom}} {{axiom | ml= (         | mo= :         | t = the left parenthesis sign }} {{axiom | ml= )         | mo= :         | t = the right parenthesis sign }} {{axiom | ml= :         | mo= :         | t = the colon }} {{axiom | ml= ,         | mo= :         | t = the comma }} {{end-axiom}} </section>


Part of specifying the language of propositional logic is to provide its formal grammar. <section notitle="true" name="def"> The following rules of formation constitute a bottom-up grammar for the language of predicate logic $\LL_1$. The definition proceeds in two steps. First, we will define terms, and then well-formed formulas. === Terms === {{:Definition:Language of Predicate Logic/Formal Grammar/Term}} === Well-Formed Formulas === The WFFs of $\LL_1$ are defined by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \mathbf W ~ \PP_n         | t = If $t_1, \ldots, t_n$ are terms, and $p \in \PP_n$ is an $n$-ary predicate symbol, then $\map p {t_1, t_2, \ldots, t_n}$ is a WFF. }} {{axiom | n = \mathbf W ~ \neg         | t = If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. }} {{axiom | n = \mathbf W ~ \lor, \land, \Rightarrow, \Leftrightarrow         | t = If $\mathbf A, \mathbf B$ are WFFs and $\circ$ is one of $\lor, \land, \mathord \implies, \mathord \iff$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF }} {{axiom | n = \mathbf W ~ \forall, \exists         | t = If $\mathbf A$ is a WFF and $x$ is a variable, then $\paren {\forall x: \mathbf A}$ and $\paren {\exists x: \mathbf A}$ are WFFs. }} {{end-axiom}} </section>


Definition:Relation Segment

Let $X$ be a set. Let $R$ be a relation on $X$. Let $x \in X$. Then the '''$R$-segment of $x$''', denoted by $x^R$, is defined by: :$x^R := \left\{ {y \in X: \left({y, x}\right) \in R}\right\}$


Definition:Remainder/Real

The '''remainder of $x$ on division by $y$''' is defined as the value of $r$ in the expression: :$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$ From the definition of the Modulo Operation: :$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$ it can be seen that the '''remainder of $x$ on division by $y$''' is defined as: : $r = x \bmod y$


Definition:Quotient (Algebra)/Real

The '''quotient of $x$ on division by $y$''' is defined as the value of $q$ in the expression: :$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$ From the definition of the Modulo Operation: :$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor = r$ it can be seen that the '''quotient of $x$ on division by $y$''' is defined as: : $q = \left \lfloor {\dfrac x y}\right \rfloor$


Definition:Increasing Mappings Satisfying Inclusion in Lower Closure

Let $R = \left({S, \preceq}\right)$ be an ordered set. Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$ Then '''ordered set $M$ of increasing mappings $f:R \to L$ satisfying''' $\forall x \in S: f\left({x}\right) \subseteq x^\preceq$ is defined by :$M = \left({F, \preccurlyeq}\right)$ where :$F = \left\{ {f: S \to {\it Ids}\left({R}\right): f}\right.$ is increasing mapping $\left.{\land \forall x \in S: f\left({x}\right) \subseteq x^\preceq}\right\}$ and :$\preccurlyeq$ is ordering on mappings generated by $\precsim$ where $x^\preceq$ denotes the lower closure of $x$.


Definition:Coprime/Notation

Let $a$ and $b$ be coprime integers, that is, such that $\gcd \left\{{a, b}\right\} = 1$. Then the notation $a \perp b$ is preferred on {{ProofWiki}}. If $\gcd \left\{{a, b}\right\} \ne 1$, the notation $a \not \!\! \mathop{\perp} b$ can be used.


Definition:Modulo Subtraction

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''subtraction modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$


Definition:Congruence (Number Theory)/Notation

The relation '''$x$ is congruent to $y$ modulo $z$''', usually denoted: :$x \equiv y \pmod z$ is also frequently seen denoted as: :$x \equiv y \ \paren {\mathop {\operatorname{modulo} } z}$ Some (usually older) sources render it as: :$x \equiv y \ \paren {\mathop {\operatorname{mod.} } z}$


Definition:Permutation/Ordered Selection/Notation

The number of $r$-permutations from a set of cardinality $n$ is denoted variously: :$P_{n r}$ :${}^r P_n$ :${}_r P_n$ :${}_n P_r$ (extra confusingly) There is little consistency in the literature). On {{ProofWiki}} the notation of choice is ${}^r P_n$.


Definition:Factorial/Definition 1

The '''factorial of $n$''' is defined inductively as: :$n! = \begin{cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end{cases}$


Definition:Factorial/Definition 2

The '''factorial of $n$''' is defined as: {{begin-eqn}} {{eqn | l = n!       | r = \prod_{k \mathop = 1}^n k       | c =  }} {{eqn | r = 1 \times 2 \times \cdots \times \paren {n - 1} \times n       | c =  }} {{end-eqn}} where $\displaystyle \prod$ denotes product notation.


Definition:Gamma Function/Hankel Form

The '''{{AuthorRef|Hermann Hankel|Hankel}} form''' of the '''Gamma function''' is: :$\displaystyle \frac 1 {\Gamma \left({z}\right)} = \dfrac 1 {2 \pi i} \oint_{\mathcal H} \frac {e^t \, \mathrm d t} {t^z}$ where $\mathcal H$ is the contour starting at $-\infty$, circling the origin in an anticlockwise direction, and returning to $-\infty$. The '''Hankel form''' is valid for all $\C$.


Definition:Permanent

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. That is, let: : $\mathbf A = \begin {pmatrix}   a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\   a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\   a_{n 1} & a_{n 2} & \cdots & a_{n n} \end {pmatrix}$ Let $\lambda: \N_{>0} \to \N_{>0}$ be a permutation on $\N_{>0}$. Then the '''permanent''' of $\mathbf A$ is defined as: :$\displaystyle \sum_{\lambda} \paren {\prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_{\lambda} a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$ where: :the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$.


Definition:Constant Symbol

Let $\LL$ be a formal language (for example, the language of predicate logic $\LL_1$). A '''constant symbol''' is a letter of $\LL$ used to describe a constant. The name '''constant symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\LL$.


Definition:Value of Term under Assignment

Let $\tau$ be a term in the language of predicate logic $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure. Let $\sigma$ be an assignment for $\tau$ in $\mathcal A$. Then the '''value of $\tau$ under $\sigma$''', denoted $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$, is defined recursively by: :If $\tau = x$, with $x \in \operatorname{dom} \left({\sigma}\right)$ a variable in the domain of $\sigma$: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({x}\right) } \left[{\sigma}\right]  :=  \sigma \left({x}\right)$ :If $\tau = f \left({\tau_1, \ldots, \tau_n}\right)$ with $\tau_i$ terms and $f \in \mathcal F_n$ an $n$-ary function symbol: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({f \left({\tau_1, \ldots, \tau_n}\right) }\right) } \left[{\sigma}\right]  :=  f_{\mathcal A} \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right], \ldots, \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_n}\right) } \left[{\sigma}\right] }\right)$ :where $f_{\mathcal A}$ denotes the interpretation of $f$ in $\mathcal A$. If $\tau$ contains no variables, one writes $\operatorname{val}_{\mathcal A} \left({\tau}\right)$ instead of $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$.


Let $\mathbf A$ be a WFF in the language of predicate logic $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure. Let $\sigma$ be an assignment for $\mathbf A$ in $\mathcal A$. Then the '''value of $\mathbf A$ under $\sigma$''', denoted $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$, is defined recursively by: :If $\mathbf A = p \left({\tau_1, \ldots, \tau_n}\right)$ with $\tau_i$ terms and $p \in \mathcal P_n$ an $n$-ary predicate symbol: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({p \left({\tau_1, \ldots, \tau_n}\right) }\right) } \left[{\sigma}\right]  :=  p_{\mathcal A} \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right], \ldots, \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_n}\right) } \left[{\sigma}\right] }\right)$ :where $p_{\mathcal A}$ denotes the interpretation of $p$ in $\mathcal A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_i}\right) } \left[{\sigma}\right]$ is the value of $\tau_i$ under $\sigma$. :If $\mathbf A = \neg \mathbf B$ with $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\neg \mathbf B}\right) } \left[{\sigma}\right]  :=  f^\neg \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right] }\right)$ :where $f^\neg$ denotes the truth function of $\neg$. :If $\mathbf A = \left({\mathbf B \circ \mathbf B'}\right)$ with $\mathbf B, \mathbf B'$ WFFs and $\circ$ one of $\land, \lor, \implies, \iff$: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B \circ \mathbf B'}\right) } \left[{\sigma}\right]  :=  f^\circ \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right], \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B'}\right) } \left[{\sigma}\right] }\right)$ :where $f^\circ$ denotes the respective truth function of $\circ$. :If $\mathbf A = \left({ \exists x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\exists x: \mathbf B}\right) } \left[{\sigma}\right]  :=  \begin{cases}  T & \text{if for some $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$ :where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$. :If $\mathbf A = \left({ \forall x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF: ::$\mathop{ \operatorname{val}_{\mathcal A} \left({\forall x: \mathbf B}\right) } \left[{\sigma}\right]  :=  \begin{cases}  T & \text{if for all $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$ :where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$. === Sentence === {{:Definition:Value of Formula under Assignment/Sentence}}


Definition:Extension of Assignment

Let $\mathcal A$ be a structure for predicate logic. Let $\sigma$ be an assignment for $\mathcal A$. Let $y \in \mathrm{VAR}$ be a variable. Let $a \in A$ be arbitrary. Then the '''extension of $\sigma$ by mapping $y$ to $a$''', denoted $\sigma + (y / a)$, is defined by: :$\forall x \in \operatorname{dom} \left({\sigma}\right) \cup \left\{{y}\right\}: \left({\sigma + (y / a)}\right)\left({x}\right) := \begin{cases} a & \text{if $x = y$} \\ \sigma \left({x}\right) & \text{otherwise} \end{cases}$ Note in particular the case where $y \in \operatorname{dom} \left({\sigma}\right)$. If $\sigma \left({y}\right) = a'$, say, then $\sigma + (y / a)$ ''overwrites'' this value to become $a$ instead.


Definition:Value of Formula under Assignment/Sentence

Let $\mathbf A$ be a sentence in the language of predicate logic. The '''value of $\mathbf A$ in $\mathcal A$''', denoted $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$, is defined as: :$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) := \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$ where $\varnothing$ is the empty mapping considered as an assignment for $\mathbf A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\varnothing}\right]$ is the value of $\mathbf A$ under $\varnothing$.


Definition:Assignment for Structure

Let $\mathcal L_1$ be the language of predicate logic. Let $\mathrm{VAR}$ be the collection of variables of $\mathcal L_1$. Let $\mathcal A$ be an $\mathcal L_1$-structure on a set $A$. An '''assignment for $\mathcal A$''' is a mapping $\sigma$ such that: :the codomain of $\sigma$ is $A$ :the domain of $\sigma$ is a subset of $\mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and maps them to elements of $A$. === Assignment for Term === {{:Definition:Assignment for Structure/Term}} === Assignment for Formula === {{:Definition:Assignment for Structure/Formula}}


Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Sentence  |section  = tc  |title    = Formal Semantics for Sentences  |header   = 3  |increase = 1  |link     = true }} {{transclude:Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula  |section  = tc  |title    = Formal Semantics for WFFs  |header   = 3  |increase = 1  |link     = true }} </section>


Definition:Semantic Consequence/Predicate Logic

<section notitle="true" name="tc"> Let $\mathcal F$ be a collection of WFFs of predicate logic. Then a WFF $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ {{iff}}: :$\mathcal A \models_{\mathrm{PL}} \mathcal F$ implies $\mathcal A \models_{\mathrm{PL}} \mathbf A$ for all structures $\mathcal A$, where $\models_{\mathrm{PL}}$ is the models relation. </section> === Notation === That $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ is denoted as: :$\mathcal F \models_{\mathrm{PL}} \mathbf A$


Definition:Structure for Predicate Logic/Formal Semantics/Sentence

Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL}$. For the purpose of this formal semantics, we consider only sentences instead of all WFFs. The structures of $\mathrm{PL}$ are said structures for $\mathcal L_1$. A sentence $\mathbf A$ is declared ($\mathrm{PL}$-)valid in a structure $\mathcal A$ {{iff}}: :$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) = T$ where $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$ is the value of $\mathbf A$ in $\mathcal A$. Symbolically, this can be expressed as: :$\mathcal A \models_{\mathrm{PL}} \mathbf A$ </section>


Definition:Structure for Predicate Logic/Formal Semantics/Well-Formed Formula

Let $\mathcal L_1$ be the language of predicate logic. <section notitle="true" name="tc"> The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL_A}$. The structures of $\mathrm{PL_A}$ are pairs $\left({\mathcal A, \sigma}\right)$, where: :$\mathcal A$ is a structure for $\mathcal L_1$ :$\sigma$ is an assignment for $\mathcal A$ A WFF $\mathbf A$ is declared ($\mathrm{PL_A}$-)valid in a structure $\mathcal A$ {{iff}}: :$\sigma$ is an assignment for $\mathbf A$ :$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$ is the value of $\mathbf A$ under $\sigma$. Symbolically, this can be expressed as one of the following: :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ :$\mathcal A \models_{\mathrm{PL_A}} \mathbf A \left[{\sigma}\right]$ </section>


Definition:Hypergeometric Function

A '''hypergeometric function''' is an infinite series power series defined as: :$\displaystyle {}_m \operatorname F_n \left({ {a_1, \ldots, a_m} \atop {b_1, \ldots, b_n} } \, \middle \vert {\, z}\right) := \sum_{k \mathop \ge 0} \dfrac { {a_1}^{\overline k} \cdots {a_m}^{\overline k} } { {b_1}^{\overline k} \cdots {b_n}^{\overline k} } \dfrac {z^k} {k!}$ where $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.


Definition:Gaussian Binomial Coefficient

Let $q \in \R_{\ne 1}$, $r \in \R$, $m \in \Z_{\ge 0}$. The '''Gaussian binomial coefficient''' is an extension of the more conventional binomial coefficient as follows: {{begin-eqn}} {{eqn | l = \binom r m_q       | o = :=       | r = \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }       | c =  }} {{eqn | r = \dfrac {\paren {1 - q^r} \paren {1 - q^{r - 1} } \cdots \paren {1 - q^{r - m + 1} } } {\paren {1 - q^m} \paren {1 - q^{m - 1} } \cdots \paren {1 - q^1} }       | c =  }} {{end-eqn}}


Definition:Multinomial Coefficient

Let $k_1, k_2, \ldots, k_m \in \Z_{\ge 0}$ be positive integers. The '''multinomial coefficient''' of $k_1, \ldots, k_m$ is defined as: :$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} := \dfrac {\left({k_1 + k_2 + \cdots + k_m}\right)!} {k_1! \, k_2! \, \ldots k_m!}$


Definition:Multinomial Coefficient/Trinomial

The '''trinomial coefficient''' of $k_1, k_2, k_3$ is a particular case of a multinomial coefficient, defined as: :$\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} := \dfrac {\left({k_1 + k_2 + k_3}\right)!} {k_1! \, k_2! \, k_3!}$


Definition:Complete Proof System/Strongly Complete

$\mathscr P$ is '''strongly complete for $\mathscr M$ {{iff}}: :Every $\mathscr M$-semantic consequence is a $\mathscr P$-provable consequence. Symbolically, this can be expressed as the statement that, for every collection $\mathcal F$ of logical formulas, and every logical formula $\phi$ of $\mathcal L$: :$\mathcal F \models_{\mathscr M} \phi$ implies $\mathcal F \vdash_{\mathscr P} \phi$


Definition:Sound Proof System/Strongly Sound

$\mathscr P$ is '''strongly sound for $\mathscr M$''' iff: :Every $\mathscr P$-provable consequence is an $\mathscr M$-semantic consequence. Symbolically, this can be expressed as the statement that, for every collection of logical formulas $\mathcal F$, and logical formula $\phi$ of $\mathcal L$: :$\mathcal F \vdash_{\mathscr P} \phi$ implies $\mathcal F \models_{\mathscr M} \phi$


Definition:Finitely Satisfiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a collection of logical formulas of $\mathcal L$. Then $\mathcal F$ is '''finitely satisfiable for $\mathscr M$''' {{iff}}: :For each finite subset $\mathcal F' \subseteq \mathcal F$, there is some $\mathscr M$-model $\mathcal M$ of $\mathcal F'$ That is, for each such $\mathcal F'$, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \mathcal F'$


Definition:Cardinality of Structure

Let $\mathcal A$ be a first-order structure. Then the '''cardinality of $\mathcal A$''', denoted $\left\vert{ \mathcal A }\right\vert$, is defined as: :$\left\vert{ \mathcal A }\right\vert := \left\vert{ A }\right\vert$ where $\left\vert{ A }\right\vert$ is the cardinality of the underlying set $A$.


Let $\mathbf A, \mathbf B$ be WFFs of predicate logic. === Definition 1 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Predicate Logic/Definition 2|Definition 2}}


Definition:Semantic Equivalence/Predicate Logic/Definition 1

Then $\mathbf A$ and $\mathbf B$ are '''equivalent''' {{iff}}: :$\mathbf A \models_{\mathrm{PL_A}} \mathbf B$ and $\mathbf B \models_{\mathrm{PL_A}} \mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\mathbf A$ and $\mathbf B$ are '''equivalent''' {{iff}}, for all structures $\mathcal A$ and assignments $\sigma$: :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ iff $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf B$ where $\models_{\mathrm{PL_A}}$ denotes the models relation.


Definition:Tautology/Formal Semantics/Predicate Logic

Let $\mathbf A$ be a WFF of predicate logic. Then $\mathbf A$ is a '''tautology''' {{iff}}: :$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$  for every structure $\mathcal A$ and assignment $\sigma$. That $\mathbf A$ is a '''tautology''' can be denoted as: :$\models_{\mathrm{PL_A}} \mathbf A$


Definition:Approximating Relation

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is '''approximating relation''' on $S$ {{iff}} :$\forall x \in S: x = \sup \left({x^{\mathcal R} }\right)$ where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.


Definition:Stirling Numbers of the First Kind/Unsigned/Definition 2

'''Unsigned Stirling numbers of the first kind''' are defined as the polynomial coefficients $\displaystyle {n \brack k}$ which satisfy the equation: :$\displaystyle x^{\underline n} = \sum_k \left({-1}\right)^{n - k} {n \brack k} x^k$ where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.


Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1

'''Unsigned Stirling numbers of the first kind''' are defined recursively by: :$\displaystyle {n \brack k} := \begin{cases} \delta_{n k} & : k = 0 \text { or } n = 0 \\ & \\ \displaystyle {n - 1 \brack k - 1} + \paren {n - 1} {n - 1 \brack k} & : \text{otherwise} \\ \end{cases}$


Definition:Stirling Numbers of the Second Kind/Definition 1

'''Stirling numbers of the second kind''' are defined recursively by: :$\displaystyle {n \brace k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \displaystyle {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$


Definition:Stirling Numbers of the Second Kind/Definition 2

'''Stirling numbers  of the second kind''' are defined as the coefficients $\displaystyle {n \brace k}$ which satisfy the equation: :$\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$ where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.


Definition:Stirling Numbers of the First Kind/Signed/Definition 1

'''Signed Stirling numbers of the first kind''' are defined recursively by: :$\map s {n, k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$


Definition:Stirling Numbers of the First Kind/Signed/Definition 2

'''Signed Stirling numbers of the first kind''' are defined as the polynomial coefficients $\map s {n, k}$ which satisfy the equation: :$\displaystyle x^{\underline n} = \sum_k \map s {n, k} x^k$ where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.


Let $\mathcal F$ be a theory in the language of predicate logic. === Relative Semantic Equivalence of WFFs === {{:Definition:Relative Semantic Equivalence/WFF}} === Relative Semantic Equivalence of Terms === {{:Definition:Relative Semantic Equivalence/Term}}


Definition:Relative Semantic Equivalence/WFF

Let $\mathbf A, \mathbf B$ be WFFs. Let $\mathbf C$ be the universal closure of $\mathbf A \iff \mathbf B$. Then $\mathbf A$ and $\mathbf B$ are '''semantically equivalent with respect to $\mathcal F$''' {{iff}}: :$\mathcal F \models_{\mathrm{PL}} \mathbf C$ That is, iff $\mathbf C$ is a semantic consequence of $\mathcal F$.


Definition:Relative Semantic Equivalence/Term

Let $\tau_1, \tau_2$ be terms. Then $\tau_1$ and $\tau_2$ are '''semantically equivalent with respect to $\mathcal F$''' {{iff}}: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_2}\right) } \left[{\sigma}\right]$ for all models $\mathcal A$ of $\mathcal F$ and assignments $\sigma$ for $\tau_1,\tau_2$ in $\mathcal A$. Here $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right]$ denotes the value of $\tau_1$ under $\sigma$.


Definition:Substitution (Formal Systems)/Term/In Term

Let $\beta, \tau$ be terms of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\beta \left({x \gets \tau}\right)$ be the term resulting from $\beta$ by replacing all occurrences of $x$ by $\tau$. Then $\beta \left({x \gets \tau}\right)$ is called the '''substitution instance of $\beta$ substituting $\tau$ for $x$'''.


Definition:Substitution (Formal Systems)/Term/In WFF

Let $\mathbf A$ be a WFF of predicate logic. Let $\tau$ be a term of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\mathbf A \left({x \gets \tau}\right)$ be the WFF resulting from $\mathbf A$ by replacing all free occurrences of $x$ by $\tau$. Then $\mathbf A \left({x \gets \tau}\right)$ is called the '''substitution instance of $\mathbf A$ substituting $\tau$ for $x$'''.


Definition:Additive Semiring/Axioms

An additive semiring is an algebraic structure $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \forall a, b \in S         | m = a * b \in S         | rc= Closure under $*$ }} {{axiom |  n = A1         |  q = \forall a, b, c \in S         | m = \left({a * b}\right) * c = a * \left({b * c}\right)         | rc= Associativity of $*$ }} {{axiom | n = A2         | q = \forall a, b \in S         | m = a * b = b * a         | rc= Commutativity of $*$ }} {{axiom | n = M0         | q = \forall a, b \in S         | m = a \circ b \in S         | rc= Closure under $\circ$ }} {{axiom | n = M1         | q = \forall a, b, c \in S         | m = \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)         | rc= Associativity of $\circ$ }} {{axiom | n = D         | q = \forall a, b, c \in S         | m = a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)         | rc= $\circ$ is distributive over $*$ }} {{end-eqn}} These criteria are called the '''additive semiring axioms'''.


Definition:Join Semilattice/Definition 1

Let $\struct {S, \preceq}$ be an ordered set. Suppose that for all $a, b \in S$: :$a \vee b \in S$ where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$. Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.


Definition:Join Semilattice/Definition 2

Let $\struct {S, \vee}$ be a semilattice. Let $\preceq$ be the ordering on $S$ defined by: :$a \preceq b \iff \paren {a \vee b} = b$ Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.


Definition:Parenthesization/Equivalent

Two parenthesizations of $a_1, \ldots, a_n$ are '''equivalent''' {{iff}} the product defined by them yields the same result.


Definition:Trivial Norm/Vector Space

Let $\struct {K, +, \circ}$ be a division ring endowed with the trivial norm. Let $V$ be a vector space over $K$, with zero $0_V$. Then the map $\norm {\cdot}: V \to \R_+ \cup \set 0$ given by: :$\norm x = \begin{cases}   0 & : \text {if $x = 0_V$} \\   1 & : \text {otherwise} \end{cases}$ defines a norm on $V$, called the '''trivial norm'''.


Definition:Trivial Norm/Division Ring

Let $\struct {R, +, \circ}$ be a division ring, and denote its zero by $0_R$. Then the map $\norm {\cdot}: R \to \R_{\ge 0}$ given by: :$\norm x = \begin{cases}   0 & : \text{if $x = 0_R$}\\   1 & : \text{otherwise} \end{cases}$ defines a norm on $R$, called the '''trivial norm'''.


Definition:Signature (Logic)/Predicate Logic

Let $\mathcal L_1$ be the language of predicate logic. Then a '''signature''' for $\mathcal L_1$ is an explicit choice of the alphabet of $\mathcal L_1$. That is to say, it amounts to choosing, for each $n \in \N$: :A collection $\mathcal F_n$ of $n$-ary function symbols :A collection $\mathcal P_n$ of $n$-ary relation symbols It is often conceptually enlightening to explicitly address the $0$-ary function symbols separately, as constant symbols.


Definition:Variable/Also known as

When it occurs in a mathematical equation, a '''variable''' is often referred to as an '''unknown'''. In the specific context of elementary algebra, the ugly misnomer '''pronumeral''' is frequently found in Australia. This was introduced by extension of the concept of a {{WP|Pronoun|pronoun}}: a symbol that '''stands in''' for a '''numeral''', by which the term number is actually meant. Thankfully the term appears not to have caught on in general.


Definition:Bipartite Graph/Partite Set

Let $G = \left({A \mid B, E}\right)$ be a bipartite graph. Then $A$ and $B$ are called the '''partite sets''' of the bipartite graph.


Definition:Conjugation (Abstract Algebra)/Conjugate

Let $a \in A$. Then $C \left({a}\right)$ is called the '''conjugate''' of $a$.


Definition:Gamma Function/Partial

Let $m \in \Z_{\ge 0}$. The '''partial Gamma function at $m$''' is defined as: :$\displaystyle \Gamma_m \left({z}\right) := \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ which is valid except for $z \in \left\{{0, -1, -2, \ldots, -m}\right\}$.


Definition:Neighborhood (Real Analysis)/Open Subset

Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$. Then $N_\alpha$ is a '''neighborhood''' of $\alpha$.


Definition:Neighborhood (Real Analysis)

Let $\alpha \in \R$ be a real number. === Open Subset Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Open Subset}} === Epsilon-Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Epsilon}}


Definition:Oscillation/Real Space/Oscillation at Point/Epsilon

The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \inf \left\{{\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right): \epsilon \in \R_{>0}}\right\}$ where $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X$.


Definition:Subsignature

Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Then $\mathcal L$ is said to be a '''subsignature''' of $\mathcal L'$''', denoted $\mathcal L \subseteq \mathcal L'$, {{iff}}, for each $n \in \N$: :$\mathcal F_n \left({\mathcal L}\right) \subseteq \mathcal F_n \left({\mathcal L'}\right)$ :$\mathcal P_n \left({\mathcal L}\right) \subseteq \mathcal P_n \left({\mathcal L'}\right)$ where $\mathcal F_n$ denotes the collection of $n$-ary function symbols, and $\mathcal P_n$ denotes the collection of $n$-ary predicate symbols. === Supersignature === {{:Definition:Subsignature/Supersignature}}


Definition:Subsignature/Supersignature

Let $\mathcal L$ be a subsignature of $\mathcal L'$. Then $\mathcal L'$ is said to be a '''supersignature''' of $\mathcal L$, denoted: :$\mathcal L' \supseteq \mathcal L$


Definition:Lower Semicontinuous/Subset

Let $A \subseteq S$, and $A \ne \varnothing$. The function $f$ is said to be '''lower semicontinuous on''' $A$ {{iff}} $f$ is lower semicontinuous at every $a \in A$.


Definition:Reduct of Structure

Let $\mathcal L, \mathcal L'$ be signatures of the language of predicate logic. Let $\mathcal L$ be a subsignature of $\mathcal L'$. Let $\mathcal A, \mathcal A'$ be structures for $\mathcal L, \mathcal L'$, respectively. Then $\mathcal A$ is called the '''reduct of $\mathcal A'$ to $\mathcal L$''' {{iff}}: :For all function symbols $f$ of $\mathcal L$, one has $f_{\mathcal A'} = f_{\mathcal A}$ :For all predicate symbols $p$ of $\mathcal L$, one has $p_{\mathcal A'} = p_{\mathcal A}$ where $f_{\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\mathcal A'$. Symbolically, one may write $\mathcal A = \mathcal A' \restriction_{\mathcal L}$.


Definition:Expansion of Structure

Let $\mathcal L, \mathcal L'$ be signatures of the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\mathcal A, \mathcal A'$ be structures for $\mathcal L, \mathcal L'$, respectively. Then $\mathcal A'$ is called an '''expansion of $\mathcal A$ to $\mathcal L'$''' {{iff}}: :For all function symbols $f$ of $\mathcal L$, one has $f_{\mathcal A'} = f_{\mathcal A}$ :For all predicate symbols $p$ of $\mathcal L$, one has $p_{\mathcal A'} = p_{\mathcal A}$ where $f_{\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\mathcal A'$.


Definition:Meet Irreducible

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $g \in S$. Then $g$ is '''meet irreducible''' {{iff}}: :$\forall x, y \in S: g = x \wedge y \implies g = x$ or $g = y$


Definition:Stencil

A quintuple $\tuple {X, S_X, \map S 0, \Delta, \delta}$ defines a '''stencil''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= Index Domain:         | t = $X$ is discrete: $X \subseteq \Z^{\size X}$ and of finite dimension: $\size X \in \N$ }} {{axiom | n = 2         | lc= State Range:         | t = $S_X$ is a well-defined set }} {{axiom | n = 3         | lc= Initial State:         | t = $\map S 0$ maps from index space to state space $\map S 0: X \to S_X$ }} {{axiom | n = 4         | lc= Neighbourhood Delta:         | t = $\Delta$ is a vector of '''index offsets''': $\Delta \in \paren {\Z^{\size X} }^{\size \Delta}$ }} {{axiom | n = 5         | lc= Transition Combinator:         | t = $\delta$ is a mapping $\delta: S_X^{\size \Delta} \to S_X$ from neighbourhood states to states }} {{end-axiom}} {{refactor|Put update sweep and stencil evolution into their own pages}} By means of the '''update sweep''' $\forall \vec x \in X: \map {\map S {n + 1} } {\vec x} = \displaystyle \map \delta {\prod_{i \mathop = 1}^{\size \Delta} \map {\map S n} {\vec x + \Delta_i} }$ this induces a '''stencil evolution''' $\N \ni n \mapsto \map S n \in \paren {X \to S_X}$ from the '''initial state''' $\map S 0$. Category:Definitions/Cellular Automata s4pak4bzdx71q0bp97qi89alappd492


Definition:Conway Life

A cellular evolution deriving from a '''stencil''' shaped as: :$(1): \quad X := \Z^2$ :$(2): \quad S_X := 2 := \set {0, 1}$ :$(3): \quad \map S 0 \in \paren {\Z^2 \to 2}$ :$(4): \quad \Delta := \tuple {\tuple {0, 0}, \tuple {0, 1}, \tuple {1, 1}, \tuple {1, 0}, \tuple {1, -1}, \tuple {0, -1}, \tuple {-1, -1}, \tuple {-1, 0}, \tuple {-1, 1} }$ :$(5): \quad \map \delta {m, n_1, n_2, n_3, n_4, n_5, n_6, n_7, n_8} := \tuple {m, \displaystyle \sum_{i \mathop = 1}^8 n_i} \in \set {\tuple {0, 3}, \tuple {1, 2}, \tuple {1, 3} } ? 1 : 0$ is called a '''Conway life'''.


Definition:Legendre Symbol/Definition 1

The '''Legendre symbol''' $\left({\dfrac a p}\right)$ is defined as: :$\left({\dfrac a p}\right) := \begin{cases} +1 & : a^{\frac{(p - 1)} 2} \bmod p = 1 \\ 0 & : a^{\frac{(p - 1)} 2} \bmod p = 0 \\ -1 & : a^{\frac{(p - 1)} 2} \bmod p = p - 1 \end{cases}$


Definition:Legendre Symbol/Definition 2

The '''Legendre symbol''' $\left({\dfrac a p}\right)$ is defined as: :$\left({\dfrac a p}\right) := a^{\frac{(p - 1)} 2} \bmod p$


Definition:Termial

Let $n \in \Z_{> 0}$ be a positive integer. The '''termial''' of $n$ is denoted $n?$ and defined as: :$\displaystyle n? = \sum_{k \mathop = 1}^n k = 1 + 2 + \cdots + n$


Definition:Way Below Open

Let $\left({S, \preceq}\right)$ a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''way below open''' {{iff}}: :$\forall x \in X: \exists y \in X: y \ll x$ where $\ll$ denotes the way below relation.


Definition:Way Above Closure

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. The '''way above closure''' of $x$, denoted by $x^\gg$, is defined by: :$x^\gg := \left\{ {y \in S: x \ll y}\right\}$ where $\ll$ denotes the way below relation.


Definition:Termial/Real Numbers

Let $x \in \R$ be a real number. The '''termial''' of $x$ is denoted $x?$ and defined as: :$\displaystyle x? = \dfrac {x \left({x + 1}\right)} 2$


Definition:Operation/Binary Operation/Product/Right

Let $x$ and $y$ be elements which are operated on by a given operation $\circ$. The '''right-hand product of $x$ by $y$''' is the product $x \circ y$.


Definition:Operation/Binary Operation/Product/Left

Let $x$ and $y$ be elements which are operated on by a given operation $\circ$. The '''left-hand product of $x$ by $y$''' is the product $y \circ x$.


Definition:Special Orthogonal Group

Let $k$ be a field. The '''special ($n$th) orthogonal group (on $k$)''', denoted $\map {\mathrm {SO} } {n, k}$, is: :the set of all proper orthogonal order-$n$ square matrices over $k$ :under (conventional) matrix multiplication. That is: $\map {\mathrm {SO} } {n, k} = \map {\mathrm O} {n, k} \cap \SL {n, k}$


Definition:Finite Semigroup

A '''finite semigroup''' is a semigroup of finite order. That is, a semigroup $\struct {S, \circ}$ is a '''finite semigroup''' {{iff}} its underlying set $G$ is finite. That is, a '''finite semigroup''' is a semigroup with a finite number of elements.


Definition:Inverse Row form of Cayley Table for Group

Let $\struct {G, \circ}$ be a finite group. The Cayley table for $\struct {G, \circ}$ can be presented in a form such that the rows are headed by the inverse elements of the elements which head the corresponding columns. This form is known as the '''inverse row form''' of the Cayley table for $\struct {G, \circ}$.


Definition:Group Generated by Reciprocal of z and 1 minus z

Let: :$S = \set {f_1, f_2, f_3, f_4, f_5, f_6}$ where $f_1, f_2, \ldots, f_6$ are complex functions defined for all $z \in \C \setminus \set {0, 1}$ as: {{begin-eqn}} {{eqn | l = \map {f_1} z       | r = z }} {{eqn | l = \map {f_2} z       | r = \dfrac 1 {1 - z} }} {{eqn | l = \map {f_3} z       | r = \dfrac {z - 1} z }} {{eqn | l = \map {f_4} z       | r = \dfrac 1 z }} {{eqn | l = \map {f_5} z       | r = 1 - z }} {{eqn | l = \map {f_6} z       | r = \dfrac z {z - 1} }} {{end-eqn}} Let $\circ$ denote composition of functions. Then $\struct {S, \circ}$ is the '''group generated by $\dfrac 1 z$ and $1 - z$'''.


Definition:Coprime Residue Class

Let $m \in \Z: m \ge 1$. Let $a \in \Z$ such that: :$a \perp m$ where $\perp$ denotes that $a$ is prime to $m$. Let $\eqclass a m$ be the residue class of $a$ (modulo $m$): :$\set {x \in \Z: \exists k \in \Z: x = a + k m}$ Then $\eqclass a m$ is referred to as a '''coprime residue class'''.


Definition:Reduced Residue System

The '''reduced residue system modulo $m$''', denoted $\Z'_m$, is the set of all residue classes of $k$ (modulo $m$) which are prime to $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Thus $\Z'_m$ is the '''set of all coprime residue classes modulo $m$''': :$\Z'_m = \set {\eqclass {a_1} m, \eqclass {a_2} m, \ldots, \eqclass {a_{\map \phi m} } m}$ where: :$\forall k: a_k \perp m$ :$\map \phi m$ denotes the Euler phi function of $m$.


Definition:Set of Residue Classes/Real Modulus

The quotient set of congruence modulo $z$ denoted $\R_z$ is: :$\R_z = \dfrac \R {\mathcal R_z}$ Thus $\R_z$ is the '''set of all residue classes modulo $z$'''. It follows from the Fundamental Theorem on Equivalence Relations that the quotient set $\R_z$ of congruence modulo $z$ forms a partition of $\R$.


Definition:Order Generating

Let $\left({S, \preceq}\right)$ be a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''order generating''' {{iff}} :$\forall x \in S: x^\succeq \cap X$ admits an infimum and $x = \inf \left({x^\succeq \cap X}\right)$


Definition:Finite Group/Axioms

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


Definition:Group of Rotation Matrices Order 4

Consider the algebraic structure $S$ of rotation matrices: :$R_4 = \set {\begin {bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix}, \begin {bmatrix} 0 & 1 \\ -1 & 0 \end {bmatrix}, \begin {bmatrix} -1 & 0 \\ 0 & -1 \end {bmatrix}, \begin {bmatrix} 0 & -1 \\ 1 & 0 \end {bmatrix} }$ under the operation of (conventional) matrix multiplication. $R_4$ is the '''group of rotation matrices of order $4$'''.


Definition:Additive Group of Integers Modulo 4

The '''additive group of integers modulo $4$''' $\struct {\Z_4, +_4}$ is the set of integers modulo $4$ under the operation of addition modulo $4$.


Definition:Dimension of Vector Space/Vector

Informally, an element of an $n$-dimensional vector space is often referred to as an '''$n$-dimensional vector'''.  It must be understood that this is no more than a convenient shorthand. It is not the ''vector'' which possesses the dimensionality, but the ''space in which it is embedded''.


Definition:Complex Root

Let $z \in \C$ be a complex number such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $w \in \C$ such that: :$w^n = z$ Then $w$ is a '''(complex) $n$th root of $z$''', and we can write: :$w = z^{1 / n}$


Definition:Root of Unity/Complex/Primitive

A '''primitive (complex) $n$th root of unity''' is an element $\alpha \in U_n$ such that: :$U_n = \set {1, \alpha, \alpha^2, \ldots, \alpha^{n - 1} }$


Definition:Prime Element (Order Theory)

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Let $p \in S$. Then $p$ is '''a prime element (of $\struct {S, \wedge, \preceq}$)''' {{iff}}: :$\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$


Definition:Relation/Relation as Subset of Cartesian Product

Most treatments of set theory and relation theory define a '''relation on $S \times T$''' to refer to just the truth set itself: :$\RR \subseteq S \times T$ where: :$S \times T$ is the Cartesian product of $S$ and $T$. Thus under this treatment, $\RR$ is a set of ordered pairs, the first coordinate from $S$ and the second coordinate from $T$. This approach leaves the precise nature of $S$ and $T$ undefined.


Definition:Airy's Equation

'''Airy's equation''' is the second order ODE: :$y'' = x y$


Definition:Game

In the context of game theory, a '''game''' is any social situation which involves more than one person. More specifically, a '''game''' is a problem of strategy between two or more parties in which all parties involved seek (usually) to maximise their outcome. A '''game''' $G$ can be specified by: :its players :the moves available to each player :the preference relation for each player: that is: :$G = \sequence {N, \sequence {A_i}, \sequence {\succsim_i} }$ $G$ can also be specified by: :its players :the moves available to each player :the payoff function for each player: that is: :$G = \sequence {N, \sequence {A_i}, \sequence {u_i} }$


Definition:Game/Player

Each of the parties involved in a '''game''' are called '''players'''.


Definition:Game/Strategy

A '''strategy''' is a complete plan of action that defines what a '''player''' will do under all circumstances in a  '''game'''.


Definition:Game/Payoff

The '''payoff''' of a '''game''' is the reward or punishment made to a '''player''' at the end of the '''game''' as a result of combination of the various strategies employed.


Definition:Game of Perfect Information

A '''game of perfect information''' is a game in which all players have complete information on the state of the game.


Definition:Two-Person Game

A '''two-person game''' is a game in which there are (exactly) two players.


Definition:Zero-Sum Game

A '''zero-sum game''' is a game in which the total of the payoffs to all players is zero.


Definition:Equilibrium Point

An '''equilibrium point''' is a stable outcome of a game associated with a particular set of strategies.


Definition:Zero-Sum Game/Examples

Examples of zero-sum game include: * [https://en.wikipedia.org/wiki/Poker Poker]: the total amount of money on the table is fixed -- what one person loses the other players win.


Definition:Stable Outcome

An '''stable outcome''' is an end point of a game such that if a players were to use a new strategy, that player would experience a lower payoff.


Definition:Two-Person Zero-Sum Game

A '''two-person zero-sum game''' is a two-person game with zero sum.


Definition:Equilibrium Strategies

A system of strategies, one for each player, is '''in equilibrium''' {{iff}} they result in an equilibrium point. Such strategies are known as '''equilibrium strategies'''. In the singular, an '''equilibrium strategy''' is one that contributes to an equilibrium point.


Definition:Linear Programming

'''Linear programming''' is the branch of mathematical programming which studies optimization of mathematical models whose requirements are represented by linear relationships.


Definition:Matching Pennies

There are two players: $A$ and $B$. Each player puts down a coin, traditionally a [https://en.wikipedia.org/wiki/Penny penny], either head or tail up, without showing it to the other player. The coins are then uncovered. If they both show the same side, $A$ is deemed to have won, and he takes both coins. If they show different sides, $B$ is deemed to have won, and he takes both coins.


Definition:Coin/Coin-Tossing

'''Coin-tossing''' is a technique to select randomly one of two options. The usual scenario is to resolve an issue between two parties. The most usual of these is to determine which of the two parties is to choose whether to take the first move, or otherwise determine the starting arrangement, in a game or sport. A coin is thrown into the air in such a manner that it turns over and over. This can be conveniently achieved by coiling the hand into a loose fist, balancing the coin on the first knuckle of the index finger, and then pinging the thumb up from under the index finger by a physical process derived from the science of catastrophe theory. The nail of the thumb will then impact smartly against the edge of the coin which projects over the edge of the index finger's knuckle, propelling it upwards in such a manner that, by an application of the physics of angular momentum, it will be caused to rotate more-or-less rapidly about a diameter. When it descends, the side which will remain uppermost will be dependent upon both the speed of rotation and the distance travelled, neither of which is easy to gauge precisely enough to make the process predictable. The descending coin will either be caught in the hand by the one flipping it (the "tosser"), to be immediately clasped to the back of the other hand, or allowed to fall directly to the ground. The other party (the "caller") is offered the chance of guessing which of the two sides of the coin is on top. The two sides are usually referred to (in the English-speaking world) as "heads" and "tails". The "heads" side tends to be the one that features the head of state of the nation to which the coin belongs, while the "tails" is the other side. Once the caller has made the call, the uppermost side of the coin is revealed. If the side matches that called by the caller, the caller has won, and is given the option of taking the choice as to the starting configuration of the game. Otherwise the tosser has won, and takes the option.<ref>The play ''{{WP|Rosencrantz_and_Guildenstern_Are_Dead|Rosencrantz and Guildenstern are Dead}}'' by {{WP|Tom_Stoppard|Tom Stoppard}} begins with an unfeasibly long sequence of "heads" calls in a game of '''coin-tossing'''.</ref>


Definition:Coin

A '''coin''' is a device for choosing a random number in the set $\left\{ {0, 1}\right\}$. It has been suggested that this device has other uses, but social anthropologists are divided as to what they might be. Perhaps they can resolve their differences by tossing a coin.


Definition:Coin/Head

The '''head''' of a coin is the side which traditionally bears an image of the head of state of the nation to which the coin is issued.


Definition:Coin/Tail

The '''tail''' of a coin is the opposite side to its head. It traditionally bears an image of cultural or historical significance to the nation to which the coin is issued.


Definition:Game/Rules

A game is specified completely by its '''rules'''. They prescribe for each player a model of rational choice: {{:Definition:Model of Rational Choice}}


Definition:Move

A '''move''' is an action by a player in a game made according to the rules.


Definition:Play

A '''play''' is a particular instance of the operation of a game.


Definition:Blotto

'''Blotto''' is a class of game of which an example has the mechanics as follows: There are two players: '''General $A$''' and '''Colonel $B$(lotto)'''. $A$ commands $4$ companies. $B$ commands $5$ companies. $A$ can reach a town on either of two different roads. It is the aim of $A$ to reach a town, which can be achieved by travelling down either of two different roads. He can send each of his $4$ companies on whatever road he chooses. $B$ can order any of his $5$ companies to defend either of the two roads. $A$ wins if, on either road, he has (strictly) more companies than $B$.


Definition:Card Game with Bluffing

There are two players: $A$ and $B$. First it is $A$'s move. $A$ receives one of two possible cards: either the '''high card''' or the '''low card'''. If he receives the '''high card''', he must bid $2$ credits. If he receives the '''low card''', he has two options, either: :$A (1): \quad$ He may pay $1$ credit, and the play is complete. or :$A (2): \quad$ He may bid $2$ credits. If $A$ has bid $2$ credits, it is $B$'s move. $B$ has two options, either: :$B (1): \quad$ He may pay $1$ credit or :$B (2): \quad$ He may challenge $A$'s bid. If $A$ had the '''high card''', $B$ must pay $2$ credits to $A$. If $A$ had the '''low card''', $A$ must pay $2$ credits to $B$. The play is complete.


Definition:Normalization of Game

The operation which reduces the structure of a game to its strategies is known as '''normalizing''' the game.


Definition:Extensive Form of Game

A game is described in its '''extensive form''' as follows: :$(1): \quad$ Each successive move of each player is analysed :$(2): \quad$ The pattern of information is also taken into account.


Definition:Non-Zero-Sum Game/Examples

Examples of non-zero-sum games include: === Trade === Two parties trading: both may simultaneously gain by a transaction. It can be assumed that the buyer attaches a value to goods purchased which is higher than the price paid. At the same time, the price paid to the vendor can also be assumed to be higher than the value, to the vendor, of those goods. Thus, at the end of this transaction, both parties have a positive payoff. === Naval Warfare === The loss of a particular ship may cause a setback of considerably greater value to the enemy than the cost of the materiel used by the attacker to destroy it.


Definition:Zero-Sum Game/Non-Zero

A '''non-zero-sum game''' is a game in which the total of the payoffs to all players is dependent upon the moves made to reach those payoffs.


Definition:Multi-Person Game

A '''multi-person game''' is a game which has (strictly) more than two players.


Definition:Cooperative Game

A '''cooperative game''' is a multi-person game in which the players may form coalitions outside the game.


Definition:Coalition

A '''coalition''' is an agreement among players in a cooperative multi-person game such that they coordinate their strategies so as to confer an advantage to all players in that '''coalition'''.


Definition:Non-Cooperative Game

A '''non-cooperative game''' is a game in which the players cannot form coalitions outside the game. Hence each player is concerned only with his / her own fortunes.


Definition:Payoff Table

Let $G$ be a two-person game. A '''payoff table''' for $G$ is an array which specifies the payoff to each player for each strategy of both players. $G$ is completely defined by its '''payoff table'''. {{PayoffTable|table = $\begin{array} {r {{|}} c {{|}} } & \text{L} & \text{R} \\ \hline \text{T} & w_1, w_2 & x_1, x_2 \\ \hline \text{B} & y_1, y_2 & z_1, z_2 \\ \hline \end{array}$}} The two numbers in the entry formed by row $r$ and column $c$ are the payoffs when the row player's moves is $r$ and the column player's moves is $c$. The first component given is the payoff to the row player. If the names of the players are $1$ and $2$, the convention is that the row player is player $1$ and the column player is player $2$. If the names of the players are $\text{A}$ and $\text{B}$, the convention is that the row player is player $\text{A}$ and the column player is player $\text{B}$.


Definition:Maximising Player

Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the first of the players listed (in this case $A$) is distinguished from the other and referred to as the '''maximising player'''.


Definition:Minimising Player

Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the second of the players listed (in this case $B$) is distinguished from the other and referred to as the '''minimising player'''.


Definition:Distance to Nearest Integer Function/Definition 1

:$\norm \alpha:= \min \set {\size {n - \alpha}: n \in \Z}$


Definition:Distance to Nearest Integer Function/Definition 2

:$\norm \alpha:= \min \set {\set \alpha, 1 - \set \alpha}$ where $\set \alpha$ is the fractional part of $\alpha$.


Definition:Refinement of Cover/Finer Cover

Let $\mathcal V$ be a refinement of $\mathcal U$. Then $\mathcal V$ is '''finer''' than $\mathcal U$.


Definition:Refinement of Cover/Coarser Cover

Let $\mathcal V$ be a refinement of $\mathcal U$. Then $\mathcal U$ is '''coarser''' than $\mathcal V$.


Definition:Maximin Strategy

Let $G$ be a two-person game between players $A$ and $B$. A '''minimax strategy''' for $A$ is one in which: :$(1): \quad A$ assumes that $B$ will make that move which makes $A$'s payoff (or average payoff, in the case of a random element) as small as possible. :$(2): \quad A$ then makes the move which makes this smallest gain the largest possible. Hence it is the strategy that will '''maxi'''mise $A$'s '''min'''imum payoff. The reasoning is: :''I fear that my opponent will make the move which makes my payoff the smallest possible.  Hence I make my move so as to make this smallest payoff as large as possible.''


Definition:Payoff Table/Entry

Each of the values in a '''payoff table''' corresponding to the payoff for a combination of a move by each player is called an '''entry'''.


Definition:Totally Pathwise Disconnected Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''totally pathwise disconnected''' {{iff}} all path components of $T$ are singletons.


Definition:Saddle Point (Game Theory)

Let $G$ be a two-player zero-sum game. Let $G$ be defined by a payoff table $T$. Let $E$ be an entry in $T$ such that: :$E$ is the smallest entry in its row :$E$ is the largest entry in its column. Then $E$ is a '''saddle point''' of $G$.


Definition:Specially Strictly Determined Game

Let $G$ be a two-person zero-sum game. Let $G$ have a saddle point. Then $G$ can be referred to as a '''specially strictly determined game'''.


Definition:Eluding Game

'''The eluding game''' is a two-person game whose mechanics are as follows. The two players are $A$ and $B$. $A$ and $B$ each have three possible moves, the values: :$1, 2, 3$ If both choose the same number, $A$ pays to $B$ the amount of the chosen number. Otherwise, $A$ receives the amount of his own number from $B$. === Payoff Table === The payoff table of the '''eluding game''' is as follows: {{:Definition:Eluding Game/Payoff Table}}


Definition:Mixed Strategy

Let $G$ be a game. A '''mixed strategy''' for $G$ is a combination of different strategies for subsequent plays of $G$ which are played with particular frequencies.


Definition:Pure Strategy

Let $G$ be a game. A '''pure strategy''' for $G$ is single strategy for consecutive plays of $G$ which is used exclusively. It can be seen that a '''pure strategy''' is a special case of a mixed strategy for which the frequencies of all strategies are zero except for one, which is used every time.


Definition:Totally Pathwise Disconnected Space/Definition 2

A topological space $T = \struct {S, \tau}$ is '''totally pathwise disconnected''' {{iff}} the only continuous mappings from the closed unit interval $\closedint 0 1$ to $T$ are constant mappings.


Definition:Solution of Game

Let $G$ be a game. A '''solution''' of $G$ is a systematic description of the outcomes that may emerge in a family of games.


Definition:Optimal Strategy

Let $G$ be a game. Let $P$ be a player of $G$. An '''optimal strategy''' for $P$ is a strategy that contributes towards a solution of $G$.


Definition:Value of Game

Let $G$ be a game. The '''value''' of $G$ is the payoff resulting from a solution of $G$.


Definition:Completely Mixed Game

Let $G$ be a game. Then $G$ is '''completely mixed''' {{iff}} all players use all their strategies in its optimal solution.


Definition:Dominating Strategy

Let $G$ be a game. Let player $P$ have pure strategies $A_1$ and $A_2$ in $G$. Then $A_1$ '''dominates''' $A_2$ {{iff}}: :for any strategy of an opposing player, $A_1$ is at least as good as $A_2$ :for at least one strategy of an opposing player, $A_1$ is strictly better than $A_2$.


Definition:Strategic Game

A '''strategic game''' is a game in which: :$(1): \quad$ Each player chooses a strategy once per play :$(2): \quad$ The move(s) of each player operates simultaneously.  That is, when choosing a move, each player has any information about the move that another player will make. It consists of: :A finite set $N$ of players :For each player $i \in N$, a non-empty set $A_i$ of moves available to player $i$ :For each player $i \in N$, a preference relation $\succsim_i$ on $A = \displaystyle \prod_{j \mathop \in N} A_j$ of player $i$.


Definition:Extensive Game

An '''extensive game''' is a game in which: :$(1): \quad$ moves are made according to an order of events :$(2): \quad$ A player can consider and reconsider a course of action at any time a decision needs to be made, throughout the play of a game.


Definition:Game of Imperfect Information

A '''game of imperfect information''' is a game in which some information about other players is not known by all the players.


Definition:Rational Player

The players of a game are generally assumed to be '''rational''' in the sense that: :$(1): \quad$ Each player is fully aware of all possible choices that can be made :$(2): \quad$ Each player may form expectations about any unknowns :$(3): \quad$ Each player deliberately chooses an action based on a process of optimization.


Definition:Model of Rational Choice

:$(1): \quad$ A set $A$ of moves from which the player may choose :$(2): \quad$ A set $C$ of consequences of each of those moves :$(3): \quad$ A consequence function $g: A \to C$ which maps a consequence to each action :$(4): \quad$ A preference relation $\succsim$, which is a total ordering on $C$.


Definition:Consequence

A '''consequence''' is a state in a game which results from a move made by a player in that game made according to the rules.


Definition:Consequence Function

Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. A '''consequence function''' for $P$ is a mapping from the set $A$ to the set $C$: :$g: A \to C$


Definition:Preference Relation

Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''preference relation''' on $P$ is a total preordering $\succsim$ on $C$ which ranks $C$ according to $P$'s eventual payoff.


Definition:Utility Function

Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''utility function''' on $C$ is a mapping from $C$ to the real numbers $\R$ so as to define a preference relation on $C$: :$U: C \to \R$ by the condition: :$\forall x, y \in C: x \succsim y \iff U \left({x}\right) \ge U \left({y}\right)$


Definition:Rational Decision-Maker

Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set $C$ of consequences of the moves available to $P$. Let $g: A \to C$ be the consequence function on $A$. Let $B \subseteq A$ be a the set of moves which are feasible. $P$ is a '''rational decision-maker''' {{iff}} $P$ chooses an element $a^* \in B$ such that: :$\forall a \in a^*: g \left({a^*}\right) \succsim g \left({a}\right)$ where $\succsim$ is the preference relation on $C$. That is, that $P$ solves the problem: :$\displaystyle \max_{a \mathop \in B} U \left({g \left({a}\right)}\right)$ where $U$ is the utility function on $C$. It is assumed that $P$ uses the same preference relation for all $B \subseteq A$.


Definition:Steady State Interpretation

Let $G$ be a game. The '''steady state interpretation''' of the solution of $G$ treats $G$ as a model whose purpose is to explain some sort of regularity in a collection of similar situations.


Definition:Deductive Interpretation

Let $G$ be a game. The '''deductive interpretation''' of the solution of $G$ treats $G$ in isolation, as a singular event, and, under the assumption that the players are rational, deduces the outcome of $G$ based on the rational model.


Definition:Positive Real Vector Space

Let $\R_{\ge 0}$ be the set of positive real numbers. Then the $\R_{\ge 0}$-module $\R_{\ge 0}^n$ is called the '''positive real ($n$-dimensional) vector space'''.


Definition:Absolute Real Vector Ordering

Let $x$ and $y$ be elements of the real vector space $\R^n$. The '''absolute real vector ordering''' is the partial ordering $\ge$ defined on the real vector space $\R^n$ as: :$\forall x, y \in \R^n: x \ge y \iff \forall i \in \left\{ {1, 2, \ldots, n}\right\}: x_i \ge y_i$


Definition:Absolute Real Vector Strict Ordering

Let $x$ and $y$ be elements of the real vector space $\R^n$. The '''absolute real vector strict ordering''' is the strict partial ordering $\ge$ defined on the real vector space $\R^n$ as: :$\forall x, y \in \R^n: x \ge y \iff \forall i \in \left\{ {1, 2, \ldots, n}\right\}: x_i \ge y_i$


Definition:Maximizers of Real Function

Let $f: X \to \R$ be a real-valued function. The '''set of maximisers''' of a $f$ are denoted: :$\displaystyle \arg \max_{x \mathop \in X} f \left({x}\right)$ {{definition wanted|The notation is defined in my source work, but not the concept.}}


Definition:Profile

Let $G$ be a game. Let $N$ be the set of players of $G$. Let $V$ be a variable defining some aspect of each of the players. The family of values of $V$ is referred to as a '''profile''', denoted: :$\family {x_i}_{i \mathop \in N}$ If the fact that $i \in N$ is understood, then $\family {x_i}$ can be used.


Definition:Profile/Less Player

Let $x = \left\langle{x_i}\right\rangle_{i \mathop \in N}$ be a profile of $V$. Then $x_{-i}$ is used to denote $x$ for all players except for $i$: :$x_{-i} := \left\langle{x_j}\right\rangle_{j \mathop \in N \setminus \left\{ {i}\right\} }$


Definition:Continuous Total Preordering

Let $S$ be a set. Let $\precsim$ be a total preordering on $S$. Let $\precsim$ be such that: :$a \precsim b$ whenever there exist sequences $\left\langle{a^k}\right\rangle_k$ and $\left\langle{b^k}\right\rangle_k$ that converge to $a$ and $b$ respectively for which $a^k \precsim b^k$ for all $k$. Then $\precsim$ is '''continuous'''. {{explain|The above has been rendered more-or-less verbatim from the source cited, but it is not well explained. Its context appears to be important in the discipline of game theory.}}


Definition:Quasi-Concave Total Preordering

Let $R^n$ be a real vector space. Let $\precsim$ be a total preordering on $\R^n$. Let $\precsim$ be such that: :for all $b \in \R^n$, the set $\left\{ {a \in \R^n: b \precsim a}\right\}$ is convex. Then $\precsim$ is '''quasi-concave'''.


Definition:Strictly Quasi-Concave Total Preordering

Let $R^n$ be a real vector space. Let $\precsim$ be a total preordering on $\R^n$. Let $\precsim$ be such that: :for all $b \in \R^n$, the set $\left\{ {a \in \R^n: b \precsim a}\right\}$ is strictly convex. Then $\precsim$ is '''strictly quasi-concave'''.


Definition:Pareto Efficiency

Let $N$ be a finite set. Let $X \subseteq \R^N$ be a set. {{explain|what does $\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \in X$ is '''Pareto efficient''' {{iff}} there exists no $y \in X$ for which $x_i < y_i$ for all $i \in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}


Definition:Strong Pareto Efficiency

Let $N$ be a finite set. Let $X \subseteq \R^N$ be a set. {{explain|what does $\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \in X$ is '''strongly Pareto efficient''' {{iff}} there exists no $y \in X$ for which $x_i \le y_i$ for all $i \in N$ and for which $x_i < y_i$ for at least one $i \in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}


Definition:Limit of Vector-Valued Function/Definition 1

Let: :$\mathbf r: t \mapsto \begin{bmatrix} f_1\left({t}\right) \\ f_2\left({t}\right) \\ \vdots \\ f_n\left({t}\right) \end{bmatrix}$ be a vector-valued function. The '''limit''' of $\mathbf r$ as $t$ approaches $c$ is defined as follows: {{begin-eqn}} {{eqn | l = \lim_{t \to c} \ \mathbf r \left({t}\right)       | o = :=       | r = \begin{bmatrix} \lim_{t \to c} \ f_1\left({t}\right) \\ \lim_{t \to c} \ f_2\left({t}\right) \\ \vdots \\ \lim_{t \to c} \ f_n\left({t}\right)\end{bmatrix} }} {{end-eqn}} where each $\lim$ on the RHS is a limit of a real function. The limit is defined to exist precisely when all the respective limits of the component functions exist.


Definition:Finite Game

A '''finite game''' is a game in which the set $A_i$ of moves available to player $i$ is finite for each player $i$.


Definition:Profile of Preference Relations

Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. The '''profile $\left\langle{\succsim_i^*}\right\rangle$ of preference relations over $C$''' is defined as: :$\forall a, b \in A: \left({a \succsim_i b}\right) \iff \left({g \left({a}\right) \succsim_i^* g \left({b}\right)}\right)$


Definition:Consequence Function with Probability

Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves. A '''consequence function''' for $P$ is a mapping from $A \times \Omega$ to $C$: :$g: A \times \Omega \to C$ interpreted as that $g \left({a, \omega}\right)$ is the consequence when the move is $a \in A$ and the realization $\omega$ of the random variable is $\omega \in \Omega$.


Definition:Lottery

A '''lottery''' is a game in which the consequence of each move is determined by the realization of a random variable. Let $X$ denote the set of prizes which a player may receive. Let $\Omega$ denote the set of possible states. A '''lottery''' is a mapping $f: X \times \Omega \to \R_{\ge 0}$ such that: :$\forall t \in \Omega: \displaystyle \sum_{x \mathop \in X} f \left({x, t}\right) = 1$ === Probability Model === {{:Definition:Lottery/Probability Model}} === State-Variable Model === {{:Definition:Lottery/State-Variable Model}}


Definition:Lottery Induced by Preference Relation

Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \in N$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves. Let $g: A \times \Omega \to C$ be the consequence function for player $i$. Then the '''lottery on $C$ induced by the profile of preference relations over $C$''' is defined by: :$\forall a, b \in A: \left({a \succsim_i b}\right) \iff \left({g \left({a, \omega_a}\right) \succsim_i^* g \left({b, \omega_b}\right)}\right)$ where $\succsim_i$ is the preference relation for player $i$.


Definition:Payoff Function

Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \in N$. A '''payoff function''' on $A$ is a mapping $u_i$ from $A$ to the real numbers $\R$: :$u_i: A \to \R$ defined by the condition: :$\forall a, b \in A: a \succsim_i b \iff u_i \left({a}\right) \ge u_i \left({b}\right)$ where $\succsim_i$ denotes the preference relation for player $i$.


Definition:Payoff Table/Zero-Sum

Let $G$ be a two-person zero-sum game. A '''payoff table''' for $G$ is an array which specifies the payoff to (conventionally) the maximising player for each strategy of both players. As $G$ is zero-sum, there is no need to specify the payoff to the minimising player, as it will be the negative of the payoff to the maximising player. {{PayoffTable|table = $\begin{array} {r {{|}} c {{|}} } & \text{L} & \text{R} \\ \hline \text{T} & w & x \\ \hline \text{B} & y & z \\ \hline \end{array}$}}


Definition:Consecutive Strategic Games

The interpretation of a strategic game $G$ can be that a particular play of $G$ is one of a sequence of such plays. In such a case, it is assumed that a player can form an expectation of the behaviour of other players through experience of previous plays.


Definition:Nash Equilibrium

Let a strategic game $G$ be modelled by: :$G = \stratgame N {A_i} {\succsim_i}$ A '''Nash equilibrium''' of $G$ is a profile $a^* \in A$ of moves which has the property that: :$\forall i \in N: \forall a_i \in A_i: \tuple {a^*_{-i}, a^*_i} \succsim_i \tuple {a^*_{-i}, a_i}$ Thus, for $a^*$ to be a '''Nash equilibrium''', no player $i$ has a move yielding a preferable outcome to that when $a^*_i$ is chosen, given that every other player $j$ has chosen his own equilibrium move. That is, no player can profitably deviate, if no other player also deviates. {{NamedforDef|John Forbes Nash|cat = Nash}}


Definition:Best-Response Function

Let a strategic game $G$ be modelled by: :$G = \left\langle{N, \left\langle{A_i}\right\rangle, \left\langle{\succsim_i}\right\rangle}\right\rangle$ Let $a^*$ be a Nash equilibrium of $G$: :$\forall i \in N: \forall a_i \in A_i: \left({a^*_{-i}, a^*_i}\right) \succsim_i \left({a^*_{-i}, a_i}\right)$ For any $a_{-1} \in A_{-i}$, let $B_i \left({a_{-i} }\right)$ be the set of player $i$'s best moves, defined as: :$B_i \left({a_{-i} }\right) = \left\{ {a_i \in A_i: \forall a'_i \in A_i: \left({a_{-i}, a_i}\right) \precsim_i  \left({a_{-i}, a'_i}\right)}\right\}$ Then $B_{-i}$ is known as the '''best-response function''' of player $i$.


Definition:Bach or Stravinsky?

There are two players: $\text A$lexis and $\text B$everley. They wish to go out together to a musical concert, but $\text A$ prefers Bach and $\text B$ prefers Stravinsky. The key points are: :$\text A$lexis and $\text B$everley wish to coordinate their behaviour but: :they have conflicting interests.


Definition:Coordination Game

There are two players: $\text A$lexis and $\text B$everley. They wish to go out together to a musical concert to experience either the music of Mozart or Mahler. Unaccountably, both $\text A$ and $\text B$ prefer Mozart.  (It takes all sorts to make a world.) The key points are: :$\text A$lexis and $\text B$everley wish to coordinate their behaviour but: :they have common interests.


Definition:Strictly Competitive Game

A '''strictly competitive game''' is a game in which the interests of each player are diametrically opposed.


Definition:Sealed-Bid Auction

A '''sealed-bid auction''' is a game in the following format: An object $F$ is to be assigned to a player in a set $\left\{ {1, 2, \ldots, n}\right\}$ in exchange for a payment. The valuation of $F$ by player $i$ is $v_i$, and: :$v_1 > v_2 > \cdots > v_n > 0$ Each player simultaneously submits a non-negative number, called a '''bid'''. $F$ is then given to the player with the lowest index among those who submit the highest bid, in exchange for a payment. === First Price Auction === {{:Definition:Sealed-Bid Auction/First Price}} === Second Price Auction === {{:Definition:Sealed-Bid Auction/Second Price}}


Definition:Sealed-Bid Auction/First Price

A '''first price auction''' is a sealed-bid auction in which the payment made by the winner is the price which is bid by that player.


Definition:Sealed-Bid Auction/Second Price

A '''second price auction''' is a sealed-bid auction in which the payment made by the winner is the highest bid made by the players who did ''not'' win. So if only one player submits the highest bid, the price paid is the second highest bid.


Definition:Weakly Dominant Move

Let $G$ be a game. Let $a$ be a move available to player $P$. Then $a$ is '''weakly dominant''' {{iff}} :for any move of an opposing player, the payoff of $a$ is not less than the payoff of any other move available to $P$.


Definition:Intelligent Player

Let $G$ be a game. Let $P$ be a player in $G$. Then $P$ is described as '''intelligent''' {{iff}} $P$ knows everything about $G$ that there is to know, and can use that knowledge to make informed decisions about what move to make. In game theory, all players are assumed to be '''intelligent'''.


Definition:Decision Theory

'''Decision theory''' is the study of the reasoning underlying an agent's choices.


Definition:Bayesian Decision Theory

'''Bayesian decision theory''' is a branch of decision theory which is informed by Bayesian probability. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs.


Definition:Subjective Probability Distribution

A '''subjective probability distribution''' is a probability distribution based on the beliefs of a rational decision-maker about all relevant unknown factors concerning a game. As new information becomes available, the '''subjective probabilities''' can then revised according to Bayes' Formula.


Definition:Lottery/Probability Model

In a '''probability model''', a lottery is a probability distribution over a set of prizes.


Definition:Lottery/State-Variable Model

In a '''state-variable model''', a lottery is defined as a mapping from a set of possible states into a set of prizes.


Definition:Objective Unknown

An '''objective unknown''' is an event which has a well-defined objective probability of occurring. Two '''objective unknowns''' with the same probability are equivalent in the field of decision theory. An '''objective unknown''' is appropriately modelled by means of a probability model.


Definition:Subjective Unknown

An '''subjective unknown''' is an event which does not have a well-defined objective probability of occurring. An '''subjective unknown''' is appropriately modelled by means of a state-variable model. === Examples === {{:Definition:Subjective Unknown/Examples}}


Definition:Prize

Let $G$ be a game. A prize in $G$ is a payoff whose utility value is strictly positive. The term is usually seen in the context of a lottery.


Definition:State

Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. Each of the possible values that can be taken by $X$ is known as a '''state''' of $G$.


Definition:True State

Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. The particular state of $G$ which $X$ actually takes is known as the '''true state (of the world)'''.


Definition:Conditional Preference

Let $G$ be a lottery. Let $P$ be a player of $G$. Let $X$ denote the set of prizes which $P$ may receive. Let $\Omega$ denote the set of possible states of $G$. Let $\Xi$ be the event space of $G$. Let $L$ be the set of all plays of $G$. Let $f$ and $g$ be two lotteries in $G$. Let $S \subseteq \Xi$ be an event. A '''conditional preference''' is a preference relation $\succsim_S$ such that: :$f \succsim_S g$ {{iff}} $f$ would be at least as desirable to $P$ as $g$, if $P$ was aware that the true state of the world was $S$. That is, $f \succsim_S g$ {{iff}} $P$ prefers $f$ to $g$ and he knows only that $S$ has occurred. The notation $a \sim_S b$ is defined as: :$a \sim_S b$ {{iff}} $a \succsim_S b$ and $b \succsim_S a$ The notation $a \succ_S b$ is defined as: :$a \succ_S b$ {{iff}} $a \succsim_S b$ and $a \not \sim_S a$ When no conditioning event $S$ is mentioned, the notation $a \succsim_\Omega b$, $a \succ_\Omega b$ and $a \sim_\Omega b$ can be used, which mean the same as $a \succsim b$, $a \succ b$ and $a \sim b$. {{handwaving|Myerson is as lax as all the other game theory writers when it comes to defining rigorous concepts. I am going to have to abandon this field of study until I really understand exactly what the underlying mathematical objects are.}}


Definition:Biquadrate

A '''biquadrate number''' (or just '''biquadrate''') is a number which can be expressed as the fourth power of an integer. The sequence of positive '''biquadrates''' begins: :$1, 16, 81, 256, 625, 1297, 2401, 4096, 6561, 10000, \ldots$ {{OEIS|A000583}}


Definition:Egyptian Fraction

An '''Egyptian fraction''' is a fraction either: :whose numerator is $1$ or :which is $\dfrac 2 3$


Definition:Primorial/Prime

Let $p_n$ be the $n$th prime number. Then the '''$n$th primorial''' $p_n \#$ is defined as: :$\displaystyle p_n \# := \prod_{k \mathop = 1}^n p_k$ That is, $p_n \#$ is the product of the first $n$ primes.


Definition:Primorial/Positive Integer

Let $n$ be a positive integer. Then: : $\displaystyle n\# := \prod_{i \mathop = 1}^{\pi \left({n}\right)} p_i = p_{\pi \left({n}\right)}\#$ where $\pi \left({n}\right)$ is the prime counting function. That is, $n\#$ is defined as the product of all primes less than or equal to $n$. Thus: :$n\# = \begin{cases} 1 & : n \le 1 \\ n \left({\left({n - 1}\right)\#}\right) & : n \mbox { prime} \\ \left({n - 1}\right)\# & : n \mbox { composite} \end{cases}$


Definition:Repunit

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$ be expressed in base $b$. $n$ is a '''repunit base $b$''' {{iff}} $n$ is a repdigit number base $b$ whose digits are $1$. That is, $n$ is a '''repunit base $b$''' {{iff}} all of the digits of $n$ are $1$. When $b$ is the usual base $10$, $n$ is merely referred to as a '''repunit'''.


Definition:Unit Fraction

A '''unit fraction''' is a fraction whose numerator is $1$.


Definition:Myriad

'''Myriad''' is an archaic word for the number $10 \ 000$ (ten thousand).


Definition:Thousand

A '''thousand''' is $1000$: $10$ to the power of $3$: :$1000 = 10^3 = 10 \times 10 \times 10$


Definition:Almost All/Set Theory/Countable

Let $S$ be a countably infinite set. Let $P: S \to \left\{ {\text{true}, \text{false} }\right\}$ be a property of $S$ such that: :$\left\{ {s \in S: \neg P \left({s}\right)}\right\}$ is finite. Then $P$ holds for '''almost all''' of the elements of $S$.


Definition:Champernowne Constant

The '''Champernowne constant''' is the real number whose decimal expansion is formed by concatenating the positive integers in ascending order: :$C_{10} = 0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$ {{OEIS|A033307}}


Definition:Absolutely Normal Real Number

A real number $r$ is '''absolutely normal''' if it is normal with respect to ''every'' number base $b$. That is, {{iff}} its basis expansion in every 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:Anomalous Cancellation

Let $r = \dfrac a b$ be a fraction where $a$ and $b$ are integers expressed in conventional decimal notation. '''Anomalous cancellation''' is a phenomenon whereby deleting (that is, cancelling) common digits from the numerator $a$ and the denominator $b$ of $r$, the value of $r$ the fraction does not change.


Definition:Newton-Mercator Series

Let $\ln x$ denote the natural logarithm function. Then: {{begin-eqn}} {{eqn | l = \map \ln {1 + x}       | r = x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n       | c =  }} {{end-eqn}} The series converges to the natural logarithm (shifted by $1$) for $-1 < x \le 1$. This is known as the '''Newton-Mercator series'''.


Definition:Filter on Set/Filtered Set

Let $\FF$ be a filter on $S$. Then $S$ is said to be '''filtered by $\FF$''', or just a '''filtered set'''.


Definition:Filter on Set/Trivial Filter

A filter $\FF$ on $S$ by definition specifically does ''not'' include the empty set $\O$. If a filter $\FF$ ''were'' to include $\O$, then from Empty Set is Subset of All Sets it would follow that ''every'' subset of $S$ would have to be in $\FF$, and so $\FF = \powerset S$. Such a "filter" is called '''the trivial filter''' on $S$.


Definition:Filter on Set/Definition 1

A '''filter on $S$''' (or '''filter of $S$''') is a set $\FF \subset \powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text F 1         | m = S \in \FF }} {{axiom | n = \text F 2         | m = \O \notin \FF }} {{axiom | n = \text F 3         | m = U, V \in \FF \implies U \cap V \in \FF }} {{axiom | n = \text F 4         | m = \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF }} {{end-axiom}}


Definition:Filter on Set/Definition 2

A '''filter on $S$''' (or '''filter of $S$''') is a set $\FF \subset \powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text F 1         | m = S \in \FF }} {{axiom | n = \text F 2         | m = \O \notin \FF }} {{axiom | n = \text F 3         | m = \forall n \in \N: U_1, \ldots, U_n \in \FF \implies \bigcap_{i \mathop = 1}^n U_i \in \FF }} {{axiom | n = \text F 4         | m = \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF }} {{end-axiom}}


Definition:Filter/Proper Filter

Let $\mathcal F$ be a filter on $\left({S, \preccurlyeq}\right)$. Then $\mathcal F$ is a '''proper filter''' on $S$ {{iff}} $\mathcal F \ne S$. That is, {{iff}} $\mathcal F$ is a proper subset of $S$.


Definition:Closed under Mapping/Arbitrary Product

Let $\phi: X^I \to T$ be a mapping or a partial mapping, taking $I$-indexed families as arguments. Denote with $\Dom \phi$ the domain of $\phi$ (if $\phi$ is a mapping, this is simply $X^I$). A set $S$ is '''closed under $\phi$''' {{iff}}: :$\forall \family {s_i}_{i \mathop \in I} \in S^I \cap \Dom \phi: \map \phi {\family {s_i}_{i \mathop \in I} } \in S$ Phrased in terms of image of a mapping, this translates to: :$\map \phi {S^I \cap \Dom \phi} \subseteq S$ Thus, in words, $S$ is '''closed under $\phi$''', {{iff}}: :Whenever $\phi$ is defined for an $I$-indexed family from $S$, it maps that indexed family into $S$ again.


Definition:Filter Basis/Equivalent Filter Bases

Two '''filter bases''' are '''equivalent''' {{iff}} they both generate the same filter.


Definition:Unit Cube

A '''unit cube''' is a cube each of whose edges is one unit long.


Definition:Cube Root/Real

Let $x \in \R_{\ge 0}$ be a positive real number. The '''cube roots of $x$''' is the real number defined as: :$x^{\paren {1 / 3} } := \set {y \in \R: y^3 = x}$ where $x^{\paren {1 / 3} }$ is the $3$rd root of $x$. The notation: :$y = \sqrt [3] x$ is usually encountered.


Definition:Archytas Curve

The '''Archytas curve''' is a curve formed as the intersection of a cone, a cylinder and a horn torus. {{DefinitionWanted}} {{NamedforDef|Archytas of Tarentum|cat = Archytas}}


Definition:Conchoid of Nicomedes

The '''conchoid of Nicomedes''' is the plane curve defined in Cartesian coordinates as: :$\paren {x - a}^2 \paren {x^2 + y^2} = b^2 x^2$ or in polar coordinates as: :$r = b + a \sec \theta$ for some real constants $a \in \R$, $b \in \R_{> 0}$. :600px The above diagram illustrates the '''conchoid of Nicomedes''' for $b = 1$ and various values of $a$ from $0$ to $3$.


Definition:Cissoid of Diocles

Let $C$ be a a circle of radius $a$ with a distinguished point $O$ on its circumference. Let $L$ be the tangent to $C$ at the other end of the diameter of $C$ through $O$. Let $R$ be a point on the circumference of $C$. Let $OR$ be produced to meet $L$ at $S$. Let $P$ be the point on $OS$ such that $OP$ = $RS$. The '''cissoid of Diocles''' is the locus of points $P$ as $R$ travels around the circumference of $C$.


Definition:Pell Numbers

The '''Pell numbers''' are a sequence $\sequence {P_n}$ which is formally defined by the recurrence relation: :$P_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$ The sequence of '''Pell numbers''' begins: :$0, 1, 2, 5, 12, 29, 70, 169, 408, 985, \ldots$


Definition:Pell-Lucas Numbers

The '''Pell-Lucas numbers''' are a sequence $\left \langle {Q_n}\right \rangle$ which is formally defined by the recurrence relation: :$P_n = \begin{cases} 2 & : n = 0 \\ 2 & : n = 1 \\ 2 P_{n -1} + P_{n - 2} & : \text {otherwise}\end{cases}$ The sequence of '''Pell-Lucas numbers''' begins: :$2, 2, 6, 14, 34, 82, 198, 478, 1154, \ldots$ {{OEIS|A002203}}


Definition:Best Rational Approximation

Let $x \in \R$ be an (irrational) real number. The rational number $a = \dfrac p q$ is a '''best rational approximation''' to $x$ {{iff}}: :$(1): \quad a$ is in canonical form, that is $p$ is coprime to $q$: $p \perp q$ :$(2): \quad \left\vert{x - \dfrac p q}\right\vert = \min \left\{ {\left\vert{x - \dfrac {p'} {q'} }\right\vert: q' \le q}\right\}$ That is: :$\left\vert{x - \dfrac p q}\right\vert$ is smaller than for any $\dfrac {p'} {q'}$ where $q' \le q$ where $\left\vert{x}\right\vert$ denotes the absolute value of $x$. === Sequence === {{definition wanted|sorted by increasing denominator}} Category:Definitions/Number Theory Category:Definitions/Real Analysis Category:Definitions/Continued Fractions k8dnfwqwqiwk8ba1vq2mz6snubc52ys


Definition:Prime Filter (Order Theory)

Let $\left({S, \preceq}\right)$ be an ordered set. Let $F$ be a filter in $\left({S, \preceq}\right)$. $F$ is a '''prime filter''' {{iff}}: :$\forall x, y \in S: \left({x \vee y \in F \implies x \in F \lor y \in F}\right)$


Definition:Quadratic Irrational/Reduced

An irrational root $\alpha$ of a quadratic equation with integer coefficients is a '''reduced quadratic irrational''' {{iff}} :$(1): \quad \alpha > 1$ :$(2): \quad$ its conjugate $\tilde{\alpha}$ satisfies: ::::$-1 < \tilde{\alpha} < 0$


Definition:Power Tower Function

The '''power tower function''' is the real function $f: \R \to \R$ defined on the closed interval $\closedint {e^{-e} } {e^{1/e} }$ as: :$\map f x = x^{x^{x^{x^{x^{\cdot^{\cdot^\cdot} } } } } }$ where the tower of powers tends to infinity.


Definition:Pentagram

A '''pentagram''' is a geometric figure formed by the line segments connecting alternate vertices of a regular pentagon: :400px


Definition:Golden Rectangle

A '''golden rectangle''' is a rectangle whose sides are in extreme and mean ratio: :500px Thus, in the above rectangle: :$AB = AD$ :$AB : BC = AC : AB$


Definition:Beatty Sequence

Let $x$ be an irrational number. The '''Beatty sequence on $x$''' is the integer sequence $\BB_x$ defined as: :$\BB_x := \sequence{\floor{n x} }_{n \mathop \in \Z_{\ge 0} }$


Definition:Beatty Sequence/Complementary

Let $\mathcal B_x$ be the '''Beatty sequence''' on $x$. The '''complementary Beatty sequence on $x$''' is the integer sequence formed by the integers which are missing from $\mathcal B_x$.


Definition:Lower Wythoff Sequence

The '''lower Wythoff sequence''' is the Beatty sequence on the golden section $\phi$. It starts: :$0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, \ldots$


Definition:Ultrafilter (Order Theory)

Let $O = \left({S, \preceq}\right)$ be an ordered set. Let $F$ be a filter in $O$. Then $F$ is '''ultrafilter''' (on $O$) {{iff}} :$F$ is proper subset of $S$ and :for all filter $G$ in $O$: $\left({F \subseteq G \implies F = G \lor G = S}\right)$


Definition:Upper Wythoff Sequence

{{:Definition:Upper Wythoff Sequence/Definition 2}} The '''upper Wythoff sequence''' starts: :$0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, \ldots$


Definition:Upper Wythoff Sequence/Definition 1

The '''upper Wythoff sequence''' is the complementary Beatty sequence on the golden section $\phi$.


Definition:Upper Wythoff Sequence/Definition 2

The '''upper Wythoff sequence''' is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.


Definition:Wythoff Pair

A '''Wythoff pair''' is an ordered pair of integers of the form: :$\tuple {\floor {k \phi}, \floor {k \phi^2} }$ where: :$\phi$ denotes the golden section: $\phi = 1 \cdotp 618 \ldots$ :$\floor x$ denotes the floor of $x$ :$k$ signifies a positive integer: $k \in \Z_{\ge 0}$. Thus the coordinates of a '''Wythoff pair''' are corresponding terms of the lower and upper Wythoff sequences. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}


Definition:Wythoff Sequence

=== Lower Wythoff Sequence === {{:Definition:Lower Wythoff Sequence}} === Upper Wythoff Sequence === {{:Definition:Upper Wythoff Sequence}}


Definition:Finite Infima Set

Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite infima set''' of $X$, denoted $\operatorname{fininfs}\left({X}\right)$, is defined by :$\left\{ {\inf A: A \in \mathit{Fin}\left({X}\right) \land A}\right.$ admits an infimum$\left.{}\right\}$ where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Definition:Wythoff's Game

'''Wythoff's game''' is a two-person game whose rules are as follows: :$(1): \quad$ The game starts with two piles of counters. :$(2): \quad$ Each player takes it in turns to remove either: ::::A number of counters from one pile ::::A number of counters from both pile such that the number of counters removed from each pile must be equal. :$(3): \quad$ The person who removes the last counter (or counters) is the winner. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}


Definition:Tribonacci Constant

The '''Tribonacci constant''' $\eta$ is the one real root of the cubic: :$x^3 - x^2 - x - 1 = 0$ Its decimal expansion starts: :$\eta = 1 \cdotp 83928 \, 67552 \, 1416 \ldots$ {{OEIS|A058265}}


Definition:Tribonacci Sequence/General

A '''general Tribonacci sequence''' is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as: :$u_n = \begin{cases} a & : n = 0 \\ b & : n = 1 \\ c & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$ where $a, b, c \in \Z$ are constants.


Definition:Tribonacci Sequence

The '''Tribonacci sequence''' is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as: :$u_n = \begin{cases} 0 & : n = 0 \\ 0 & : n = 1 \\ 1 & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$


Definition:Brun's Constant

'''Brun's constant''' is the sum of the series consisting of the reciprocals of the twin primes: :$B_2 := \paren {\dfrac 1 3 + \dfrac 1 5} + \paren {\dfrac 1 5 + \dfrac 1 7} + \paren {\dfrac 1 {11} + \dfrac 1 {13} } + \paren {\dfrac 1 {17} + \dfrac 1 {19} } + \paren {\dfrac 1 {29} + \dfrac 1 {31} } + \cdots$ Its approxmiate decimal expansion is: :$B_2 \approx 1 \cdotp 90216 \, 05831 \, 04 \ldots$ {{OEIS|A065421}} Estimates of its value are occasionally refined as further work is done to establish its nature.


Definition:Bilateral Symmetry

Let $F$ be a geometric figure. Let there exists a reflection $R$ in the such that $\map R F$ is a symmetry. Then $F$ has '''bilateral symmetry'''. === Axis of Symmetry === {{:Definition:Bilateral Symmetry/Axis}}


Definition:Metre/Square Metre

The '''square metre''' is the SI unit of area. The symbol for the '''square metre''' is $\mathrm m^2$.


Definition:Centimetre/Square Centimetre

The '''square centimetre''' is the CGS unit of area. The symbol for the '''square centimetre''' is $\mathrm {cm}^2$ or (the informal and ugly) $\mathrm {sq}. \ \mathrm {cm}$.


Definition:SI Units/Derived

The units derived from the SI base units include the following: {{begin-axiom}} {{axiom | lc= Square metre:         | m = \mathrm m^2 }} {{axiom | lc= Cubic metre:         | m = \mathrm m^3 }} {{axiom | lc= Newton:         | m = \mathrm {kg} \, \mathrm m \, \mathrm s^{-2} }} {{axiom | lc= Coulomb:         | m = \mathrm A \, \mathrm s }} {{end-axiom}} {{stub}}


Definition:FPS/Base Units

{| class="wikitable" style="margin:1em auto 1em auto" |+ FPS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! foot | $\mathrm {ft}$ | Length | $l$ |- ! pound | $\mathrm {lb}$ | Mass | $m$ |- ! second | $\mathrm s$ | Time | $t$ |- |} === Foot === {{:Definition:FPS/Foot}} === Pound === {{:Definition:FPS/Pound}} === Second === {{:Definition:FPS/Second}}


Definition:FPS/Pound

The '''pound''' is the FPS unit of measurement of mass, which has a number of different standards. The most common standard is for it to be defined in kilograms to be exactly $0.453 \, 592 \, 37 \ \mathrm{kg}$.


Definition:Imperial/Volume

The imperial units of volume are based on a binary system, in which each unit is a factor of $2$ larger than the next smaller unit. === Fluid Ounce === {{Definition:Imperial/Volume/Fluid Ounce}} === Gill === {{Definition:Imperial/Volume/Gill}} === Chopin === {{Definition:Imperial/Volume/Chopin}} === Pint === {{Definition:Imperial/Volume/Pint}} === Quart === {{Definition:Imperial/Volume/Quart}} === Pottle === {{Definition:Imperial/Volume/Pottle}} === Gallon === {{Definition:Imperial/Volume/Gallon}} === Peck === {{Definition:Imperial/Volume/Peck}} === Demi-Bushel === {{Definition:Imperial/Volume/Demi-Bushel}} === Bushel === {{Definition:Imperial/Volume/Bushel}} === Kilderkin === {{Definition:Imperial/Volume/Kilderkin}} === Barrel === {{Definition:Imperial/Volume/Barrel}} === Hogshead === {{Definition:Imperial/Volume/Hogshead}} === Pipe === {{Definition:Imperial/Volume/Pipe}} === Tun === {{Definition:Imperial/Volume/Tun}}


Definition:Imperial/Volume/Tun

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''tun''' }} {{eqn | r = 2       | c = pipes }} {{eqn | r = 1 \cdotp 16379 \, 904       | c = cubic metres }} {{eqn | r = 1 \, 163 \cdotp 79904       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Pipe

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pipe''' }} {{eqn | r = 2       | c = hogsheads }} {{eqn | r = 0 \cdotp 58189 \, 952       | c = cubic metres }} {{eqn | r = 581 \cdotp 89952       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Hogshead

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hogshead''' }} {{eqn | r = 2       | c = barrels }} {{eqn | r = 0 \cdotp 29094 \, 976       | c = cubic metres }} {{eqn | r = 290 \cdotp 94976       | c = litres }} {{end-eqn}}