Definition:Converse Statement

The '''converse''' of the conditional: :$p \implies q$ is the statement: :$q \implies p$


Definition:Complete Metric Space

=== Definition 1 === {{:Definition:Complete Metric Space/Definition 1}} === Definition 2 === {{:Definition:Complete Metric Space/Definition 2}}


Definition:Golden Mean

=== Definition 1 === {{:Definition:Golden Mean/Definition 1}} === Definition 2 === {{:Definition:Golden Mean/Definition 2}}


Definition:Teichmüller Annulus

Let $R \in \R_{>0}$. The set: :$A := \C \setminus \paren {\closedint{-1} 0 \cup \hointr R {+\infty} }$ is a '''Teichmüller annulus'''. The modulus of $A$ is denoted $\map \Lambda R$.


Definition:Degenerate Connected Set

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $T$. $H$ is a '''degenerate connected set''' of $T$ {{iff}} it is a connected set of $T$ containing exactly one element. === Non-Degenerate Connected Set === {{:Definition:Degenerate Connected Set/Non-Degenerate}} === Degenerate Connected Space === When $H = S$ itself, the entire space can be referred to in this way: {{:Definition:Degenerate Connected Set/Topological Space}}


Definition:Riemann Zeta Function

The '''Riemann Zeta Function''' $\zeta$ is the complex function defined on the half-plane $\map \Re s > 1$ as the series: :$\displaystyle \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$


Definition:Cevian

Let $\triangle ABC$ be a triangle. :400px A '''cevian''' is a line segment from a vertex of the triangle to the opposite side. In the above diagram, $CD$ is a '''cevian'''.


Definition:Median of Triangle

Let $\triangle ABC$ be a triangle. :400px A '''median''' is a cevian which bisects the opposite.


Definition:Planar Graph

A '''planar graph''' is a graph which can be drawn in the plane (for example, on a piece of paper) without any of the edges crossing over, that is, meeting at points other than the vertices. This is a '''planar graph''': :300px === Face === {{:Definition:Planar Graph/Face}}


Definition:Graph (Graph Theory)/Vertex

Let $G = \struct {V, E}$ be a graph. The '''vertices''' (singular: '''vertex''') are the elements of $V$. Informally, the '''vertices''' are the points that are connected by the edges.


Definition:Area

'''Area''' is the measure of the extent of a surface. It has $2$ dimensions and is specified in units of length squared.


Definition:Euler Characteristic of Finite Graph

Let $X = \struct {V, E}$ be a graph. Let $X$ be embedded in a surface. The '''Euler characteristic''' of $X$ is written $\map \chi X$ and is defined as: :$\map \chi x = v - e + f$ where: :$v = \size V$ is the number of vertices :$e = \size E$ is the number of edges :$f$ is the number of faces. === Generalized Formula === {{DefinitionWanted}}


Definition:Deleted Neighborhood

=== Real Analysis === {{:Definition:Deleted Neighborhood/Real Analysis}} === Complex Analysis === {{:Definition:Deleted Neighborhood/Complex Analysis}} === Metric Space === {{:Definition:Deleted Neighborhood/Metric Space}} === Normed Vector Space === {{:Definition:Deleted Neighborhood/Normed Vector Space}} === Topology === {{:Definition:Deleted Neighborhood/Topology}}


Definition:Depressed Polynomial

Let $f \left({x}\right)$ be a polynomial over a field $k$: :$f \left({x}\right) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0$ If $a_{n-1} = 0_k$, then we call $f$ a '''depressed polynomial'''. It has been suggested that a polynomial with further zero terms might be referred to as "downright despondent", though this convention has yet to gain widespread usage by the community. {{refactor|Put the below into its own page}}


Definition:Cubic Equation

A '''cubic equation''' is a polynomial equation of the form: : $a x^3 + b x^2 + c x + d = 0$


Definition:Distinct

=== Of Two or More Objects === {{:Definition:Distinct/Plural}} === Of a Single Object === {{:Definition:Distinct/Singular}}


Definition:Quartic Equation

A '''quartic equation''' is a polynomial equation of the form: :$a x^4 + b x^3 + c x^2 + d x + e$


Definition:Quintic Equation

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


Definition:Mersenne Prime

A '''Mersenne prime''' is a Mersenne number which happens to be prime. That is, it is a prime number of the form $2^p - 1$. The number $2^p - 1$ is, in this context, often denoted $M_p$.


Definition:Net (Metric Space)

Let $M = \struct {A, d}$ be a metric space. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. An '''$\epsilon$-net for $M$''' is a subset $S \subseteq A$ such that: :$\displaystyle A \subseteq \bigcup_{x \mathop \in S} \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.


Definition:Lebesgue Number

Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal U$ be an open cover of $M$. A fixed strictly positive real number $\epsilon \in \R_{>0}$ is called a '''Lebesgue number for $\mathcal U$''' {{iff}}: :$\forall x \in A: \exists U \left({x}\right) \in \mathcal U: B_\epsilon \left({x}\right) \subseteq U \left({x}\right)$ where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$ in $M$.


Definition:Circular Definition

A '''circular definition''' is a definition of a concept $A$ in terms of another concept $B$ such that the definition of $B$ itself refers (either directly or indirectly) to the definition of $A$. It is a truism that '''circular definitions''' are always to be avoided.


Definition:Basic Null Sequence

The sequences defined as: :$\sequence {\dfrac 1 {n^r} }_{n \mathop \in \N}$ for $r \in \R: r > 0$ :$\sequence {\alpha^n}_{n \mathop \in \N}$ for $\alpha \in \C: \cmod {\alpha} < 1$ are known as the '''basic null sequences'''.


Definition:Dot Product

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions: :$\mathbf a = \displaystyle \sum_{k \mathop = 1}^n a_k \mathbf e_k$ :$\mathbf b = \displaystyle \sum_{k \mathop = 1}^n b_k \mathbf e_k$ where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.


Definition:Legendre Symbol

Let $p$ be an odd prime. Let $a \in \Z$ be an integer.


Definition:Diophantine Equation

A '''Diophantine equation''' is an indeterminate polynomial equation that allows the variables to take integer values only.


Definition:Perfect Number

=== Definition 1 === {{:Definition:Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Perfect Number/Definition 3}} === Definition 4 === {{:Definition:Perfect Number/Definition 4}}


Definition:Fermat Prime

A '''Fermat prime''' is a Fermat number, that is a number of the form $2^{\paren {2^n} } + 1$, which happens to be prime.


Definition:Mersenne Number

A '''Mersenne number''' is a natural number of the form $2^p - 1$, where $p$ is prime. The number $2^p - 1$, in this context, can be denoted $M_p$.


Definition:Carmichael Number

An integer $n > 0$ is a '''Carmichael number''' {{iff}}: : $(1): \quad n$ is composite : $(2): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$, or, equivalently, that $a^{n - 1} \equiv 1 \pmod n$. That is, a '''Carmichael number''' is a composite number $n$ which satisfies $a^n \equiv a \pmod n$ for all integers $a$ which are coprime to it.


Definition:Divisor Counting Function

Let $n$ be an integer such that $n \ge 1$. The '''divisor counting function''' is defined on $n$ as being the total number of positive integer divisors of $n$. It is denoted on {{ProofWiki}} as $\tau$ (the Greek letter '''tau'''). That is: :$\displaystyle \map \tau n = \sum_{d \mathop \divides n} 1$ where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.


Definition:Sum Over Divisors

Let $n$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be a mapping on the positive integers. Let $d \divides n$ denote that $d$ is a divisor of $n$. Then the '''sum of $\map f d$ over all the divisors of $n$''' is denoted: :$\displaystyle \sum_{d \mathop \divides n} \map f d$ Thus, for example: :$\displaystyle \sum_{d \mathop \divides 10} \map f d = \map f 1 + \map f 2 + \map f 5 + \map f {10}$ Category:Definitions/Number Theory iatkl4ppzsm04lgginpa6k7bligu0hl


Definition:Sigma Function

Let $n$ be an integer such that $n \ge 1$. The '''sigma function''' $\map \sigma n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$. That is: :$\displaystyle \map \sigma n = \sum_{d \mathop \divides n} d$ where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.


Definition:Abundancy Index

Let $n$ be a positive integer. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Then the '''abundancy index''' of $n$ is defined as $\dfrac {\map \sigma n} n$.


Definition:Abundance

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $\sigma \left({n}\right)$ be the sigma function of $n$. That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$. Then the '''abundance''' of $n$ is defined as $A \left({n}\right) = \sigma \left({n}\right) - 2 n$.


Definition:Multiplicative Order of Integer

Let $a$ and $n$ be integers. Let there exist a positive integer $c$ such that: :$a^c \equiv 1 \pmod n$ Then the least such integer is called '''order of $a$ modulo $n$'''.


Definition:Euler Phi Function

Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Euler $\phi$ (phi) function''' is the arithmetic function $\phi: \Z_{>0} \to \Z_{>0}$ defined as: :$\map \phi n = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$ That is: :$\map \phi n = \card {S_n}: S_n = \set {k: 1 \le k \le n, k \perp n}$


Definition:Reduced Residue System/Least Positive

The '''least positive reduced residue system modulo $m$''' is the set of integers: :$\set {a_1, a_2, \ldots, a_{\map \phi m} }$ with the following properties: :$\map \phi m$ is the Euler $\phi$ function :$\forall i: 0 < a_i < m$ :each of which is prime to $m$ :no two of which are congruent modulo $m$.


Definition:Divisor Function

The '''divisor function''': :$\displaystyle \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$ (meaning the summation is taken over all $m \le n$ such that $m$ divides $n$).


Definition:Von Mangoldt Function

The '''von Mangoldt function''' $\Lambda: \N \to \R$ is defined as: :$\Lambda \left({n}\right) = \begin{cases} \ln \left({p}\right) & : \exists m \in \N, p \in \mathbb P: n = p^m \\ 0 & : \text{otherwise} \end{cases}$ where $\mathbb P$ is the set of all prime numbers.


Definition:Knot (Knot Theory)

Let $Y$ be a manifold and $X \subset Y$ a submanifold of $Y$. Let $i: X \to Y$ be an inclusion, that is a mapping such that $i \sqbrk X = X$. Then a '''knotted embedding''' is an embedding $\phi: X \to Y$ (or the image of such an embedding) such that $\phi \sqbrk X$ is not freely homotopic to $i \sqbrk X$.


Definition:Reidemeister Move

The '''Reidemeister moves''' are one of three operations on knots which encapsulate the intuitive notion of '''equivalent knots'''.   The three moves are shown below. :File:Reidemeister move 1.png  :File:Reidemeister move 2.png  :File:Reidemeister move 3.png {{namedfor|Kurt Reidemeister}} {{SUBPAGENAME}} 768qxtheox2rkueocc9cn88uv7mu0ff


Definition:Alexander Polynomial

For a knot $K$ with Seifert matrix $V$, the '''Alexander polynomial''' of $K$ is defined as: :$\Delta_K \left({t}\right) = \det \left({V - t V^T}\right)$ {{NamedforDef|James Waddell Alexander II|cat=Alexander}} Category:Definitions/Knot Theory 46i64iuexv28nredfqs7sslyuvx0ksf


Definition:Seifert Matrix

For a knot $K$ with Seifert surface $S$, the '''Seifert matrix''' $V$ of $K$ is defined by its entries as: :$v_{ij} = \operatorname{lk} \left({ x_i, x_k^* }\right)$ where: : the $x_a$ are the generators of the fundamental group $\pi_1(S)$ : $x_a^*$ is the positive push-off of $x_a$ : $\operatorname{lk}$ is the linking number of the two loops. {{NamedforDef|Karl Johannes Herbert Seifert}} Category:Definitions/Knot Theory d1dkw982gw80dm2m2egtdme1ebvceii


Definition:Completely Multiplicative Function

Let $K$ be a field. Let $f: K \to K$ be a function on $K$. Then $f$ is described as '''completely multiplicative''' {{iff}}: :$\forall m, n \in K: \map f {m n} = \map f m \map f n$


Definition:Embedding (Topology)

Let $A, B$ be topological spaces. Let $f: A \to B$ be a mapping. Let the image of $f$ be given the subspace topology. Let the restriction $f {\restriction_{A \times f \left({A}\right)}}$ of $f$ to its image be a homeomorphism. Then $f$ is an '''embedding''' (of $A$ into $B$).


Definition:Primitive Root

Let $a, n \in \Z_{>0}$, that is, let $a$ and $n$ be strictly positive integers. Let the multiplicative order of $a$ modulo $n$ be $\phi \left({n}\right)$, where $\phi \left({n}\right)$ is the Euler phi function of $n$. Then $a$ is a '''primitive root of $n$''' or a '''primitive root modulo $n$'''.


Definition:Square-Free Integer

Let $n \in \Z$. Then $n$ is '''square-free''' {{iff}} $n$ has no divisor which is the square of a prime. That is, {{iff}} the prime decomposition $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$ is such that: :$\forall i: 1 \le i \le r: k_i = 1$


Definition:Quadratic Residue

Let $p$ be an odd prime. Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$. Then $a$ is a '''quadratic residue of $p$''' {{iff}} $x^2 \equiv a \pmod p$ has a solution. That is, {{iff}}: :$\exists x \in \Z: x^2 \equiv a \pmod p$


Definition:Set of Residue Classes/Least Positive

Let $\eqclass a m$ be the residue class of $a$ (modulo $m$). Let $r$ be the smallest non-negative integer in $\eqclass a m$. Then from Integer is Congruent to Integer less than Modulus: :$0 \le r < m$ and: :$a \equiv r \pmod m$ Then $r$ is called the '''least positive residue''' of $a \pmod m$.


Definition:Lowest Common Multiple

=== Integral Domain === {{:Definition:Lowest Common Multiple/Integral Domain}} === Integers === {{:Definition:Lowest Common Multiple/Integers}}


Definition:Greatest Common Divisor

=== Integral Domain === {{:Definition:Greatest Common Divisor/Integral Domain}} === Integers === When the integral domain in question is the integers $\Z$, the GCD is often defined differently, as follows: {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Polynomial Ring over Field === {{:Definition:Greatest Common Divisor/Polynomial Ring over Field}}


Definition:Prime-Counting Function

The '''prime-counting function''' is the function $\pi: \R \to \Z$ which counts the number of primes less than or equal to some real number. That is: :$\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$ where $\mathbb P$ denotes the set of prime numbers.


Definition:Primorial

There are two definitions for '''primorials''': one for primes and one for positive integers. === Definition for Primes === {{:Definition:Primorial/Prime}} === Definition for Positive Integers === {{:Definition:Primorial/Positive Integer}}


Definition:Euclid Number

A '''Euclid number''' is a natural number of the form: :$E_n := p_n\# + 1$ where $p_n\#$ is the primorial of the $n$th prime number.


Definition:O Notation

'''$\OO$ notation''' is a type of order notation, typically used in computer science for comparing 'run-times' of algorithms, or in analysis for comparing growth rates between two growth functions.


Definition:Residue (Complex Analysis)

Let $f: \C \to \C$ be a complex function. Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$. Then by Existence of Laurent Series, there is a Laurent series: :$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$ such that the sum converges to $f$ in $U - \left\{{z_0}\right\}$.   The '''residue''' at a point $z = z_0$ of a function $f: \C \to \C$ is defined as $a_{-1}$ in that Laurent series. It is denoted $\operatorname{Res} \left({f, z_0}\right)$ or just $\operatorname{Res}\left({z_0}\right)$ when $f$ is understood. Category:Definitions/Complex Analysis 9wpg3ebiqofw56oz9g2cyuu4velr859


Definition:Laurent Series

Let $f: \C \to \C$ be a complex function. Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$. A '''Laurent series''' is a sum: :$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$ such that the sum converges to $f$ in $U \setminus \left\{{z_0}\right\}$. {{NamedforDef|Pierre Alphonse Laurent|cat = Laurent}}


Definition:Residue Class

Let $m \in \Z$. Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$: :$\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ We have that congruence modulo $m$ is an equivalence relation. So for any $m \in \Z$, we denote the equivalence class of any $a \in \Z$ by $\eqclass a m$, such that: {{begin-eqn}} {{eqn | l = \eqclass a m       | r = \set {x \in \Z: a \equiv x \pmod m}       | c =  }} {{eqn | r = \set {x \in \Z: \exists k \in \Z: x = a + k m}       | c =  }} {{eqn | r = \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}       | c =  }} {{end-eqn}} The equivalence class $\eqclass a m$ is called the '''residue class of $a$ (modulo $m$)'''.


Definition:Integral Polynomial

An '''integral polynomial''' is a polynomial over the ring of integers $\Z$. Category:Definitions/Polynomial Theory 9r6gum4pxtxzllljm37cc6hw64ps770


Definition:Polynomial Congruence

Let $P \left({x}\right)$ be an integral polynomial. Then the expression: :$P \left({x}\right) \equiv 0 \pmod n$ is known as a '''polynomial congruence'''. === Linear Congruence === {{:Definition:Linear Congruence}}


Definition:Linear Congruence

A '''linear congruence''' is a polynomial congruence of the form: :$a_0 + a_1 x \equiv 0 \pmod n$ That is, one where the degree of the integral polynomial is $1$. Such a congruence is frequently encountered in the equivalent form: :$a x \equiv b \pmod n$


Definition:Simultaneous Congruences

A '''system of simultaneous congruences''' is a set of polynomial congruences: :$\forall i \in \left[{1 \,.\,.\, r}\right]: P_i \left({x}\right) \equiv 0 \pmod {n_i}$ That is: {{begin-eqn}} {{eqn | l = P_1 \left({x}\right)       | r = 0       | rr= \pmod {n_1}       | c =  }} {{eqn | l = P_2 \left({x}\right)       | r = 0       | rr= \pmod {n_2}       | c =  }} {{eqn | o = \cdots       | c =  }} {{eqn | l = P_r \left({x}\right)       | r = 0       | rr= \pmod {n_r}       | c =  }} {{end-eqn}}


Definition:Jacobi Symbol

Let $m \in \Z$ be any integer and $n \in \Z$ be any odd integer such that $n \ge 3$. Let the prime decomposition of $n$ be: : $\displaystyle n = \prod_{i \mathop = 1}^r p_i^{k_i}$. Then the '''Jacobi symbol''' $\left({\dfrac m n}\right)$ is defined as: :$\displaystyle \left({\frac m n}\right) = \prod_{i \mathop = 1}^r \left({\frac m {p_i}}\right)^{k_i}$ where $\left({\dfrac m {p_i}}\right)$ is defined as the Legendre symbol.


Definition:Continued Fraction

{{:Definition:Continued Fraction/Finite}}


Definition:Exterior Derivative

Let an exact $n$-form $\omega$ be given on an $m$-manifold, with local coordinates $x_1, x_2, \dots, x_m$. Let a local coordinate expression for $\omega$ be given: :$\omega = \map f {x_1, \ldots, x_m} \rd x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \cdots \wedge \d x_{\map \phi n}$ where: :$\phi: \set {1, \ldots, n} \to \set {1, \ldots, m}$ is an injection which determines which coordinate vectors $\omega$ acts on. :$\wedge$ denotes the wedge product. The '''exterior derivative''' $\d \omega$ is the $\paren {n + 1}$-form defined as: :$\displaystyle \d \omega = \paren {\sum_{k \mathop = 1}^m \frac {\partial f} {\partial x_k} \rd x_k} \wedge \d x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \dots \wedge \d x_{\map \phi n}$ For inexact forms: :$\map \d {a + b} = \d a + \d b$


Definition:Wedge Product

Let $\alpha$ and $\beta$ be two differential forms. Let $\alpha$ be an $x$-form. Let $\beta$ be a $y$-form. The '''wedge product''' $\alpha \wedge \beta$ is defined as the linear antisymmetric map from $F^x \times F^y \to F^{x + y}$, where $F^a$ is the set of $a$-forms in some manifold. Let $x_0$ be a specific point in a manifold $X$. Let $\alpha$ be an $x$-form. Let $\phi$ be a $1$-form. Let there be a set of vectors $\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v \in \map {T_{x_0} } X$. The '''wedge product''' is defined as: :$\alpha \wedge \map \phi {\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v} := \sum_P \map \varepsilon P \, \map \alpha {P_1} \, \map \phi {P_2}$ where: :$P$ is some permutation of $\mathbf u_1, \mathbf u_2, \dots, \mathbf u_x, \mathbf v$ :$P_1$ is the first $x$ terms of the permutation $P$ :$P_2$ the final term of permutation $P$ :$\varepsilon$ is the permutation symbol of $P$. The sum is taken over all possible permutations. This definition extends to wedge products of arbitrary forms through the linearity and antisymmetric conditions. Category:Definitions/Topology 0iso3pdnb5hi2zud1lsrw02j5zjl4la


Definition:Permutation Symbol

The '''permutation symbol''' $\varepsilon$ of a permutation $P$ of a set of elements is defined as: :$+1$ for even permutations (permutations that are an even number of pair swaps) :$-1$ for odd permutations :$0$ if the list of elements is not a permutation (that is, contains a repeated value). Frequently, the permutation will be explicit, for example: :$\varepsilon_{i j k \ldots}$ :$\varepsilon^{i j \ldots}_{k l \ldots}$ :$\varepsilon^{i j k \ldots}$ This notation is especially useful when raising and lowering indices (that is, converting between forms and vectors).


Definition:Local Coordinates

Let $X$ be an $n$-dimensional manifold. Let $p \in X$, and let $U \subset X$ be a neighbourhood of $p$. Then a set of mappings $x_i: U \to \R$, $1 \le i \le n$, satisfying: :$a = b \iff \forall i: \map {x_i} a = \map {x_i} b$ is called a set of '''local coordinates'''. When the neighbourhood $U$ is to be stressed, one may also say '''local coordinates for $U$'''. Similarly, when the element $p$ is to be stressed, one may also say '''local coordinates around $p$'''.


Definition:Partial Derivative

=== Real Analysis === {{:Definition:Partial Derivative/Real Analysis}} === Complex Analysis === {{:Definition:Partial Derivative/Complex Analysis}}


Definition:Riemann Surface

A '''Riemann surface''' is a connected complex manifold of dimension $1$.


Definition:Riemann Sphere

Let $f_1: \C \to \R^2$ be defined as: :$\forall z \in \C: \map {f_1} z = \tuple {\map \Re z, \map \Im z}$ Let $f_2: \R^2 \to \R^3$ be the inclusion map: :$\forall \tuple {a, b} \in \C^2: \map {f_2} {a, b} = \tuple {a, b, 0}$ Let $f = f_2 \circ f_1$. Let $F: \C \to \map {\PP} {\R^3}$ be defined as the mapping which takes $z$ to the closed line interval from $\tuple {0, 0, 1}$ to $\map f z$ for all $z \in \C$. Let $G = \set {x, y, z: x^2 + y^2 + z^2 = 1}$. Then the Riemann map $R: \C \to \mathbb S^2$ is defined as: :$\map R x = \map F z \cap G$ The set $R \sqbrk \C \cup \set {\tuple {0, 0, 1} } $ is called the '''Riemann sphere''', with the understanding that $\map f \infty = \tuple {0, 0, 1}$. {{expand|According to the definition in Clapham & Nicholson, include the definition as the extended complex plane under a stereographic projection}}


Definition:Measure (Measure Theory)

Let $\struct {X, \Sigma}$ be a measurable space. Let $\mu: \Sigma \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers. Then $\mu$ is called a '''measure''' on $\Sigma$ {{iff}} $\mu$ has the following properties: {{begin-axiom}} {{axiom | n = 1         | q = \forall E \in \Sigma         | ml= \map \mu E         | mo= \ge         | mr= 0 }} {{axiom | n = 2         | q = \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O         | ml= \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}         | mo= =         | mr= \sum_{n \mathop = 1}^\infty \map \mu {S_n}         | rc= that is, $\mu$ is a countably additive function }} {{axiom | n = 3         | q = \exists E \in \Sigma         | ml= \map \mu E         | mo= \in         | mr= \R         | rc= that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite }} {{end-axiom}}


Definition:Periodic Continued Fraction

Let $\left[{a_1, a_2, a_3, \ldots}\right]$ be a simple infinite continued fraction. Let the partial quotients be of the form: :$\left[{r_1, r_2, \ldots, r_m, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, \ldots}\right]$ that is, ending in a block of partial quotients which repeats itself indefinitely. Such a SICF is known as a '''periodic continued fraction'''. The notation used for this is $\left[{r_1, r_2, \ldots, r_m, \left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$, where the repeating block is placed in angle brackets. === Purely Periodic Continued Fraction === {{:Definition:Periodic Continued Fraction/Purely Periodic}} === Cycle === {{:Definition:Periodic Continued Fraction/Cycle}} Category:Definitions/Continued Fractions d8e3wu8mzkg2wgroy2qfbtovdmhlsus


Definition:Fractional Part

Let $x \in \R$ be a real number. Let $\floor x$ be the floor function of $x$. The '''fractional part''' of $x$ is the difference: :$\fractpart x := x - \floor x$


Definition:Analytic Continuation

Let $U \subset \C$ be an open set. Let $f: U \to \C$ be an analytic function. Let $V$ be an open subset of $\C$ such that $U \subset V$. An '''analytic continuation of $f$ to $V$''' is an analytic function $F: V \to \C$ such that $\map F z = \map f z$ for $z \in U$.


Definition:Gamma Function

=== Integral Form === {{:Definition:Gamma Function/Integral Form}} === Weierstrass Form === {{:Definition:Gamma Function/Weierstrass Form}} === Hankel Form === {{:Definition:Gamma Function/Hankel Form}} === Euler Form === {{:Definition:Gamma Function/Euler Form}}


=== Complex Numbers === Let $\sequence {a_n}$ be a sequence in $\C$. {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === General Definition === Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\sequence {a_n}$ be a sequence in $\mathbb K$. {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}}


Definition:Fourier Series

=== Formulation 1 === {{:Definition:Fourier Series/Formulation 1}} === Formulation 2 === {{:Definition:Fourier Series/Formulation 2}}


Definition:Quadratic Irrational

A '''quadratic irrational''' is an irrational number of the form: :$r + s \sqrt n$ where $r, s$ are rational and $n$ is a positive integer which is not a square.


Definition:Pythagorean Triple

A '''Pythagorean triple''' is an ordered triple of positive integers $\tuple {x, y, z}$ such that $x^2 + y^2 = z^2$. That is, a '''Pythagorean triple''' is a solution to the Pythagorean equation.


Definition:Projective Space

=== Real Projective Space === {{:Definition:Projective Space/Real Projective Space}} === Over a Field === {{:Definition:Projective Space/Over a Field}} Category:Definitions/Topology Category:Definitions/Projective Geometry t7mbpaqdvrj4t5lg8ftdxiyrain9yiz


Definition:Dirichlet Series

Let $a_n: \N \to \C$ be an arithmetic function. Its '''Dirichlet series''' is a complex function $f: \C \to \C$ defined by a series: :$\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$


Definition:Pythagorean Equation

The '''Pythagorean equation''' is the Diophantine equation: :$x^2 + y^2 = z^2$ where $x, y, z$ are integers such that $x, y, z > 0$. Solutions of this equation are known as Pythagorean triples. If, in addition, $\left({x, y, z}\right)$ is a primitive Pythagorean triple, then $\left({x, y, z}\right)$ is known as a '''primitive solution''' of $x^2 + y^2 = z^2$.


Definition:Triangular Number

'''Triangular numbers''' are those denumerating a collection of objects which can be arranged in the form of an equilateral triangle.


Definition:Geometric Sequence

A '''geometric sequence''' is a sequence $\sequence {x_n}$ in $\R$ defined as: :$x_n = a r^n$ for $n = 0, 1, 2, 3, \ldots$ Thus its general form is: :$a, ar, ar^2, ar^3, \ldots$ and the general term can be defined recursively as: :$x_n = \begin{cases} a & : n = 0 \\ r x_n & : n > 0 \\ \end{cases}$


Definition:Square Number

'''Square numbers''' are those denumerating a collection of objects which can be arranged in the form of a square. They can be denoted: :$S_1, S_2, S_3, \ldots$ === Definition 1 === {{:Definition:Square Number/Definition 1}} === Definition 2 === {{:Definition:Square Number/Definition 2}} === Definition 3 === {{:Definition:Square Number/Definition 3}} === Definition 4 === {{:Definition:Square Number/Definition 4}}


Definition:Arithmetic Sequence

An '''arithmetic sequence''' is a finite sequence $\sequence {a_k}$ in $\R$ or $\C$ defined as: :$a_k = a_0 + k d$ for $k = 0, 1, 2, \ldots, n - 1$ Thus its general form is: :$a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \paren {n - 1} d$


Definition:Polygonal Number

'''Polygonal numbers''' are those denumerating a collection of objects which can be arranged in the form of an regular polygon.


Definition:Pentagonal Number

'''Pentagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular pentagon.


Definition:Hexagonal Number

'''Hexagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular hexagon.


Definition:Pyramidal Number

The '''pyramidal numbers''' are integers defined as sums of sequences of polygonal numbers as follows. Let $P \left({k, n}\right)$ be the $n$th $k$-gonal number. Then the '''$n$th $k$gonal-based pyramidal number''' $Y \left({k, n}\right)$ is defined as: :$\displaystyle Y \left({k, n}\right) = \sum_{i \mathop = 1}^n P \left({k, i}\right)$


Definition:Set of Residue Classes/Least Absolute

Let $\eqclass a m$ be the residue class of $a$ (modulo $m$). Except when $r = \dfrac m 2$, we can choose $r$ to be the integer in $\eqclass a m$ which has the smallest absolute value. In that exceptional case we have: : $-\dfrac m 2 + m = \dfrac m 2$ and so: : $-\dfrac m 2 \equiv \dfrac m 2 \pmod m$ Thus $r$ is defined as the '''least absolute residue''' of $a$ (modulo $m$) {{iff}}: : $-\dfrac m 2 < r \le \dfrac m 2$


Definition:Balanced Incomplete Block Design

A '''Balanced Incomplete Block Design''' or '''BIBD''' with parameters $v, b, r, k, \lambda$ is a block design where: * $v$ is the number of points in the design * $b$ is the number of blocks * $k$ is the size of each block * $r$ is the number of blocks any point can be in * $\lambda$ is the number of times any two points can occur in the same block and has the following properties: * Each block is of size $k$ * All of the $\displaystyle \binom v 2$ pairs occur together in exactly $\lambda$ blocks. A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example: * $\operatorname{BIBD} \left({v, k, \lambda}\right)$ * $\left ({v, k, \lambda}\right)$-$\operatorname{BIBD}$


Definition:Odd Prime

Every even integer is divisible by $2$ (because this is the definition of '''even'''). Therefore, apart from $2$ itself, all primes are odd. So, referring to an '''odd prime''' is a convenient way of specifying that a number is a prime number, but not equal to $2$.


Definition:Combination

Let $S$ be a set containing $n$ elements. An '''$r$-combination of $S$''' is a subset of $S$ which has $r$ elements.


Definition:Generating Function

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


Definition:Fermat Pseudoprime

Let $q$ be a composite number such that $\exists n \in N: n^q \equiv n \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $n$'''.


Definition:Algorithm

An '''algorithm''' is a finite set of '''instructions''' (or '''rules''') that defines a sequence of '''operations''' for solving a particular '''computational problem''' for all '''problem instances''' for some '''problem set'''.


Definition:Algorism

'''Algorism''' is an archaic term which means '''the process of doing arithmetic using Arabic numerals'''. === Algorist === {{:Definition:Algorism/Algorist}}


Definition:Abacus

An '''abacus''' (plural: '''abacuses''' or '''abaci''') is a tool for performing arithmetical calculations. It consists of: : a series of lines (for example: grooves in sand, or wires on a frame), upon which are: : a number of items (for example: pebbles in the grooves, or beads on the wires), which are manipulated by hand so as to represent numbers. As such, it is the earliest known machine for mathematics, and can be regarded as the earliest ancestor of the electronic computer.


Definition:Abacism

'''Abacism''' means '''the process of doing arithmetic using an abacus'''. === Abacist === {{:Definition:Abacism/Abacist}}


Definition:Arabic Numerals

The '''Arabic numerals''' are: {{begin-eqn}} {{eqn | l = 1:       | o = \)'''one'''\(:       | r = \circ       | c =  }} {{eqn | l = 2:       | o = \)'''two'''\(:       | r = \circ \, \circ       | c =  }} {{eqn | l = 3:       | o = \)'''three'''\(:       | r = \circ \circ \circ       | c =  }} {{eqn | l = 4:       | o = \)'''four'''\(:       | r = \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 5:       | o = \)'''five'''\(:       | r = \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 6:       | o = \)'''six'''\(:       | r = \circ \circ \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 7:       | o = \)'''seven'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 8:       | o = \)'''eight'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 9:       | o = \)'''nine'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 0:       | o = \)'''zero'''\(:       | r = \)or '''nought''', '''naught''', '''nil''', '''nothing''', '''cypher''', '''zilch''', '''kabutnik''', '''sweet Fanny Adams''', etc.\(       | c =  }} {{end-eqn}} They are used in conjunction with the positional decimal system of numeric representation: see Basis Representation Theorem.


Definition:Roman Numerals

The '''Roman numerals''' are: {{begin-eqn}} {{eqn | l = \mathrm I:       | o = \)'''unus'''\(:       | r = 1 }} {{eqn | l = \mathrm V:       | o = \)'''quinque'''\(:       | r = 5 }} {{eqn | l = \mathrm X:       | o = \)'''decem'''\(:       | r = 10 }} {{eqn | l = \mathrm L:       | o = \)'''quinquaginta'''\(:       | r = 50 }} {{eqn | l = \mathrm C:       | o = \)'''centum'''\(:       | r = 100 }} {{eqn | l = \mathrm D:       | o = \)'''quingenti'''\(:       | r = 500 }} {{eqn | l = \mathrm M:       | o = \)'''mille'''\(:       | r = 1000 }} {{end-eqn}}


Definition:Unlimited Register Machine

An '''unlimited register machine''', abbreviated URM, is an abstraction (or idealization) of a computing device with the following characteristics: === Registers === A '''URM''' has a number of locations called '''registers''' which can store natural numbers: $\set {0, 1, 2, \ldots}$. Any given URM program may make use of only a finite number of registers. Registers are usually referred to by the subscripted uppercase letters $R_1, R_2, R_3, \ldots$. The subscript (which is a natural number) is called the '''index''' of the register. The number held at any one time by a register is usually referred to by the corresponding lowercase letter $r_1, r_2, r_3, \ldots$. The registers are '''unlimited''' in the following two senses: :$(1): \quad$ Although a URM program may make use of only a finite number of registers, there is no actual upper bound on how many a particular program ''can'' actually use. :$(2): \quad$ There is no upper bound on the size of the natural numbers that may be stored in any register. === Program === The numbers held in the registers of a URM are manipulated according to a '''program'''. A '''URM program''' is a finite list of basic instructions. The instructions are written in a fixed order and numbered $1, 2, 3, \ldots$. For historical reasons, the number of the instruction is called its '''line''' number. We can refer either to the '''line of the program''' or the '''line in the URM'''. It is convenient to use $\Bbb U$ to stand for the set of all URM programs. ==== Length of Program ==== Let $P \in \Bbb U$ be a URM program. By definition, $P$ is a finite list of basic instructions. We define the function $\lambda: \Bbb U \to \N$ as follows: :$\forall P \in \Bbb U: \map \lambda P = \text { the number of basic instructions that comprise } P$. ==== Number of Registers Used ==== Let $P \in \Bbb U$ be a URM program. By definition, $P$ uses a finite number of registers. We define the function $\rho: \Bbb U \to \N$ as follows: :$\forall P \in \Bbb U: \map \rho P = \text{ the highest register number used by } P$. So in any URM program $P$, no instruction refers to any register with index greater than $R_u$, where $u = \map \rho P$. === Basic Instructions === {| border="1" |- ! align="left" | Name  ! align="left" | Notation ! align="left" | Effect ! align="left" | Description |- | <tt>Zero</tt>  | $\map Z n$ | $0 \to R_n$ | Replace the number in $R_n$ by $0$. |- | <tt>Successor</tt>  | $\map S n$ | $r_n + 1 \to R_n$ | Add $1$ to the number in $R_n$. |- | <tt>Copy</tt>  | $\map C {m, n}$ | $r_m \to R_n$ | Replace the number in $R_n$ by the number in $R_m$ (leaving the one in $R_m$ as it was). |- | <tt>Jump</tt>  | $\map J {m, n, q}$ | $r_m = r_n ? \Rightarrow q$ | If the numbers in $R_m$ and $R_n$ are equal, go to instruction number $q$, otherwise go to the next instruction. |} Basic instructions are also (and more commonly) known as '''commands''', because the word's shorter and quicker to say. === Operation === When a URM runs a program, it always starts by executing the first instruction of the program. When it has carried out an instruction, it moves to the next instruction and executes that one, unless required otherwise by a <tt>Jump</tt> instruction. ==== Instruction Pointer ==== The line number which is currently about to be executed is known as the '''instruction pointer'''. It can be imagined as a special-purpose register in the URM that holds the line number. ==== Stage of Computation ==== The '''stage of computation''' (or just '''stage''') of a URM program is the count of how many basic instructions have been carried out. Thus each stage corresponds to the processing of one instruction. ==== State ==== The '''state''' (or '''situation''') of a URM program at a particular point in time is defined as: :the value of the instruction pointer :the value, at that point, of each of the registers that are used by the program. === Termination === A URM program '''stops''', or '''terminates''', or '''halts''', when there are no more instructions to carry out. This can happen in either of two ways: :$(1): \quad$ If the program carries out the last instruction, and this does not involve a <tt>Jump</tt> to an earlier instruction, the program will stop. :$(2): \quad$ If the program carries out a <tt>Jump</tt> instruction to a non-existent instruction, the program will stop. Such a <tt>Jump</tt> instruction is known as an '''exit jump''' . The line on which a particular run of a URM program stops is called the '''exit line'''. If the program, when running, never reaches such a state, then it is said to be in an '''endless loop''' and will '''never terminate'''. Note that whether a program terminates or not may depend on its input. It may terminate perfectly well for one input, but go into an endless loop on another. === Input === The '''input''' to a URM program is: :either an ordered $k$-tuple $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$ :or a natural number $n \in \N$. In the latter case, it is convenient to consider a single natural number as an ordered $1$-tuple $\tuple {n_1} \in \N^1 = \N$. Hence we can discuss inputs to URM programs solely as instances of tuples, and not be concerned with cumbersome repetition for the cases where $k = 1$ and otherwise. The convention usually used is for a URM program $P$ to start computation with: :the input $\left({n_1, n_2, \ldots, n_k}\right)$ in registers $R_1, R_2, \ldots, R_k$ :$0$ in all other registers used by $P$. That is, the initial state of the URM is: :$\forall i \in \closedint 1 k: r_i = n_i$ :$\forall i > k: r_i = 0$. It is usual for the input (either all or part) to be overwritten during the course of the operation of a program. That is, at the end of a program, $R_1, R_2, \ldots, R_k$ are not guaranteed still to contain $n_1, n_2, \ldots, n_k$ unless the program has been explicitly written so as to ensure that this is the case. === Output === At the end of the running of a URM program, the '''output''' will be found in register $R_1$. === Null Program === A '''null program''' or '''empty program''' is a URM program which contains no instructions.


Definition:URM Computability

Let $P$ be a URM program, and let $k$ be any positive integer. === Program === $P$ is said to '''compute the function $f: \N^k \to \N$''' {{iff}}: :for all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$, the computation of a URM using the program $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ produces the output $\map f {n_1, n_2, \ldots, n_k}$. If there are any inputs such that either of the following happens: :the output fails to equal $\map f {n_1, n_2, \ldots, n_k}$ :the program will never terminate, then the program does ''not'' compute the function $f: \N^k \to \N$. === Function === The function $f: \N^k \to \N$ is said to be '''URM computable''' if there exists a URM program which computes it. ==== Partial Function ==== $P$ is said to '''compute the partial function $f: \N^k \to \N$''' {{iff}}: :For all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$: ::If the computation of $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ '''halts''', it produces the output $\map f {n_1, n_2, \ldots, n_k}$. ::If the computation of $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ '''does not halt''', $\map f {n_1, n_2, \ldots, n_k}$ is undefined. The partial function $f: \N^k \to \N$ is said to be '''URM computable''' if there exists a URM program which computes it. Note that a URM program can be used with any number of input variables. For any positive integer $k$, the input consists of the state of the registers $R_1, R_2, \ldots, R_k$. Thus a given URM program $P$ computes a partial function $f: \N^k \to \N$ for ''each'' positive integer $k$. In this context, it is convenient to use the notation $f^k_P$ to denote the partial function of $k$ variables computed by $P$. === Set === Let $A \subseteq \N$. Then $A$ is a '''URM computable set''' {{iff}} its characteristic function $\chi_A$ is a URM computable function. === Relation === Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''URM computable relation''' {{iff}} its characteristic function $\chi_\mathcal R$ is a URM computable function. Category:Definitions/Mathematical Logic 76rt3tca4y9ub6otxrsed9y89sdhumh


Definition:Indeterminate Variable

An '''indeterminate variable''' is a variable whose  value has not been specified. Informally, it "could be anything". Category:Definitions/Algebra 8l2sa3neea2rgq1qtpuzgbspyqe9oea


Definition:Turing Machine

A '''Turing machine''' is an idealization of a computing machine. The idea goes as follows. A '''Turing machine''' works by manipulating symbols on an imaginary piece of paper by means of a specific set of algorithmic rules.


Definition:Substitution (Mathematical Logic)

=== Mapping === Let $S$ be a set. Let $f: S^t \to S$ be a mapping. Let $\left\{{g_1: S^k \to S, g_2: S^k \to S, \ldots, g_t: S^k \to S}\right\}$ be a set of mappings. Let the mapping $h: S^k \to S$ be defined as: :$h \left({s_1, s_2, \ldots, s_k}\right) = f \left({g_1 \left({s_1, s_2, \ldots, s_k}\right), g_2 \left({s_1, s_2, \ldots, s_k}\right), \ldots, g_t \left({s_1, s_2, \ldots, s_k}\right)}\right)$ Then $h$ is said to be '''obtained from $f, g_1, g_2, \ldots, g_k$ by substitution'''. The definition can be generalized in the following ways: * It can apply to mappings which operate on variously different sets. * Each of $g_1, g_2, \ldots, g_t$ may have different arities. If $g$ is a mapping of $m$ variables where $m > k$, we can always consider it a mapping of $k$ variables in which the additional variables play no part. So if $g_i$ is a mapping of $k_i$ variables, we can take $k = \max \left\{{k_i: i = 1, 2, \ldots, t}\right\}$ and then each $g_i$ is then a mapping of $k$ variables. === Partial Function === Let $f: \N^t \to \N$ be a partial function. Let $\left\{{g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N}\right\}$ be a set of partial functions. Let the partial function $h: \N^k \to \N$ be defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) \approx f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$ where $\approx$ is as defined in Partial Function Equality. Then $h$ is said to be '''obtained from $f, g_1, g_2, \ldots, g_k$ by substitution'''. Note that $h \left({n_1, n_2, \ldots, n_k}\right)$ is defined only when: * All of $g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)$ are defined * $f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$ is defined. {{SUBPAGENAME}} 902scgeuilpski6x8f3xrrjypbjez3o


Definition:Legendre's Constant

'''Legendre's constant''' (or the '''Legendre constant''') is a mathematical constant conjectured by {{AuthorRef|Adrien-Marie Legendre}} to specify the prime-counting function $\map \pi n$. Legendre conjectured in $1796$ that $\map \pi n$ satisfies: :$\displaystyle \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$ where $B$ is '''Legendre's constant'''. If such a number $B$ exists, then this implies the Prime Number Theorem. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s guess for $B$ was about $1 \cdotp 08366$. {{OEIS|A228211}} Later, {{AuthorRef|Carl Friedrich Gauss}} looked at this problem and thought that $B$ might actually be lower. In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s first guess of $1 \cdotp 08366 \ldots$ is still (incorrectly) referred to as '''Legendre's constant''', even though its "correct" value is in fact exactly $1$. Hence it is only now of historical importance. {{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}


Definition:Concatenation of URM Programs

Let $P$ and $Q$ be URM programs. The '''concatenation of $P$ and $Q$''' is denoted $P * Q$. It consists of program $P$ followed immediately by program $Q$. If $P$ and $Q$ compute particular functions, then it is usually necessary to make modifications to either $P$ or $Q$, or both, to make sure that the concatenated program continues to compute a function. The simplest concatenation is illustrated in Composition of One-Variable URM Computable Functions.


Definition:Primitive Recursion

=== Primitive Recursion on Several Variables === {{:Definition:Primitive Recursion/Several Variables}} === Primitive Recursion on One Variable === {{:Definition:Primitive Recursion/One Variable}} === Primitive Recursion on Partial Functions === {{:Definition:Primitive Recursion/Partial Function}} Category:Definitions/Mathematical Logic nuimqic7c09jqixl4gfoq47l6ub7fur


The '''basic primitive recursive functions''' are: === Zero Function === {{:Definition:Basic Primitive Recursive Function/Zero Function}} === Successor Function === {{:Definition:Basic Primitive Recursive Function/Successor Function}} === Projection Function === {{:Definition:Basic Primitive Recursive Function/Projection Function}} === Identity Function === {{:Definition:Basic Primitive Recursive Function/Identity Function}}


Definition:Primitive Recursive

=== Function === {{:Definition:Primitive Recursive/Function}} === Set === {{:Definition:Primitive Recursive/Set}} === Relation === {{:Definition:Primitive Recursive/Relation}} Category:Definitions/Mathematical Logic 4ff3875born2il0kz9vdw2xhm4au51x


Definition:Logarithmic Integral

The '''logarithmic integral''' is defined as: :$\displaystyle \map {\operatorname {li} } x = \PV_0^x \frac {\d t} {\map \ln t}$ where: :$\ln$ denotes the natural logarithm function :$\operatorname {PV}$ denotes the Cauchy principal value of the proceeding integral. {{mistake|"Proceeding" is wrong here. Suspected malapropism. Recommended a rewrite to make it more accurate.}} That is, as $\dfrac 1 {\ln t}$ has discontinuities at $t = 0$ and $t = 1$: :$\map {\operatorname {li} } x = \begin{cases}\displaystyle \lim_{\varepsilon \to 0^+} \paren {\int_{\varepsilon}^x \frac {\rd t} {\ln t} } & 0 < x < 1 \\ \displaystyle \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & x > 1\end{cases}$ === Eulerian Logarithmic Interval === {{:Definition:Logarithmic Integral/Eulerian}}


Definition:Big-Omega

'''Big-Omega notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Let $f, g$ be two functions. {{explain|Domain and range of $f$ and $g$ to be specified: reals?}} Then: : $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ iff: : $\exists c > 0, k \ge 0: \forall n > k: f \left({n}\right) \ge c g \left({n}\right)$ This is read as: :'''$f \left({n}\right)$ is big omega of $g \left({n}\right)$'''. {{refactor|Put the below into separate pages, either as a 2nd definition or just an equivalence proof}} Another method of determining the condition is the following limit: :$\displaystyle \lim_{n \to \infty} {\frac{f \left({n}\right)} {g \left({n}\right)}} = c > 0$ where $0 < c \le \infty$. If such a $c$ does exist, then: : $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ {{refactor|The below into its own page}} To say that $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ is equivalent to: : $g \left({n}\right) \in \mathcal O \left({f \left({n}\right)}\right)$ where $\mathcal O$ is the big-O notation.


Definition:Big-Theta

'''Big-Theta notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Big-Theta is a stronger statement than big-O and big-omega. Suppose $f: \N \to \R, g: \N \to \R$ are two functions. Then: :$f \left({n}\right) \in \Theta \left({g \left({n}\right)}\right)$  iff: :$(f \left({n}\right) \in O \left({g \left({n}\right)}\right) \land (f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ where $O \left({g \left({n}\right)}\right)$ is big-O and $\Omega \left({g \left({n}\right)}\right)$ is Big-Omega. This is read as: : $f \left({n}\right)$ is '''big-theta''' of $g \left({n}\right)$. Another method of determining the condition is the following limit: :$\displaystyle \lim_{n \to \infty} \frac{f \left({n}\right)}{g \left({n}\right)} = c$, where $0 < c < \infty$ If such a $c$ does exist, then $f \left({n}\right) \in \Theta (g \left({n}\right))$. Category:Definitions/Order Notation gq89r79059rqjllgejs4j80517jrrpx


Definition:Signum Function

Let $X \subseteq \R$ be a subset of the real numbers. The '''signum function''' $\sgn: X \to \set {-1, 0, 1}$ is defined as: :$\forall x \in X: \map \sgn x := \sqbrk {x > 0} - \sqbrk {x < 0}$ where $\sqbrk {x > 0}$ etc. denotes Iverson's convention. That is: :$\forall x \in X: \map \sgn x := \begin{cases} -1 & : x < 0 \\ 0 & : x = 0 \\ 1 & : x > 0 \end{cases}$


Definition:Hall Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a Hall subgroup of $G$ {{iff}} the index and order of $H$ in $G$ are coprime: :$\index G H \perp \order H$


Definition:Hall Divisor

Let $n \in \Z$ be an integer. A '''Hall divisor''' of $n$ is a divisor $d$ of $n$ such that $n$ and $\dfrac n d$ are coprime. {{NamedforDef|Philip Hall|cat=Hall}} Category:Definitions/Number Theory nz9dq89jb25nbrz7jpbr6i1lz2yh7y2


Definition:Fermat Number

A '''Fermat number''' is a natural number of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$. The number $2^{\paren {2^n} } + 1$ is, in this context, often denoted $F_n$.


Definition:Bounded Minimization

=== Function === Let $f: \N^{k+1} \to \N$ be a function. Let $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ and let $z \in \N$ be fixed. Then the '''bounded minimization operation on $f$''' is written as: :$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$ and is specified as follows: :$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right) = \begin{cases} \text{the smallest } y \in \N \text{ such that } f \left({n_1, n_2, \ldots, n_k, y}\right) = 0 & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$ === Relation === Let $\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) $ be a $k+1$-ary relation on $\N^{k+1}$. Let $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ and let $z \in \N$ be fixed. Then the '''bounded minimization operation on $\mathcal R$''' is written as: :$\mu y \le z \mathcal R\left({n_1, n_2, \ldots, n_k, y}\right)$ and is specified as follows: :$\mu y \le z \mathcal R\left({n_1, n_2, \ldots, n_k, y}\right) = \begin{cases} \text{the smallest } y \in \N \text{ for which } \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) \text{ holds} & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$ We can consider the definition for a function to be a special case of this. === The no-solution case === The choice of $z + 1$ for the value when there is no solution $y$ less than or equal to $z$ is arbitrary, but convenient. It ensures a well-defined solution for every $z$. {{SUBPAGENAME}} rc6242xnjliugkuy1a959k2vrh36zdn


Definition:Sequence Coding

Let $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$ be a finite sequence in $\N^*$ (that is, $\forall i \in \left\{{1, 2, \ldots, k}\right\}: a_i > 0$). Let $p_i$ be the $i$th prime number, so that: * $p_1 = 2$ * $p_2 = 3$ * $p_3 = 5$ * $p_4 = 7$ etc. Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_i$ is the $i$th prime number. Then $n \in \N$ '''codes the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$''', or $n$ is the '''code number for the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$'''. The set of all code numbers of finite sequences in $\N$ is denoted $\operatorname{Seq}$. Note that $n \in \operatorname{Seq}$ {{iff}} $n$ is divisible by all the primes $p_1, p_2, \ldots, p_k$ , where $p_k$ is the ''largest'' prime dividing $n$. Category:Definitions/Mathematical Logic 8nm3rv1b0qojnbcrur1aqlhaj5ey4pr


Definition:Length of Integer

Let $n \in \Z$ such that $n \ge 2$. The '''length of $n$''' is defined as the number of distinct prime factors of $n$ and is denoted $\operatorname{len} \left({n}\right)$. Category:Definitions/Number Theory bypkuv4z8a6bx5f0ieg97dy06q1s1ll


Definition:Prime Enumeration Function

Let the function $p: \N \to \N$ be defined as: {| | align="left" | $p(0) = 1$, |  |- | align="left" | $p(n) =$ the $n^\text{th}$ prime number, | align="left" | if $n > 0$ |} Thus for example: * $p \left({0}\right) = 1$ * $p \left({1}\right) = 2$ * $p \left({2}\right) = 3$ * $p \left({3}\right) = 5 \ldots$ This function is called the '''prime enumeration function'''. Note, of course, that although $p \left({0}\right) = 1$, there is no suggestion of treating $1$ as prime (it definitely isn't). {{SUBPAGENAME}} 5znwmmg9j7fhvw5u2zzwk6wi00uvbsc


Definition:Prime Exponent Function

Let $n \in \N$ be a natural number. Let the prime decomposition of $n$ be given as: :$\displaystyle n = \prod_{j \mathop = 1}^k \left({p \left({j}\right)}\right)^{a_j}$ where $p \left({j}\right)$ is the prime enumeration function. Then the exponent $a_j$ of $p \left({j}\right)$ in $n$ is denoted $\left({n}\right)_j$. If $p \left({j}\right)$ does not divide $n$, then $\left({n}\right)_j = 0$. We also define: : $\forall n \in \N: \left({n}\right)_0 = 0$ : $\forall j \in \N: \left({0}\right)_j = 0$ : $\forall j \in \N: \left({1}\right)_j = 0$ Category:Definitions/Mathematical Logic omf6e7p52tjekowf8q9ct16nkt68wv4


Definition:Partial Function

Let $S \subset \N^k$. Let $f: S \to \N$ be a function. Suppose that $\forall x \in \N^k \setminus S$, $f$ is undefined at $x$. Then $f$ is known as a '''partial function from $\N^k$ to $\N$'''. Thus we can specify a function that has values for some, but not all, elements of $\N$. It can be seen that the definition of a '''partial function''' as given here is compatible with that of a partial mapping.


Definition:Fibonacci Prime

A '''Fibonacci prime''' is a Fibonacci number which happens to be prime.


Definition:Minimization

=== Function === {{:Definition:Minimization/Function}} === Partial Function === {{:Definition:Minimization/Partial Function}} === Relation === {{:Definition:Minimization/Relation}} Category:Definitions/Mathematical Logic k03ox0qvog152icfb0q4bgpixvoifwz


Definition:Total Function

Let $f: \N^k \to \N$ be a partial function such that $f$ is defined on $S \subseteq \N^k$. Now let $S = \N^k$. That is, the domain of $f$ is then the ''whole'' of $\N^k$. Then $f$ is a '''total function'''. {{SUBPAGENAME}} 9cdhchwm3nvl55y2i8wd6ja4wi7ryyk


Definition:Partial Function Equality

Let $g: \N^k \to \N$ and $h: \N^k \to \N$ be partial functions. We write: :$\map g {n_1, n_2, \ldots, n_k} \approx \map h {n_1, n_2, \ldots, n_k}$ {{iff}} either: :both $\map g {n_1, n_2, \ldots, n_k}$ and $\map h {n_1, n_2, \ldots, n_k}$ are defined and equal or: :neither $\map g {n_1, n_2, \ldots, n_k}$ nor $\map h {n_1, n_2, \ldots, n_k}$ are defined. That is, {{iff}} $\map g {n_1, n_2, \ldots, n_k} = \map h {n_1, n_2, \ldots, n_k}$ wherever either are defined. Thus, '''$g$ is equal to $h$''', and we can write $g = h$, {{iff}}: :$\forall x \in \N^k: \map g x \approx \map h x$ Category:Definitions/Mathematical Logic 61x59x4xn76f8r3x33886hhbxaeoql8


Definition:Degree (Vertex)

Let $G = \struct {V, E}$ be an undirected graph. Let $v \in V$ be a vertex of $G$. The '''degree of $v$ in $G$''' is the number of edges to which it is incident. It is denoted $\map {\deg_G} v$, or just $\map \deg v$ if it is clear from the context which graph is being referred to. That is: :$\map {\deg_G} v = \card {\set {u \in V : \set {u, v} \in E} }$


Definition:Recursive

=== Function === {{:Definition:Recursive/Function}} === Set === {{:Definition:Recursive/Set}} === Relation === {{:Definition:Recursive/Relation}}


Definition:Total Recursive Function

In general, a recursive function is a partial function. A '''total recursive function''' is a recursive function which is total. {{SUBPAGENAME}} d62xoedqu8ncmk22ggtbm02u9t3xw08


Definition:Trace Table

Let $P$ be a URM program. The '''trace table''' of $P$ consists of: * The stage of computation; * The number of the instruction of $P$ that is about to be performed; * A list of the contents of all the registers used by $P$ at this point. Thus the trace table is a list of the states of the URM program at each stage.


Definition:Algorithmic Computability

The concept of '''algorithmic computability''' is an intuitive one. An '''algorithmically computable function''' is a function which can be carried by means of an algorithm, theoretically by a person using pencil and paper. The concept arose in the decades before the invention of digital computers. Much of the theoretical groundwork was done in the 1930s by such as {{AuthorRef|Alan Mathison Turing}}, {{AuthorRef|Alonzo Church}} and {{AuthorRef|Stephen Cole Kleene}}. The term used by {{AuthorRef|Alonzo Church|Church}} when he discussed the issue in his how famous Church's Thesis was '''effectively calculable'''. Category:Definitions/Mathematical Logic 46qfqqk5ue9dtu54ziryz5971matx0s


Definition:Mistake

A '''mistake''' (as opposed to an error) is an inaccuracy or (deduced) falsehood caused by (ultimately) human fault. Synonyms include such terms as '''boob''', '''bloomer''', '''blooper''', '''blunder''', '''cock-up''', '''bug''' (computer science), '''mess-up''', '''oversight''', '''howler''' and other less polite words frequently encountered at the backs of schoolrooms worldwide. Please feel free to add your own.


Definition:Error

Let $x$ be an approximation to a (true) value $X$. The '''error''' $\varepsilon$ is a measure of how much difference there is between $x$ and $X$.


Definition:Transversal (Geometry)

A '''transversal''' of two lines lying in the same plane is a line which intersects them in two different points. The transversal is said to '''cut''' the two lines that it crosses. :400px In the above diagram, $EF$ is a '''transversal''' of the lines $AB$ and $CD$.


Definition:Supplementary Angles

:500px Let $\angle ACB$ be a straight angle. Let $\angle BCD + \angle DCA = \angle ACB$. That is, $\angle DCA = \angle ACB - \angle BCD$. Then $\angle DCA$ is the '''supplement''' of $\angle BCD$. Hence, for any angle $\alpha$ (whether less than a straight angle or not), the '''supplement''' of $\alpha$ is $\pi - \alpha$. Measured in degrees, the '''supplement''' of $\alpha$ is $180^\circ - \alpha$. If $\alpha$ is the '''supplement''' of $\beta$, then it follows that $\beta$ is the '''supplement''' of $\alpha$. Hence we can say that $\alpha$ and $\beta$ are '''supplementary'''. It can be seen from this that the '''supplement''' of a reflex angle is negative. Thus, '''supplementary angles''' are two angles whose measures add up to the measure of $2$ right angles. That is, their measurements add up to $180$ degrees or $\pi$ radians. Another (equivalent) definition is to say that two angles are '''supplementary''' which, when set next to each other, form a straight angle.


Definition:Complementary Angles

:300px Let $\angle BAC$ be a right angle. Let $\angle BAD + \angle DAC = \angle BAC$. That is, $\angle DAC = \angle BAC - \angle BAD$. Then $\angle DAC$ is the '''complement''' of $\angle BAD$. Hence, for any angle $\alpha$ (whether less than a right angle or not), the complement of $\alpha$ is $\dfrac \pi 2 - \alpha$. Measured in degrees, the complement of $\alpha$ is $90^\circ - \alpha$. If $\alpha$ is the complement of $\beta$, then it follows that $\beta$ is the complement of $\alpha$. Hence we can say that $\alpha$ and $\beta$ are '''complementary'''. It can be seen from this that the '''complement''' of an angle greater than a right angle is negative. Thus '''complementary angles''' are two angles whose measures add up to the measure of a right angle. That is, their measurements add up to $90$ degrees or $\dfrac \pi 2$ radians.


Definition:Lucas Number

=== Definition 1 === {{:Definition:Lucas Number/Definition 1}} === Definition 2 === {{:Definition:Lucas Number/Definition 2}} === Sequence === {{:Definition:Lucas Number/Sequence}}


Definition:Exclusive Or

'''Exclusive Or''' is a binary connective which can be written symbolically as $p \oplus q$ whose behaviour is as follows: :$p \oplus q$ means: : '''Either $p$ is true ''or'' $q$ is true ''but not both''.''' or symbolically: : $p \oplus q := \paren {p \lor q} \land \neg \paren {p \land q}$ where $\land$ denotes the ''and'' operator and $\lor$ denotes the ''or'' operator.


Definition:Doubleton

A '''doubleton''' is a set that contains exactly two elements.


Definition:Connected Relation

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is '''connected''' {{iff}}: :$\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ That is, {{iff}} every pair of distinct elements is related (either or both ways round).


Definition:Well-Founded Ordered Set

Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is '''well-founded''' {{iff}} it satisfies the '''minimal condition''': :Every non-empty subset of $S$ has a minimal element. The term '''well-founded''' can equivalently be said to apply to the ordering $\preceq$ itself rather than to the ordered set $\struct {S, \preceq}$ as a whole.


Definition:Partial Ordering

Let $\struct {S, \preceq}$ be an ordered set. Then the ordering $\preceq$ is a '''partial ordering''' on $S$ {{iff}} $\preceq$ is not connected. That is, {{iff}} $\struct {S, \preceq}$ has at least one pair which is non-comparable: :$\exists x, y \in S: x \npreceq y \land y \npreceq x$


Definition:Well-Ordering

{{:Definition:Well-Ordering/Definition 1}}


Definition:Ordinal

{{:Definition:Ordinal/Definition 2}}


Definition:Mathematical Theory

A '''mathematical theory''', or just '''theory''', is a concept in mathematical logic. Let $U$ be a set of logical formulas. Let $\map \TT U$ be the set of all logical formulas $P$ such that $P$ is a semantic consequence of $U$. That is, let $\map \TT U = \set {P: U \models P}$. Then $\TT$ is called '''the (mathematical) theory of $U$'''. The elements of $\map \TT U$ are called theorems of $U$. The elements of $U$ are called the axioms of $\map \TT U$.


Definition:Sign (Mathematical Theory)

A '''sign''' of a mathematical theory is a symbol which has a specific meaning in the theory, as follows: # A logical sign; # A letter; # A specific sign. A logical sign is a sign which has a specific meaning and definition in a particular class of mathematical theory. A letter is a more or less arbitrary symbol whose definition depends on the specific context. For a given mathematical theory, there are signs which are specific to the use to which the theory is being put. These are the specific signs. {{wtd|This may need to go, as it covers the same ground as Sign (Formal Systems).}}


Definition:Logical Sign (Mathematical Theory)

A '''logical sign''' of a mathematical theory is a symbol which is specifically pre-defined for the class of the theory.


Definition:Letter (Mathematical Theory)

A '''letter''' of a mathematical theory is a more or less arbitrary symbol whose definition depends on the specific context. Depending on the nature of the mathematical theory, and the particular location in whatever assembly in which it occurs, the interpretation of the letter will vary.


Definition:Specific Sign (Mathematical Theory)

A '''specific sign''' of a mathematical theory is a symbol which is defined according to the use to which the theory is being put.


Definition:Assembly (Mathematical Theory)

An '''assembly''' in a mathematical theory is a succession of signs written one after another, along with other delineating marks according to the specific nature of the theory under consideration.


Definition:Link (Bourbaki Theory)

An assembly in {{AuthorRef|Nicolas Bourbaki|Bourbaki}}'s exposition of a mathematical theory is specified as being: * A succession of signs written one after another; * Certain signs which are not letters can be joined in pairs by '''links''', as follows: :$\overbrace {\tau A \Box}^{} A'$<ref>The specific symbology used by {{AuthorRef|Nicolas Bourbaki|Bourbaki}} in {{BookLink|Theory of Sets|Nicolas Bourbaki}} has not been rendered accurately here, as the author of this page has not been able to establish a method by which to do it. However, the intent has been expressed as accurately as possible.</ref> {{stub}}


Definition:Word (Formal Systems)

Let $\mathcal A$ be an alphabet. Then a '''word in $\mathcal A$''' is a juxtaposition of finitely many (primitive) symbols of $\mathcal A$. '''Words''' are the most ubiquitous of collations used for formal languages.


Definition:Concatenation (Formal Systems)

Let $\AA$ be an alphabet of symbols. '''Concatenation''' is the process of placing elements of $\AA$ and words in $\AA$ next to each other to form a longer word.


Definition:String

Let $\mathcal A$ be an alphabet of symbols. A '''string (in $\mathcal A$)''' is a sequence of symbols from $\mathcal A$. There is no limit to the number of times a particular symbol may appear in a given '''string'''. === Finite String === {{:Definition:String/Finite}} === Infinite String === {{:Definition:String/Infinite}}


Definition:Formal Grammar

Let $\LL$ be a formal language whose alphabet is $\AA$. The '''formal grammar''' of $\LL$ comprises of rules of formation, which determine whether collations in $\AA$ belong to $\LL$ or not. Roughly speaking, there are two types of '''formal grammar''': top-down grammar and bottom-up grammar.


Definition:Well-Formed Formula

Let $\FF$ be a formal language whose alphabet is $\AA$. A '''well-formed formula''' is a collation in $\AA$ which can be built by using the rules of formation of the formal grammar of $\FF$. That is, a collation in $\AA$ is a '''well-formed formula''' in $\FF$ {{iff}} it has a parsing sequence in $\FF$.


Definition:Formal System

A '''formal system''' is a formal language $\mathcal L$ together with a deductive apparatus for $\mathcal L$.


Definition:Deductive Apparatus

Let $\mathcal L$ be a formal language. A '''deductive apparatus for $\mathcal L$''' is a formally specified system for deriving conclusions about the well-formed formulas of $\mathcal L$. In mathematics and logic, '''deductive apparatuses''' can by and large be divided into proof systems and formal semantics. === Proof System === {{:Definition:Proof System}} === Formal Semantics === {{:Definition:Formal Semantics}}


Definition:Axiom/Formal Systems/Axiom Schema

An '''axiom schema''' is a well-formed formula $\phi$ of $\LL$, except for it containing one or more variables which are ''outside'' $\LL$ itself. This formula can then be used to represent an infinite number of individual axioms in one statement. Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs. Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.


Definition:Proof System/Rule of Inference

A '''rule of inference''' is a specification of a valid means to conclude new theorems in $\mathscr P$ from given theorems and axioms of $\mathscr P$. Often, the formulation of '''rules of inference''' also appeals to the notion of provable consequence.


Definition:LAST

'''LAST''' stands for '''LAnguage of Set Theory'''. It is a formal system designed for the description of sets.


Definition:Zenzizenzizenzic

The '''zenzizenzizenzic''' of a number is its eighth power.


Definition:Predicate Symbol

Let $\mathcal L$ be a formal language (for example, the language of predicate logic $\mathcal L_1$). A '''predicate symbol''' is a letter of $\mathcal L$ used to describe a predicate or a relation. The name '''predicate symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\mathcal L$.


Definition:Singular Statement

A '''singular statement''' is a statement whose subject is identified by means of a proper name. More generally, it is a statement which contains no variables, either bound or free. === Individuating Description === {{:Definition:Singular Statement/Individuating Description}} === Designatory Function === {{:Definition:Singular Statement/Designatory Function}}


Definition:Universal Statement

A '''universal statement''' is one which expresses the fact that all objects (in a particular universe of discourse) have a particular property. That is, a statement of the form: :$\forall x: P \paren x$ where: : $\forall$ is the universal quantifier : $P$ is a predicate symbol. It means: :All $x$ (in some given universe of discourse) have the property $P$. Note that if there exist no $x$ in this particular universe, $\forall x: P \paren x$ is always true: see vacuous truth. === Bound Variable === {{:Definition:Bound Variable/Examples/Universal Statement}}


Definition:Existential Statement

An '''existential statement''' is one which expresses the existence of at least one object (in a particular universe of discourse) which has a particular property. That is, a statement of the form: :$\exists x: P \paren x$ where: : $\exists$ is the existential quantifier : $P$ is a predicate symbol. It means: :There exists at least one $x$ (in some given universe of discourse) which has the property $P$. === Bound Variable === {{:Definition:Bound Variable/Examples/Existential Statement}}


Definition:Universe of Discourse

The '''universe of discourse''', or just '''universe''', is the term used to mean '''everything we are talking about'''. When introducing the symbols $\forall$ (the universal quantifier) or $\exists$ (the existential quantifier), it is understood that the objects referred to are those in the specified '''universe'''. It is usual to define that '''universe'''.


Definition:Logical Complement

The '''(logical) complement''' of a propositional formula $\mathbf A$ is the negation of $\mathbf A$, that is, $\neg \mathbf A$. Conversely, the '''complement''' of $\neg \mathbf A$ is defined to be $\mathbf A$. === Complementary Pair === {{:Definition:Logical Complement/Complementary Pair}}


Definition:Logical NAND

'''NAND''' (that is, '''not and'''), is a binary connective, written symbolically as $p \uparrow q$, whose behaviour is as follows: :$p \uparrow q$ is defined as: :'''it is not the case that $p$ and $q$ are both true.''' $p \uparrow q$ is voiced: :'''$p$ nand $q$'''


Definition:Logical NOR

'''NOR''' (that is, '''not or'''), is a binary connective, written symbolically as $p \downarrow q$, whose behaviour is as follows: :$p \downarrow q$ is defined as:  :'''neither $p$ nor $q$ is true.''' $p \downarrow q$ is voiced: : '''$p$ nor $q$'''


Definition:Backus-Naur Form

'''Backus-Naur Form''' (abbrevated '''BNF''') is a (formal) metalanguage for defining the syntax of a formal language $\LL$. As such, it is a formal grammar for $\LL$. '''BNF''' is only applicable to formal languages that use the collation system of words and concatenation. {{transclude:Definition:Backus-Naur Form/Alphabet   | section  = definition   | increase = 1   | title    = Alphabet   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Rules of Formation   | section  = definition   | increase = 1   | title    = Rules of Formation   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Specification   | section  = definition   | increase = 1   | title    = Specification   | header   = 3   | link     = true }} === Terminals and Non-Terminals === ==== Non-Terminal ==== {{:Definition:Backus-Naur Form/Non-Terminal}} ==== Terminal ==== {{:Definition:Backus-Naur Form/Terminal}} That is, non-terminals are analogous to grammatical clauses of a natural language, while terminals are analogous to its words.


Definition:Metalanguage

A '''metalanguage''' is a language (either formal or natural) which is used to make statements about another language (again, either formal or natural). === Formal Systems === {{:Definition:Metalanguage/Formal Systems}} === Object Language === {{:Definition:Metalanguage/Object Language}} === Metasyntax === {{:Definition:Metalanguage/Metasyntax}} === Metasymbol === {{:Definition:Metalanguage/Metasymbol}}


Definition:Syntax

The '''syntax''' of a language (either natural or formal) is its ''structure''.


Definition:Parsing Sequence

Let $\FF$ be a formal language with alphabet $\AA$. Let $S$ be a collation in $\AA$. A '''parsing sequence''' for $S$ in $\FF$ is a sequence of collations in $\FF$ formed by application of rules of formation of $\FF$ from previous collations in this sequence, and ending in the collation $S$. If $S$ has no '''parsing sequence''' in $\FF$, then it is not a well-formed formula in $\FF$. A parsing sequence for a given well-formed formula in any formal language is usually not unique. Thus, we can determine whether $S$ is a well-formed formula in any formal language by using a sequence of rules of formation of that language. To '''parse''' a collation in a formal language is to find a parsing sequence for that collation, and thereby to determine whether or not it is a well-formed formula.


Definition:Metatheorem

A '''metatheorem''' of a formal language $\mathcal F$ is a theorem about $\mathcal F$ which is not part of $\mathcal F$. {{mistake|It would seem that a formal system is intended instead of formal language.}} {{NoSources}} Category:Definitions/Formal Systems ltkl28rbseo5rcqxg0q5hjai2kkrg1c


Definition:Multigraph

A '''multigraph''' is a graph that can have more than one edge between a pair of vertices. That is, $G = \left({V, E}\right)$ is a '''multigraph''' if $V$ is a set and $E$ is a multiset of 2-element subsets of $V$.


Definition:Block Incidence Matrix of a Design

The '''block incidence matrix''' of a $(v,b,r,k,\lambda)$-BIBD, is a $v\times b$ incidence matrix $A=(a_{ij})$ such that: $a_{ij}=\begin{cases} 1 & : \text{ if the } i^{th} \text { point is in the } j^{th} \text { block} \\ 0 & : \text{ otherwise} \end{cases}$ {{SUBPAGENAME}} gdc8tmjisbj7dr1loz33eonpe8ox3kz


Definition:Contrapositive Statement

The '''contrapositive''' of the conditional: :$p \implies q$ is the statement: :$\neg q \implies \neg p$


Definition:Walk (Graph Theory)

A '''walk''' on a graph is: : an alternating series of vertices and edges : beginning and ending with a vertex : in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it. A '''walk''' between two vertices $u$ and $v$ is called a '''$u$-$v$ walk'''.


Definition:Connected (Graph Theory)

=== Vertices === {{:Definition:Connected (Graph Theory)/Vertices}} === Graph === {{:Definition:Connected (Graph Theory)/Graph}}


Definition:Isomorphism (Graph Theory)

Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs. Let there exist a bijection $F: \map V G \to \map V H$ such that for each edge $\set {u, v} \in \map E G$, there is an edge $\set {\map F u, \map F v} \in \map E H$. That is, that: :$F: \map V G \to \map V H$ is a homomorphism, and :$F^{-1}: \map V H \to \map V G$ is a homomorphism. Then $G$ and $H$ are '''isomorphic''', and this is denoted $G \cong H$. The function $F$ is called an '''isomorphism''' from $G$ to $H$.


Definition:Eulerian Graph

A loop-multigraph or loop-multidigraph is called '''Eulerian''' {{iff}} it contains an Eulerian circuit.


Definition:Symmetric Design

A block design with parameters $v, b, r, k, \lambda$ is said to be '''symmetric''' if $b = v$, and (consequently) $r = k$. === Note === The '''symmetry''' does not mean the incidence matrix is symmetric (it often is not). {{explain|Why $b = v$ has as a consequence $r = k$.}}


Definition:Isomorphism (Abstract Algebra)/Group Isomorphism

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


Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism

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


Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures. Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism. Then $\phi$ is an $R$-algebraic structure isomorphism {{iff}} $\phi$ is a bijection.


Definition:Isomorphism (Abstract Algebra)/Isomorphic Copy

Let $\phi: S \to T$ be an isomorphism. Let $x \in S$. Then $\map \phi x \in T$ is known as '''the isomorphic copy of $x$ (under $\phi$)'''.


Definition:Hamiltonian Graph

A '''Hamiltonian graph''' is an undirected graph that contains a Hamilton cycle.


Definition:Subgraph

A graph $H = \struct {\map V H, \map E H}$ is called a '''subgraph''' of a graph $G = \struct {\map V G, \map E G}$ {{iff}} $\map V H \subseteq \map V G$ and $\map E H \subseteq \map E G$.  That is to say, it contains no vertices or edges that are not in the original graph. If $H$ is a '''subgraph''' of $G$, then: :$G$ '''contains''' $H$; :$H$ '''is contained in''' $G$


Definition:Hamilton Cycle

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


Definition:Eulerian Circuit

An '''Eulerian circuit''' (or '''eulerian circuit''') is a circuit that passes through every vertex of a graph and uses every edge exactly once. It follows that every Eulerian circuit is also an Eulerian trail.


Definition:Decision Procedure

Let $S$ be a set of propositional formulas. A '''decision procedure for $S$''' is an algorithm which, given a propositional formula $\mathbf A$, always terminates, returning the answer: * '''Yes''' if $\mathbf A \in S$; * '''No''' if $\mathbf A \notin S$. === Decision Procedure for Satisfiability === {{:Definition:Decision Procedure/Satisfiability}} === Decision Procedure for Tautologies === {{:Definition:Decision Procedure/Tautologies}} === Refutation Procedure === {{:Definition:Decision Procedure/Refutation Procedure}}


Definition:Semantic Consequence

Let $\mathscr M$ be a formal semantics for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of WFFs of $\mathcal L$. Let $\map {\mathscr M} {\mathcal F}$ be the formal semantics obtained from $\mathscr M$ by retaining only the structures of $\mathscr M$ that are models of $\mathcal F$. Let $\phi$ be a tautology for $\map {\mathscr M} {\mathcal F}$. Then $\phi$ is called a '''semantic consequence of $\mathcal F$''', and this is denoted as: :$\mathcal F \models_{\mathscr M} \phi$ That is to say, $\phi$ is a '''semantic consequence of $\mathcal F$''' {{iff}}, for each $\mathscr M$-structure $\mathcal M$: :$\mathcal M \models_{\mathscr M} \mathcal F$ implies $\mathcal M \models_{\mathscr M} \phi$ where $\models_{\mathscr M}$ is the models relation. Note in particular that for $\mathcal F = \O$, the notation agrees with the notation for a $\mathscr M$-tautology: :$\models_{\mathscr M} \phi$ The concept naturally generalises to sets of formulas $\mathcal G$ on the {{RHS}}: :$\mathcal F \models_{\mathscr M} \mathcal G$ {{iff}} $\mathcal F \models_{\mathscr M} \phi$ for every $\phi \in \mathcal G$.


Definition:Basis Element

Let $S^n$ be a cartesian space. An element $s \in S$ is known as a '''basis element of $S^n$'''. Alternatively, let the ordered tuple $X = \left({x_1, x_2, \ldots, x_n}\right)$ be any element of $S^n$. Let $x_j \in \left({x_1, x_2, \ldots, x_n}\right)$. Then $x_j$ is a '''basis element of $S^n$'''. Category:Definitions/Set Theory hrgjfky62qrvwcm3mrgw1dcs7onr003


Definition:Logical Complement/Complementary Pair

For any propositional formula $\mathbf A$, the set $\left\{{\mathbf A, \neg \mathbf A}\right\}$ is called a '''complementary pair of formulas'''.


Definition:Disjunctive Normal Form

A propositional formula $P$ is in '''disjunctive normal form''' {{iff}} it consists of a disjunction of: :$(1):\quad$ conjunctions of literals and/or: :$(2):\quad$ literals.


Definition:Multiset

A '''multiset''' is a pair $\struct {S, \mu}$ where: :$S$ is a set :$\mu: S \to \N_{>0}$ is a mapping to the strictly positive natural numbers For $s \in S$ the natural number $\map \mu s$ is called the '''multiplicity''' of $s$. Note that the '''multiplicities''' of elements is finite: we do not allow infinitely many occurrences of the same element, though the set $S$ itself may be finite, countably infinite or uncountably infinite.


Definition:Semantics

The '''semantics''' of a language (either natural or formal) is its '''meaning''' in a linguistic sense.


Definition:Trail

A '''trail''' is a walk in which all edges are distinct. A '''trail''' between two vertices $u$ and $v$ is called a '''$u$-$v$ trail'''. The set of vertices and edges which go to make up a '''trail''' form a subgraph. This subgraph itself is also referred to as a '''trail'''.


Definition:Circuit

A '''circuit''' is a closed trail with at least one edge. The set of vertices and edges which go to make up a circuit form a subgraph. This subgraph itself is also referred to as a '''circuit'''.


Definition:Walk (Graph Theory)/Closed

A '''closed walk''' is a walk whose first vertex is the same as the last. That is, it is a walk which ends where it starts.


Definition:Tree (Graph Theory)

=== Definition 1=== {{:Definition:Tree (Graph Theory)/Definition 1}} === Definition 2=== {{:Definition:Tree (Graph Theory)/Definition 2}}


Definition:Path (Graph Theory)

A '''path''' is a trail in which all vertices (except perhaps the first and last ones) are distinct. A path between two vertices $u$ and $v$ is called a '''$u$-$v$ path'''. The set of vertices and edges which go to make up a '''path''' form a subgraph. This subgraph itself is also referred to as a '''path'''. === Open Path === {{:Definition:Path (Graph Theory)/Open}}


Definition:Rooted Tree

A '''rooted tree''' is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the '''root node''': :300px


Definition:Cycle (Graph Theory)

A '''cycle''' is a circuit in which no vertex except the first (which is also the last) appears more than once. An '''$n$-cycle''' is a cycle with $n$ vertices. The set of vertices and edges which go to make up a cycle form a subgraph. This subgraph itself is also referred to as a '''cycle'''. === Odd Cycle === {{:Definition:Cycle (Graph Theory)/Odd}} === Even Cycle === {{:Definition:Cycle (Graph Theory)/Even}}


Definition:Walk (Graph Theory)/Length

The '''length''' of a walk is the number of edges it has, counting repeated edges as many times as they appear. A walk is said to be of '''infinite length''' {{iff}} it has infinitely many edges.


Definition:Finite Graph

A '''finite graph''' is a graph with a finite number of edges and a finite number of vertices.


Definition:Simple Graph

A '''simple graph''' is a graph which is: :An undirected graph, that is, the edges are defined as doubleton sets of vertices and not ordered pairs :Not a multigraph, that is, there is no more than one edge between each pair of vertices :Not a loop-graph, that is, there are no loops, that is, edges which start and end at the same vertex :Not a weighted graph, that is, the edges are not mapped to a number.


Definition:Root of Polynomial

Let $R$ be a commutative ring with unity. Let $f \in R \sqbrk x$ be a polynomial over $R$. A '''root''' in $R$ of $f$ is an element $x \in R$ for which $\map f x = 0$, where $\map f x$ denotes the image of $f$ under the evaluation homomorphism at $x$.


Definition:Root of Mapping

Let $f: R \to R$ be a mapping on a ring $R$. Let $x \in R$. Then the values of $x$ for which $f \left({x}\right) = 0_R$ are known as the '''roots of the mapping $f$'''.


Definition:Tree (Graph Theory)/Leaf Node

Let $v$ be a node of a tree $T$. Then $v$ is a '''leaf node''' of a $T$ {{iff}} $v$ is of degree $1$. If $T$ is a rooted tree, this is equivalent to saying that $v$ has no child nodes.


Definition:Rooted Tree/Ancestor Node

Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. An '''ancestor node''' of $t$ is a node in the path from $t$ to $r_T$. This path is indeed unique, by Path in Tree is Unique. === Proper Ancestor === {{:Definition:Rooted Tree/Ancestor Node/Proper}}


Definition:Propositional Tableau

'''Propositional tableaus''' are a specific subset of the labeled trees for propositional logic. === Identification === {{:Definition:Propositional Tableau/Identification}} {{transclude:Definition:Propositional Tableau/Construction  |section = def  |increase = 1  |link = true  |title = Construction  |header = 3 }}


Definition:Rooted Tree/Parent Node

Let $T$ be a rooted tree whose root is $r_T$. Let $t$ be a node of $T$. From Path in Tree is Unique, there is only one path from $t$ to $r_T$. Let $\pi: T \setminus \left\{{r_T}\right\} \to T$ be the mapping defined by: :$\pi \left({t}\right) := \text{the node adjacent to $t$ on the path to $r_T$}$ Then $\pi \left({t}\right)$ is known as the '''parent node''' of $t$. The mapping $\pi$ is called the '''parent mapping'''.


Definition:Labeled Tree for Propositional Logic/Hypothesis Set

The countable set $\mathbf H$ of WFFs of propositional logic is called the '''hypothesis set'''. The elements of $\mathbf H$ are known as '''hypothesis WFFs'''. The '''hypothesis set''' $\mathbf H$ is considered to be attached to the root node of $T$.


Definition:Finished Set of WFFs of Propositional Logic

Let $\Delta$ be a set of WFFs of propositional logic. Then $\Delta$ is '''finished''' iff: * $\Delta$ is not contradictory * For each WFF $\mathbf C \in \Delta$, either $\mathbf C$ is basic or one of the following is true: ** $\mathbf C$ has the form $\neg \neg \mathbf A$ where $\mathbf A \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \land \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \land \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \lor \mathbf B}\right)$ where either $\mathbf A \in \Delta$ or $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \lor \mathbf B}\right)$ where both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \implies \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \implies \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \iff \mathbf B}\right)$ where either: *** both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$, or: *** both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$; ** $\mathbf C$ has the form $\neg \left({\mathbf A \iff \mathbf B}\right)$ where either: *** both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** or: *** both $\neg \mathbf A \in \Delta$ and $\mathbf B \in \Delta$.


Definition:Finished Branch of Propositional Tableau

Let $T$ be a propositional tableau. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is '''finished''' iff: :$(1): \quad \Gamma$ is not contradictory :$(2): \quad$ Every non-basic WFF on $\Gamma$ is used at some node of $\Gamma$. That is, $\Gamma$ is '''finished''' iff the set $\Delta$ of WFFs of propositional logic which occur along $\Gamma$ is a finished set.


Definition:Finished Propositional Tableau

Let $T$ be a propositional tableau. Then $T$ is '''finished''' iff every branch of $T$ is either finished or contradictory.


Definition:Infix Notation

=== Binary Relations === {{:Definition:Infix Notation/Binary Relation}} === Binary Operations === {{:Definition:Infix Notation/Binary Operation}} Category:Definitions/Language Definitions tk1e76yudsuwl9jb67v7rmy4rymavvw


Definition:Steiner Triple System

A '''Steiner triple system of order $v$''' is a BIBD with block size $3$, and each pair of points occurring together in exactly $1$ block (called a '''triple''').


Definition:Quantifier

The universal quantifier $\forall$ and the existential quantifier $\exists$ are referred to collectively as '''quantifiers'''.


Definition:Atomic WFF of Predicate Logic

An '''atomic WFF''' of predicate logic is a WFF not containing any connectives or quantifiers. In the formal grammar of predicate logic, the '''atomic WFFs''' are precisely those  WFFs which can be formed by only applying the rule $\mathbf W ~ \mathcal P_n$.


Definition:Modulus of Complex-Valued Function

Let $f: S \to \C$ be a complex-valued function. Then the '''(complex) modulus of $f$''' is written $\left|{f}\right|: S \to \R$ and is the real-valued function defined as: :$\forall z \in S: \left|{f}\right| \left({z}\right) = \left|{f \left({z}\right)}\right|$.


Definition:Modulus (Geometric Function Theory)

In geometric function theory, the term '''modulus''' is used to denote certain conformal invariants of configurations or curve families. More precisely, the modulus of a curve family $\Gamma$ is the reciprocal of its extremal length: :$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$ === Modulus of a Quadrilateral === Consider a quadrilateral; that is, a Jordan domain $Q$ in the complex plane (or some other Riemann surface), together with two disjoint closed boundary arcs $\alpha$ and $\alpha'$. Then the '''modulus''' of the quadrilateral $\map Q {\alpha, \alpha'}$ is the extremal length of the family of curves in $Q$ that connect $\alpha$ and $\alpha'$. Equivalently, there exists a rectangle $R = \set {x + i y: \cmod x < a, \cmod y < b}$ and a conformal isomorphism between $Q$ and $R$ under which $\alpha$ and $\alpha'$ correspond to the vertical sides of $R$. Then the modulus of $\map Q {\alpha, \alpha'}$ is equal to the ratio $a/b$. See Modulus of a Quadrilateral. === Modulus of an Annulus === Consider an annulus $A$; that is, a domain whose boundary consists of two Jordan curves. Then the '''modulus''' $\mod A$ is the extremal length of the family of curves in $A$ that connect the two boundary components of $A$. Equivalently, there is a round annulus $\tilde A = \set {z \in \C: r < \cmod z < R}$ that is conformally equivalent to $A$. Then: :$\mod A := \dfrac 1 {2 \pi} \map \ln {\dfrac R r}$ The modulus of $A$ can also be denoted $\map M R$.


Definition:Occurrence (Predicate Logic)

Let $\mathbf A$ be a WFF of predicate logic. Let $S$ be a string in the alphabet of predicate logic. Each place where $S$ appears in $\mathbf A$ is called an '''occurrence of $S$ in $\mathbf A$'''. Note that $S$ may consist of a single symbol, but may not be null. === Scope === {{:Definition:Scope (Logic)/Quantifier}} === Bound Occurrence === {{:Definition:Bound Occurrence}} === Free Occurrence === {{:Definition:Free Occurrence}} === Alphabetic Substitution === {{:Definition:Alphabetic Substitution}}


Definition:Well-Formed Part

Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf A$ be a well-formed formula of $\FF$. Let $\mathbf B$ be a subcollation of $\mathbf A$. Then $\mathbf B$ is a '''well-formed part of $\mathbf A$''' {{iff}} $\mathbf B$ is a well-formed formula of $\FF$. === Proper Well-Formed Part === {{:Definition:Well-Formed Part/Proper Well-Formed Part}}


Definition:Scope (Logic)/Quantifier

Let $\mathbf A$ be a WFF of the language of predicate logic. Let $Q$ be an occurrence of a quantifier in $\mathbf A$. Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$. That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$. $\mathbf B$ is called the '''scope of the quantifier $Q$'''.


Definition:Bound Occurrence

Let $Q x$ be an occurrence of a quantifier in $\mathbf A$. Any occurrence of the variable $x$ in the scope of $Q$ is called a '''bound occurrence'''.


Definition:Freely Substitutable

Let $\mathbf C$ be a WFF of predicate logic. Let $x$ be a variable in $\mathbf C$. Let $\phi \left({y_1, \ldots, y_n}\right)$ be a term in which the variables $y_1, \ldots y_n$ occur. Then $\phi \left({y_1, \ldots, y_n}\right)$ is '''freely substitutable for $x$ in $\mathbf C$''' {{iff}} no free occurrence of $x$ occurs in a well-formed part of $\mathbf C$ which is of the form: :$( Q y_i: \mathbf B )$ where $Q$ is a quantifier and $\mathbf B$ is a WFF.


Definition:Free Occurrence

An occurrence of a variable $x$ in $\mathbf A$ is said to be a '''free occurrence''' {{iff}} it is not bound.


Definition:Classes of WFFs/Plain WFF

A '''plain WFF''' of predicate logic is a WFF with no parameters. Thus $WFF \left({\mathcal P, \mathcal F, \varnothing}\right)$ is the set of all '''plain WFFs''' with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$. {{refactor|However immediate, this goes on a proof page}} It is immediate that a '''plain WFF''' is a WFF with parameters from $\mathcal K$ for ''all'' choices of $\mathcal K$.


Definition:Free Variable

Let $x$ be a variable in an expression $E$. $x$ is a '''free variable in $E$''' {{iff}} it is not a bound variable. In the context of predicate logic, $x$ is a '''free variable in $E$''' {{iff}} it has not been introduced by a quantifier, either: :the universal quantifier $\forall$ or :the existential quantifier $\exists$.


Definition:Wilson Prime

A '''Wilson prime''' is a prime number $p$ such that: :$p^2 \divides \paren {p - 1}! + 1$ where: :$\divides$ signifies divisibility :$!$ is the factorial operator.


Definition:Instance

Let $\mathbf C$ be a plain WFF in the language of predicate logic. Let $x_1, x_2, \ldots, x_n$ be the free variables of $\mathbf C$. Let $\mathcal M$ be a structure for predicate logic of type $\mathcal P$ whose universe set is $M$. Then an '''instance of $\mathbf C$ in $M$''' is the sentence with parameters from $M$ formed by choosing $a_1, a_2, \ldots, a_n \in M$ and replacing all free occurrences of $x_k$ in $\mathbf C$ by $a_k$ for $k = 1, \ldots, n$. The resulting sentence is denoted: :$\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right)$ Thus $\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right) \in SENT \left({\mathcal P, M}\right)$. If $\mathbf C$ is a plain sentence, then no parameters are needed, and $\mathbf C$ is already an instance of itself.


Definition:Heronian Triangle

=== Definition 1 === {{:Definition:Heronian Triangle/Definition 1}} === Definition 2 === {{:Definition:Heronian Triangle/Definition 2}}


Definition:Pseudometric

A '''pseudometric''' on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 3         | q = \forall x, y \in A         | m = \map d {x, y} = \map d {y, x} }} {{end-axiom}} The difference between a '''pseudometric''' and a metric is that a '''pseudometric''' does not insist that the distance function between distinct elements is ''strictly'' positive. === Pseudometric Space === {{:Definition:Pseudometric/Pseudometric Space}}


Definition:Derived Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X \subseteq S$ be a subset of $S$. The '''derived set''' of $X$ is the set of all limit points of $X$. It is often denoted $X'$.


Definition:Perfect Set

=== Definition 1 === {{:Definition:Perfect Set/Definition 1}} === Definition 2 === {{:Definition:Perfect Set/Definition 2}} === Definition 3 === {{:Definition:Perfect Set/Definition 3}}


Definition:Independent Variable

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