Definition:Proper Mapping

Let $X$ and $Y$ be topological spaces. A mapping $f: X \to Y$ is '''proper''' {{iff}} for every compact subset $K \subset Y$, its preimage $f^{-1} \left({K}\right)$ is also compact. {{explain|Here and elsewhere: is it worth replacing "compact subset" with "compact subspace" wherever it appears, as "subset" implies just the set of elements -- "subspace" makes it clear that the topology imposed on that subset is important. I don't know whether this flies against the prevailing convention, but it would help to make the language of this complicated area, where extreme clarity is required, more easily understood.}}


Definition:Generator Set of Filter

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $F$ be a filter on $L$. The '''generator set''' $G$ of $F$ is defined as follows: :$F = \left({\operatorname{fininfs}\left({G}\right)}\right)^\succeq$ where :$\operatorname{fininfs}\left({G}\right)$ denotes the finite infima set of $G$, :$G^\succeq$ denotes the upper closure of $G$.


Definition:Proper Group Action

Let $G$ be a topological group. Let $X$ be a topological space. A group action $\phi: G \times X \to X$ is called '''proper''' {{iff}} $\phi$ is a proper mapping. Here $G\times X$ is equipped with the product topology.


Definition:Continuous Group Action

Let $G$ be a topological group. Let $X$ be a topological space. A group action $\phi: G \times X \to X$ is defined as '''continuous''' {{iff}} $\phi$ is continuous.


Definition:Loop (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $\gamma: \closedint 0 1 \to S$ be a path in $T$. $\gamma$ is a '''loop (in $T$)''' {{iff}}: :$\map \gamma 0 = \map \gamma 1$ === Base Point === {{Definition:Loop (Topology)/Base Point}}


Definition:Concatenation of Paths

Let $X$ be a topological space. Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be paths. Suppose that $f \left({1}\right) = g \left({0}\right)$. The '''concatenation''' of $f$ and $g$ is the mapping $f * g: \left[{0 \,.\,.\, 1}\right] \to X$ defined by: :$\displaystyle \left({f * g}\right) \left({s}\right) = \begin{cases} f \left({2 s}\right) & : 0 \le s \le \dfrac 1 2 \\ g \left({2 s - 1}\right) & : \dfrac 1 2 \le s \le 1 \end{cases}$


Definition:Loop (Topology)/Base Point

The '''base point''' of $\gamma$ is $\gamma \left ({0}\right)$.


Definition:Homotopy/Path

Let $X$ be a topological space. Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be paths. We say that $f$ and $g$ are '''path-homotopic''' if they are homotopic relative to $\left\{ {0, 1}\right\}$.


Definition:Homotopy Class/Path

Let $T = \left({S, \tau}\right)$ be a topological space. Let $f: \left[{0 \,.\,.\, 1}\right] \to S$ be a path in $T$. The '''homotopy class''' of the path $f$ is the homotopy class of $f$ relative to $\left\{ {0, 1}\right\}$. That is, the equivalence class of $f$ under the equivalence relation defined by path-homotopy.


Definition:Nontotient

A '''nontotient''' is a positive even integer $n$ such that: :$\nexists m \in \Z_{>0}: \phi \left({m}\right) = n$ where $\phi \left({m}\right)$ denotes the Euler $\phi$ function. That is, a '''nontotient''' is a positive even integer which is not the totient of any positive integer.


Definition:Compact-Open Topology

Let $X$ and $Y$ be topological spaces. Let $\map \CC {X, Y}$ be the set of continuous maps from $X$ to $Y$. For all compact subsets $K \subset X$ and all open subsets $U \subset Y$, let: :$\map V {K, U} = \set {f \in \map \CC {X, Y}: f \sqbrk K \subset U}$ Let: :$\BB = \set {\map V {K, U}: K \subset X \text{ compact}, U \subset Y \text{ open} }$ The '''compact-open topology''' on $\map \CC {X, Y}$ is the topology generated by $\BB$.


Definition:Multiplication of Homotopy Classes of Paths

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\alpha, \beta$ be homotopy classes of paths in $T$. Let $f, g: \left[{0 \,.\,.\, 1}\right] \to S$ be representative paths for $\alpha$ and $\beta$ respectively. Let $f \left({1}\right) = g \left({0}\right)$. The '''product of the homotopy classes''' $\alpha$ and $\beta$ is the homotopy class of the concatenated path $f * g$.


Definition:Legendre Transform

Let $\map f x$ be a strictly convex real function. Let $p = \map {f'} x$. Let $\map {f^*} p = - \map f{\map x p} + p \map x p$. {{explain|The above seems to imply that $x$ is a function.
Yes it does, doesn't it. Exactly what it does mean is to be added to this page, and if necessary a new definition page is needed to specify it.}} The '''Legendre Transform on $x$ and $f$''' is the mapping of the variable and function pair: :$\paren{x, \map f x} \to \paren{p, \map {f^*} p}$ {{expand|generalise, add properties and connect with Young's inequality
But not in here, do it somewhere else.}} {{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}


Definition:Maximal Normal Subgroup

Let $G$ be a group. Let $N \le G$ be a proper normal subgroup. Then $N$ is a '''maximal normal subgroup of $G$''' iff: :For every normal subgroup $M$ of $G$, $N \subseteq M \subseteq G$ implies $N = M$ or $M = G$. That is, if there is no normal subgroup of $G$, except $N$ and $G$ itself, which contains $N$.


Definition:Ascending Chain Condition

Let $\left({P, \leq}\right)$ be an ordered set. Then $S$ is said to have the '''ascending chain condition (ACC)''' if every increasing sequence $x_1 \leq x_2 \leq x_3 \leq \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n+1} = \cdots$. === ACC on submodules === {{:Definition:Ascending Chain Condition/Module}} === ACC on ideals === {{:Definition:Ascending Chain Condition/Ideals}} === ACC on principal ideals === {{:Definition:Ascending Chain Condition/Principal Ideals}}


Definition:Ascending Chain Condition/Ideals

Let $R$ be a commutative ring. Then $R$ is said to have the '''ascending chain condition on ideals''' {{iff}} every increasing sequence of ideals stabilizes.


Definition:Nontotient/Sequence

The sequence of '''nontotients''' begins: :$14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, \ldots$


Definition:Cofactor Matrix

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $A_{r s}$ denote the cofactor of the element whose indices are $\tuple {r, s}$. The '''cofactor matrix''' of $\mathbf A$ is the square matrix of order $n$: :$\mathbf C = \begin {bmatrix} A_{1 1} & A_{1 2} & \cdots & A_{1 n} \\ A_{2 1} & A_{2 2} & \cdots & A_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ A_{n 1} & A_{n 2} & \cdots & A_{n n} \\ \end {bmatrix}$


Definition:Fibonacci-Like Sequence

Let $A = \left({a_0, a_1, \ldots, a_{n - 1} }\right)$ be an ordered tuple of numbers. The '''Fibonacci-like sequence formed from $A$''' is defined as: :$F_A \left({k}\right) = \begin{cases} \qquad \qquad a_k & : 0 \le k < n \\ & \\ \displaystyle \sum_{k - n \mathop \le j \mathop < k} a_j & : k \ge n \end{cases}$ That is, apart from the first $n$  terms, every term is the sum of the previous $n$ terms. The main term can also be expressed as: :$F_A \left({k}\right) = 2 F_A \left({k - 1}\right) - F_A \left({k - n}\right)$


Definition:Keith Number

Let $K \in \Z_{>9}$ be an $n$-digit integer, where $n > 1$. Let $A = \left({a_1, a_2, \ldots, a_n}\right)$ be the digits of $K$ in order. Let $F_A$ be the Fibonacci-like sequence on $A$. Then $K$ is a '''Keith number''' {{iff}} $K$ occurs somewhere in $F_A$.


Definition:Keith Number/Sequence

The sequence of '''Keith numbers''' begins: :$14, 19, 28, 47, 61, 75, 197, 742, 1104, \ldots$


Definition:Dorroh Extension

Let $R$ be a ring. We define two operations on the cartesian product $R \times \Z$ as: :$\tuple {r, n} + \tuple {s ,m} = \tuple {r + s, n + m}$ :$\tuple {r, n} \cdot \tuple {s, m} = \tuple {r s + n s + m r, n m}$ The '''Dorroh extension''' of $R$ is the ring $\struct {R \times \Z, +, \cdot}$.


Definition:Ramanujan-Nagell Equation

The '''Ramanujan-Nagell equation''' is the Diophantine equation: :$x^2 + 7 = 2^n$


Definition:Canonical Transformation

Let $\paren{x,\mathbf y,\mathbf p,H}$ be canonical variables. Let $\paren{x,\mathbf Y,\mathbf P,H^*}$ be another set of canonical variables. A mapping between these is a '''canonical transformation''' {{iff}}: :$\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i},\quad\dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i}$ imply: :$\dfrac {\d Y_i} {\d x}=\dfrac {\partial H^*} {\partial P_i},\quad\dfrac {\d P_i} {\d x}=-\dfrac {\partial H^*} {\partial Y_i}$


Definition:Irreducible Subset (Topology)

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then $A$ is '''irreducible (subset)''' {{iff}} :$A$ is non-empty and closed and for all closed subsets $B, C$ of $S$: $A = B \cup C \implies A = B$ or $A = C$


Definition:Invariant Functional under Transformation

Let $y_i$, $F$, $\Phi$, $\Psi$ be real functions. Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$. Let $\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional. Let: :$X = \map \Phi {x, \mathbf y, \mathbf y'}$ :$\mathbf Y = \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'}$ Let curve $\gamma$ defined by:  :$\mathbf y = \map {\mathbf y} x, \quad x_0 \le x \le x_1$  be transformed into a curve $\Gamma$ defined by: :$\mathbf Y = \map {\mathbf Y} X, \quad X_0 \le X \le X_1$ Then the functional $J \sqbrk {\mathbf y}$ is '''invariant''' under the given transformation {{iff}}: :$J \sqbrk \Gamma = J \sqbrk \gamma$ That is, {{iff}}: :$\displaystyle \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{X_0}^{X_1} \map F {X, \mathbf Y, \mathbf Y'} \rd X$


Definition:Dense (Order Theory)/Element

Let $x \in S$. Then $x$ is '''dense''' {{iff}} :$\forall y \in S: y \ne \bot \implies x \wedge y \ne \bot$ where $\bot$ denotes the smallest element in $L$.


Definition:Dense (Order Theory)/Subset

Let $A$ be a subset of $S$. Then $A$ is '''dense''' {{iff}} it includes only dense elements. That means that iff $\forall x \in A: x$ is a dense element.


Definition:Dense (Order Theory)

Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded below meet semilattice. === Dense Element === {{:Definition:Dense (Order Theory)/Element}} === Dense Subset === {{:Definition:Dense (Order Theory)/Subset}} Category:Definitions/Order Theory iuhfbmay1t7m60yyawdxgnb34uzgi9a


Definition:Almost Perfect Number/Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''almost perfect''' {{iff}} $A \left({n}\right) = -1$.


Definition:Almost Perfect Number/Definition 2

$n$ is '''almost perfect''' {{iff}}: :$\sigma \left({n}\right) = 2 n - 1$ where $\sigma \left({n}\right)$ denotes the $\sigma$ function of $n$.


Definition:Almost Perfect Number/Definition 3

$n$ is '''almost perfect''' {{iff}} it is exactly one greater than the sum of its aliquot parts.


Definition:Geodetic Distance

Let $y_i$, $F$ be real functions. Let $\mathbf y = \sequence {y_i}_ {1 \mathop \le i \mathop \le n}$ be a vector.  Let: :$\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional with only one extremal passing any two points: :$A = \map A {x_0, \mathbf y^0}$ :$B = \map B {x_1, \mathbf y^1}$ Suppose a curve $\gamma$ is an extremal of $J$. {{explain|Extremum or extremal?}} Then: :$\ds \map S {x_0, x_1, \mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \big \rvert_{\gamma}\rd x$ is called a '''geodetic distance''' between $A$ and $B$.


Definition:Hamilton-Jacobi Equation

Let $\map S {x_0, x_1, \mathbf y} = \map S {x, \mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1 = x$. Let $H$ be Hamiltonian. The differential equation: :$\displaystyle \frac {\partial S} {\partial x} + \map H {x, \mathbf y, \nabla_{\mathbf y} S} = 0$ is known as the '''Hamilton-Jacobi Equation'''. {{NamedforDef|William Rowan Hamilton|name2 = Carl Gustav Jacob Jacobi|cat = Hamilton|cat2 = Jacobi}}


Definition:Quasiperfect Number/Definition 2

$n$ is '''quasiperfect''' {{iff}}: :$\sigma \left({n}\right) = 2 n + 1$ where $\sigma \left({n}\right)$ denotes the $\sigma$ function of $n$.


Definition:Quasiperfect Number/Definition 3

$n$ is '''quasiperfect''' {{iff}} it is exactly one less than the sum of its aliquot parts.


Definition:Principal Ideal of Ordered Set/Definition 1

Then $I$ is '''principal (ideal)''' {{iff}} :$\exists x \in I: x$ is upper bound for $I$


Definition:Principal Ideal of Ordered Set/Definition 2

Then $I$ is '''principal (ideal)''' {{iff}} :$\exists x \in S: I = x^\preceq$ where $x^\preceq$ denotes the lower closure of $x$.


Let $\left({S, \preceq}\right)$ be a preordered set. Let $I$ be an ideal in $S$. === Definition 1 === {{:Definition:Principal Ideal of Ordered Set/Definition 1}} === Definition 2 === {{:Definition:Principal Ideal of Ordered Set/Definition 2}}


Definition:Wallpaper Pattern

A '''wallpaper pattern''' is a plane geometrical figure which has translational symmetry in two non-parallel lines.


Definition:Euler Lucky Number

=== Definition 1 === {{:Definition:Euler Lucky Number/Definition 1}} === Definition 2 === {{:Definition:Euler Lucky Number/Definition 2}}


Definition:Stern Prime

A '''Stern prime''' is a prime number which can not be represented in the form: :$2 a^2 + p$ where: :$a \in \Z_{>0}$ is a (strictly) positive integer :$p$ is a prime number. === Sequence === {{:Definition:Stern Prime/Sequence}}


Definition:Stern Number

A '''Stern number''' is an odd number which can not be represented in the form: :$2 a^2 + p$ where: :$a \in \Z_{>0}$ is a (strictly) positive integer :$p$ is a prime number.


Definition:Centered Hexagonal Number

A '''centered hexagonal number''' is a figurate number which is represented by: :a point in the center :all other points surrounding the center at the points of a triangular lattice.


Definition:Centered Hexagonal Number/Sequence

The sequence of centered hexagonal numbers, for $n \in \Z_{> 0}$, begins: :$1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, \ldots$


Definition:Continuous Lattice Subframe

Let $L = \left({X, \preceq}\right)$ be an ordered set. Let $S = \left({Y, \preceq'}\right)$ be an ordered subset of $L$. Then $S$ is '''continuous lattice subframe''' of $L$ {{iff}} :$S$ inherits infima and directed suprema.


Definition:Magic Hexagon

A magic hexagon is a centered hexagonal pattern of cells containing the natural numbers from $1$ to the number of cells such that the numbers in all the cells in a given straight line all add up to the same number.


Definition:Magic Hexagon/Order

Let $H$ be an magic hexagon created on the $n$th centered hexagonal number. That is, let $H$ have $n$ cells along each side. Then $H$ is an '''order $n$ magic hexagon'''.


Definition:Magic Hexagon/Line

A '''line''' of a magic hexagon is a set of consecutively adjacent cells, in any of the three directions parallel to the sides of the underlying hexagon.


Definition:Semiperfect Number

A '''semiperfect number''' is a positive integer which is equal to the sum of some or all of its aliquot parts.


Definition:Semiperfect Number/Sequence

The sequence of '''semiperfect numbers''' begins: :$6, 12, 18, 20, 24, 28, 30, \ldots$


Definition:Vigesimal System

The '''vigesimal system''' is base $20$ notation. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 20^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, \ldots, 19}$


Definition:Bilinear Functional

Let $y_1$, $y_2$, $z$ be mappings, belonging to some normed linear space. {{explain|See talk page.}} Let $S$ be a set of ordered pairs $\tuple {y_1, y_2}$. Let $B: S \to \R$ be a mapping defined as: :$\forall \tuple {y_1, y_2} \in S: \exists x \in \R: B \sqbrk {y_1, y_2} = x$ {{explain|Establish the precise meaning of $B \sqbrk {y_1, y_2}$, and see whether it actually might mean the same as $\map B {y_1, y_2}$, in which case use the latter. If not, explain what it does mean.}} Let $B$ be linear {{WRT}} $y_1$ for fixed $y_2$, and linear {{WRT}} $y_2$ for fixed $y_1$: :$B \sqbrk {\alpha y_1 + \beta z, y_2} = \alpha B \sqbrk {\alpha y_1, y_2} + \beta B \sqbrk {z, y_2}$ :$B \sqbrk {y_1, \alpha y_2 + \beta z} = \alpha B \sqbrk {y_1, y_2} + \beta B \sqbrk {y_1, z}$ where $\alpha, \beta \in \R$. Then $B: S \to \R$ is known as a '''bilinear functional''', denoted by $B \sqbrk {y_1, y_2}$


Definition:Quadratic Functional

Let $B \sqbrk {x, y}$ be a bilinear functional. Let $x = y$. Then the functional $A \sqbrk x = B \sqbrk {x, x}$ is called a '''quadratic functional'''.


Definition:Bilinear Form

Let $R$ be a ring. Let $M$ be an $R$-module. A '''bilinear form''' on $M$ is a bilinear mapping $b : M \times M \to R$.


Definition:Twice Differentiable/Functional

Let $\Delta J \sqbrk {y; h}$ be an increment of a functional. Let: :$\Delta J \sqbrk {y; h} = \phi_1 \sqbrk {y; h} + \phi_2 \sqbrk {y; h} + \epsilon \size h^2$ where: :$\phi_1 \sqbrk {y; h}$ is a linear functional :$\phi_2 \sqbrk {y; h}$ is a quadratic functional {{WRT}} $h$ :$\epsilon \to 0$ as $\size h \to 0$. Then the functional $J\sqbrk y$ is '''twice differentiable'''. {{refactor|Extract the following two into their own definition pages, appropriately named and transcluded.}} The linear part $\phi_1$ is the '''first variation''', denoted: :$\delta J \sqbrk {y; h}$ $\phi_2$ is called the '''second variation''' (or '''differential''') of a functional, and is denoted: :$\delta^2 J \sqbrk {y; h}$


Definition:Integer Square

An '''integer square''' is a square whose sides are integers. Category:Definitions/Geometry Category:Definitions/Recreational Mathematics 0jvtx0iavnfxizw8ornszqt2o2nd0t4


Definition:Perfect Square Dissection

A '''perfect square dissection''' is a dissection of an integer square into a number of smaller integer squares all of different sizes.


Definition:Finite Projective Plane/Order 4

The finite projective plane of order $4$. It can be denoted: :$\operatorname {PG} \left({4, 4}\right)$ where: :$\operatorname {PG}$ stands for '''projective geometry''' :the first parameter $2$ specifies the dimension :the second parameter $2$ specifies the order. $\operatorname {PG} \left({4, 4}\right)$ has the following properties: : It has $21$ points : It has $21$ lines : There are $5$ points on each line : There are $5$ lines through each point.


Definition:Minimum Value of Functional

Let $S$ be a set of mappings. Let $y, \hat y \in S: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R$ be a functional. Let $J$ have a (relative) extremum for $y = \hat y$. Let $J \sqbrk y - J \sqbrk {\hat y} \ge 0$ in the neighbourhood of $y = \hat y$. {{Disambiguate|Definition:Neighborhood}} Then this extremum is called the '''minimum''' of the functional $J$.


Definition:Maximum Value of Functional

Let $S$ be a set of mappings. Let $y, \hat y \in S: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R $ be a functional. Let $J$ have a (relative) extremum for $y = \hat y$. Suppose, $J \sqbrk y - J \sqbrk {\hat y} \le 0$ in the neighbourhood of $y = \hat y$. {{explain|Make sure the link to Definition:Neighborhood invokes the correct subpage -- if there is no such appropriate subpage, then write it}} Then this extremum is called the '''maximum''' of the functional $J$.


Definition:Strongly Positive Quadratic Functional

Let $J\sqbrk y$ be a quadratic functional {{WRT}} $y$, defined on normed linear space. Suppose there exists $k\in\R_{> 0}$ such that: :$J\sqbrk y\ge k\size {y}^2$ Then the quadratic functional $J$ is '''strongly positive'''.


Definition:Second Pentagonal Number/Definition 1

The '''second pentagonal numbers''' are the integers obtained by applying the Closed Form for Pentagonal Numbers to negative $n$: :$\map {P'} n = \dfrac {-n \paren {-3 n - 1} } 2$


Definition:Second Pentagonal Number

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


=== Definition 1 === {{:Definition:Generalized Pentagonal Number/Definition 1}} === Definition 2 === {{:Definition:Generalized Pentagonal Number/Definition 2}}


Definition:Generalized Pentagonal Number/Definition 1

Recall the sequence of pentagonal numbers: :$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$ and the sequence of second pentagonal numbers: :$0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, \ldots$ The sequence of '''generalized pentagonal numbers''' consists of the elements of both of these sequences merged and arranged into ascending order.


Definition:Generalized Pentagonal Number/Definition 2

The '''generalized pentagonal numbers''' are the integers obtained from the formula: :$GP_n = \begin{cases} \dfrac {m \left({3 m + 1}\right)} 2 & : n = 2 m \\ \dfrac {m \left({3 m - 1}\right)} 2 & : n = 2 m - 1 \end{cases}$ for $n = 0, 1, 2, \ldots$


Definition:Conjugate Point

=== Definition 1 === {{:Definition:Conjugate Point/Definition 1}} === Definition 2 === {{:Definition:Conjugate Point/Definition 2}} === Definition 3 === {{:Definition:Conjugate Point/Definition 3}}


Definition:Factorial Number System

The '''factorial number system''' is a mixed radix positional numeral system in which the digit in the $i$th column from the right is of base $i$.


Definition:Cullen Number

A '''Cullen number''' is a positive integer of the form: :$n \times 2^n + 1$


Definition:Cullen Prime

A '''Cullen prime''' is a Cullen number: :$n \times 2^n + 1$ which is also prime.


Definition:Palindromic Number/Sequence

The sequence of '''palindromic integers''' in base 10 begins: :$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, \ldots$


Definition:Polygon/Chord

A '''chord''' of a polygon $P$ is a straight line connecting two non-adjacent vertices of $P$: :300px In the above diagram, $DF$ is a '''chord''' of polygon $ABCDEFG$.


Definition:Convex Polygon

Let $P$ be a polygon. $P$ is a '''convex polygon''' {{iff}}: :For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$.


Definition:Aliquot Sum

Let $n \in \Z$ be a positive integer. The '''aliquot sum''' of $n$ is defined as the sum of its aliquot parts.


Definition:Sociable Chain

Let $m$ be a positive integer. Let $\map s m$ be the aliquot sum of $m$. Define the sequence $\sequence {a_k}$ recursively as: :$a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$ A '''sociable chain''' is such a sequence $\sequence {a_k}$ where: :$a_r = a_0$ for some $r > 0$.


Definition:Sociable Chain/Order

The '''order''' of $a_k$ is the '''smallest''' $r \in \Z_{>0}$ such that :$a_r = a_0$


Definition:Sociable Number

A '''sociable number''' is a positive integer which occurs as a term in a sociable chain.


Definition:Topological Group Homomorphism

Let $G$ and $H$ be topological groups. A mapping $\phi:G \to H$ is a '''homomorphism of topological groups''' if it is a continuous group homomorphism. Category:Definitions/Topological Groups Category:Definitions/Group Homomorphisms fj4jz7ytxgcrn863hjj99fqcearnq20


Definition:Quotient Space by Group Action

Let $G$ be a group acting on a topological space $X$. The '''quotient of $X$ by the action of $G$''', denoted $X/G$, is the set of orbits equipped with the quotient topology.


Definition:Group Action by Homeomorphisms

Let $G$ be a group. Let $X$ be a topological space. Let $\phi: G \times X \to X$ be a group action Then $G$ '''acts by homeomorphisms''' {{iff}} for all $g \in G$, the mapping: :$\phi_g : X \to X : x \mapsto \phi \left({g, x}\right)$ is a homeomorphism.


Definition:Locally Connected Space/Definition 1

A topological space $T = \struct{S, \tau}$ is '''locally connected''' {{iff}} each point of $T$ has a local basis consisting entirely of connected sets in $T$.


Definition:Locally Connected Space/Definition 2

A topological space $T = \struct{S, \tau}$ is '''locally connected''' {{iff}} $T$ is weakly locally connected at each point of $T$.


Definition:Locally Connected Space/Definition 3

A topological space $T = \struct{S, \tau}$ is '''locally connected''' {{iff}} it has a basis consisting of connected sets in $T$.


Definition:Locally Path-Connected Space/Definition 3

A topological space $T = \struct {S, \tau}$ is a '''locally path-connected space''' {{iff}} it has a basis consisting of path-connected sets  in $T$.


Definition:Locally Path-Connected Space/Definition 1

A topological space $T = \struct{S, \tau}$ is a '''locally path-connected space''' {{iff}} each point of $T$ has a local basis consisting of path-connected sets in $T$.


Definition:Locally Path-Connected Space/Definition 2

A topological space $T = \struct{S, \tau}$ is a '''locally path-connected space''' {{iff}} each point of $T$ has a neighborhood basis consisting of path-connected sets in $T$.


Definition:Locally Compact Hausdorff Space/Definition 1

$T$ is a '''locally compact Hausdorff space''' {{iff}} each point of $S$ has a compact neighborhood. That is, {{iff}} $T$ is weakly locally compact.


Definition:Locally Compact Hausdorff Space/Definition 2

$T$ is a '''locally compact Hausdorff space''' {{iff}} each point has a neighborhood basis consisting of compact sets. That is, {{iff}} $T$ is locally compact (in the general sense).


Let $T = \left({S, \tau}\right)$ be a Hausdorff topological space. === Definition 1 === {{:Definition:Locally Compact Hausdorff Space/Definition 1}} === Definition 2 === {{:Definition:Locally Compact Hausdorff Space/Definition 2}}


Definition:Relatively Closed Set/Definition 1

'''$A$ is relatively closed in $B$''' {{iff}} $A$ is closed in the relative topology of $B$.


Definition:Relatively Closed Set/Definition 2

'''$A$ is relatively closed in $B$''' {{iff}} there is a closed set $C \subseteq S$ with $C \cap B = A$.


Definition:Zeroth

'''Zeroth''' is the adjectival form of the ordinal number zero. Category:Definitions/Zero 5jvx0jgag2e6s22dppm6xc4a6y772eg


Definition:Polynomial in Ring Element

Let $R$ be a commutative ring. Let $S$ be a subring with unity of $R$. Let $x\in R$. === Definition 1 === {{Definition:Polynomial in Ring Element/Definition 1}} === Definition 2 === {{Definition:Polynomial in Ring Element/Definition 2}}


Definition:Polynomial over Ring as Sequence

A '''polynomial''' over $R$ can be defined as a sequence in $R$ whose domain is $\N$ and which has finite support. That is, it is an element of the ring of sequences of finite support.


Definition:Polynomial Ring/Monoid Ring on Natural Numbers

Let $\N$ denote the additive monoid of natural numbers. Let $R \left[{\N}\right]$ be the monoid ring of $\N$ over $R$. The '''polynomial ring over $R$''' is the ordered triple $\left({R \left[{\N}\right], \iota, X}\right)$ where: :$X \in R \left[{\N}\right]$ is the standard basis element associated to $1\in \N$. :$\iota : R \to R \left[{\N}\right]$ is the canonical mapping.


Let $\sim$ be an equivalence relation on a set $S$. Let $T\subset S$ be a subset. === Definition 1 === {{:Definition:Saturation (Equivalence Relation)/Definition 1}} === Definition 2 === {{:Definition:Saturation (Equivalence Relation)/Definition 2}} === Definition 3 === {{:Definition:Saturation (Equivalence Relation)/Definition 3}}


Definition:Saturation (Equivalence Relation)/Definition 1

The '''saturation of $T$''' is the set of all elements that are equivalent to some element in $T$: :$\overline T = \{s \in S : \exists t\in T : s\sim t\}$


Definition:Saturation (Equivalence Relation)/Definition 3

The '''saturation of $T$''' is the preimage of its image under the quotient mapping: :$\overline T = q^{-1} \sqbrk {q \sqbrk T}$


Definition:Saturation (Equivalence Relation)/Definition 2

The '''saturation of $T$''' is the union of the equivalence classes of its elements: :$\displaystyle \overline T = \bigcup_{t \mathop \in T} \eqclass t \sim$


Let $\sim$ be an equivalence relation on a set $S$. Let $T\subset S$ be a subset. === Definition 1 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 1}} === Definition 2 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 2}} === Definition 3 === {{:Definition:Saturated Set (Equivalence Relation)/Definition 3}}


Definition:Saturated Set (Equivalence Relation)/Definition 1

$T$ is '''saturated''' {{iff}} it equals its saturation: :$T = \overline T$


Definition:Saturated Set (Equivalence Relation)/Definition 3

$T$ is '''saturated''' {{iff}} it is the preimage of some set under the quotient mapping: :$\exists V \subset S / \sim \; : T = q^{-1} \left[{V}\right]$


Definition:Saturated Set (Equivalence Relation)/Definition 2

$T$ is '''saturated''' {{iff}} it is a union of equivalence classes: :$\displaystyle \exists U \subset S : T = \bigcup_{u \mathop \in U} \left[\!\left[{u}\right]\!\right]$


Definition:Quotient Topology/Definition 1

Let $\tau_\RR$ be the identification topology on $S / \RR$ by $q_\RR$: :$\tau_\RR := \set {U \subseteq S / \RR: q_\RR^{-1} \sqbrk U \in \tau}$ Then $\tau_\RR$ is the '''quotient topology on $S / \RR$ by $q_\RR$'''.


Definition:Saturation (Group Action)

Let $G$ be a group acting on a set $X$. Let $S\subset X$ be a subset. === Definition 1 === {{:Definition:Saturation (Group Action)/Definition 1}} === Definition 2 === {{:Definition:Saturation (Group Action)/Definition 2}}


Definition:Saturation (Group Action)/Definition 1

The '''saturation of $S$''' is its saturation by the equivalence relation induced by the action.


Definition:Equivalence Relation Induced by Group Action

Let $G$ be a group. Let $X$ be a set. Let $\phi : G\times X \to X$ be a group action. The '''equivalence relation on $X$ induced by (the action) $\phi$''' is the relation $\mathcal R_G$ defined as: :$x \mathrel {\mathcal R_G} y \iff y \in \Orb x$ where: :$\Orb x$ denotes the orbit of $x \in X$. That is: :$x \mathrel {\mathcal R_G} y \iff \exists g \in G: y = g*x$


Definition:Saturation (Group Action)/Definition 2

The '''saturation of $S$''' is the union of its images under the group action: :$\overline S = \displaystyle \bigcup_{g \mathop \in G} g S$


Definition:Primitive Semiperfect Number

A '''primitive semiperfect number''' is a semiperfect number which is not a multiple of a smaller semiperfect number.


Definition:Primitive Semiperfect Number/Sequence

The sequence of '''primitive semiperfect numbers''' begins: :$6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, \ldots$


Definition:Primitive Abundant Number

A '''primitive abundant number''' is an abundant number whose aliquot parts are all deficient.


Definition:Primitive Abundant Number/Sequence

The sequence of '''primitive abundant numbers''' begins: :$20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, \ldots$


Definition:Ring of Polynomial Forms

Let  $R$ be a  commutative ring with unity. Let $I$ be a set Let $\family {X_i: i \in I}$ be an indexed set. Let $A = R \sqbrk {\family {X_i: i \in I} }$ be the set of all polynomial forms over $R$ in $\family {X_i: i \in I}$. Let $+$ and $\circ$ denote the standard addition and multiplication of polynomial forms. The '''ring of polynomial forms''' is the ordered triple  $\struct {A, +, \circ}$.


Definition:Induced Homomorphism Between Fundamental Groups

Let $X,Y$ be topological spaces. Let $f:X\to Y$ be a continuous map. Let $x_0\in X$ and $y_0=f(x_0)\in Y$. Let $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ be their fundamental groups. The '''homomorphism induced by $f$''' is the group homomorphism $f_* : \pi_1(X,x_0) \to \pi_1(Y,y_0)$ defined by: :$f_*([\gamma]) = [f\circ\gamma]$


Definition:Topological Covering Map

Let $E$ and $B$ be topological spaces. Let $p:E\to B$ be a continuous surjection. Then $p:E\to B$ is a '''covering map''' if every $b\in B$ has an open neighborhood whose preimage is a disjoint union of open sets such that the restriction of $p$ to each of them is a homeomorphism That is: :Every $b \in B$ has an evenly covered open neighborhood.


Definition:Evenly Covered

Let $E$ and $B$ be topological spaces. Let $p: E \to B$ be a continuous surjection. An open set $U \subset B$ is '''evenly covered by $p$''' if its preimage is a disjoint union of open sets such that the restriction of $p$ to each of them is a homeomorphism to $U$.


Definition:Contractible Space/Definition 2

$X$ is called '''contractible''' {{iff}} it is homotopy equivalent to a point.


Definition:Fourth Power

A '''fourth power''' is an integer which can be expressed as the $4$th power of an integer.


Definition:Separable Polynomial/Definition 1

$P$ is '''separable''' {{iff}} its roots are distinct in an algebraic closure of $K$.


Definition:Separable Polynomial/Definition 2

$P$ is '''separable''' {{iff}} it has no double roots in every field extension of $K$.


Definition:Separable Polynomial/Definition 3

$P$ is '''separable''' {{iff}} it has $n$ distinct roots in every field extension where $P$ splits.


Definition:Separable Element

Let $L/K$ be a field extension. Let $\alpha\in L$. Then $\alpha$ is '''separable over $K$''' {{iff}} its minimal polynomial over $K$ is separable.


Definition:Quadratic Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers. The '''quadratic mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$Q_n := \displaystyle \sqrt {\frac 1 n \sum_{k \mathop = 1}^n x_k^2}$


Definition:Automorphism Group

=== Group of Automorphisms === {{:Definition:Automorphism Group/Group}}


Definition:Galois Extension/Finite/Definition 2

$L/K$ is a '''Galois extension''' {{Iff}} it is normal and separable.


Definition:Galois Extension/Finite/Definition 1

$L/K$ is a '''Galois extension''' {{Iff}} the fixed field of its automorphism group is $K$: :$\operatorname{Fix}_L(\operatorname{Gal}(L/K)) = K$


Definition:Frobenius Endomorphism

Let $p$ be a prime number. Let $R$ be a commutative ring with unity of characteristic $p$. The '''Frobenius endomorphism''' is the ring endomorphism $\Frob: R \to R$ defined as: :$\forall x \in R: \map \Frob x = x^p$


Definition:Perfect Field

Let $F$ be a field. === Definition 1 === {{:Definition:Perfect Field/Definition 1}} === Definition 2 === {{:Definition:Perfect Field/Definition 2}}


Definition:Giuga Number

Let $n$ be a positive integer. $n$ is a '''Giuga number''' {{iff}}: :$\displaystyle \sum_{p \mathop \divides n} \frac 1 p - \prod_{p \mathop \divides n} \frac 1 p \in \Z_{\ge 0}$


Definition:Area/Dimension

'''Area''' is a dimension of measurement of physics. The dimension of '''area''' is $L^2$: length squared.


Definition:Imperial/Area

The imperial units of area are based on a system of units in which each unit is the square of a corresponding imperial length unit. The imperial units of area are as follows: === Square Inch === {{:Definition:Imperial/Area/Square Inch}} === Square Foot === {{:Definition:Imperial/Area/Square Foot}} === Square Yard === {{:Definition:Imperial/Area/Square Yard}} === Square Rod, Pole or Perch === {{:Definition:Imperial/Area/Square Rod, Pole or Perch}} === Square Chain === {{:Definition:Imperial/Area/Square Chain}} === Rood === {{:Definition:Imperial/Area/Rood}} === Acre === {{:Definition:Imperial/Area/Acre}} === Square Mile === {{:Definition:Imperial/Area/Square Mile}}


Definition:Imperial/Area/Square Rod, Pole or Perch

One '''square rod, pole or perch''' is equal to a square of side $1$ rod, pole or perch in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square rod, pole or perch''' }} {{eqn | r = 30 \dfrac 1 4 = \paren {5 \dfrac 1 2}^2       | c = square yards }} {{eqn | r = 272 \dfrac 1 4 = \paren {16 \dfrac 1 2}^2       | c = square feet }} {{end-eqn}}


Definition:Jacobi's Equation of Functional

Let: :$(1): \quad \ds \int_a^b \map F {x, y, y'} \rd x$ be a functional such that: :$\map y a = A$ :$\map y b = B$ Let: :$(2): \quad \displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x$ be a quadratic functional such that: :$P = \dfrac 1 2 F_{y'y'}$ :$Q = \dfrac 1 2 \paren {F_{yy} - \dfrac \d {\d x} F_{yy'} }$ Then Euler's equation of functional $(2)$: :$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$ is called '''Jacobi's Equation''' of functional $(1)$. {{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}


Definition:Happy Number

Let $n \in \Z_{>0}$ be a positive integer. Square the digits of $n$ and add the results. Repeat. If this process eventually reaches $1$, then $n$ is a '''happy number'''.


Definition:Variational Equation of Differential Equation

For $n, i \in \N$, let: :$\map F {x, \sequence {\map {y^{\paren i} } x}_{0 \mathop \le i \mathop \le n} } = 0$ where $\sequence {\map {y^{\paren i} } x}_{0 \mathop \le i \mathop \le n}$ is a sequence of derivatives of a function $y$, be a differential equation. Let $\map y x, \map g x$ be real functions which solve the given differential equation, such that :$\map g x = \map y x + \map h x$ Then, neglecting $\map {\mathcal O} {h^2}$, the differential equation satisfied by $h$ is called the '''variational equation''' of the differential equation $F = 0$.


Definition:Fahrenheit

'''Fahrenheit''' is a temperature scale. Its two reference points are: : $32 \fahr$, which is set at the melting point of water. : $212 \fahr$, which is set at the boiling point of water, as defined at sea level and standard atmospheric pressure. A temperature measured in '''Fahrenheit''' is often referred to as so many '''degrees Fahrenheit'''. === Symbol === {{:Definition:Fahrenheit/Symbol}} {{NamedforDef|Daniel Gabriel Fahrenheit|cat = Fahrenheit}}


Definition:Fahrenheit/Symbol

The symbol for the '''degree Fahrenheit''' is $\fahr$.


Definition:Temperature/Scales

There are several scales against which '''temperature''' is measured. Each one has two reference points. {| class="wikitable" style="margin:1em auto 1em auto" |+ Temperature Scales |- ! Name ! Unit symbol ! Absolute Zero ! Melting point of water ! Boiling point of water |- ! Celsius | ${}^\circ \mathrm C$ | $-273.15 \ {}^\circ \mathrm C$ | $0 \ {}^\circ \mathrm C$ | $99.9839 \ {}^\circ \mathrm C$ |- ! Fahrenheit | ${}^\circ \mathrm F$ | $-459.67 \ {}^\circ \mathrm F$ | $32 \ {}^\circ \mathrm F$ | $211.9710 \ {}^\circ \mathrm F$ |- ! Kelvin | $\mathrm K$ | $0 \ \mathrm K$ | $273.15 \ \mathrm K$ | $373.1339 \ \mathrm K$ |- ! [http://en.wikipedia.org/wiki/Rankine_scale Rankine] | ${}^\circ \mathrm R$ | $0 \ {}^\circ \mathrm R$ | $491.67 \ {}^\circ \mathrm R$ | $671.641 \ {}^\circ \mathrm R$ |- |} There are others. As a general rule, only Kelvin is used in physics nowadays. Celsius is usually used in the domestic context, for weather reporting and so on, in most nations, and sometimes seen in the teaching of physics, but usually at the most elementary levels in schools. Fahrenheit is still used as the official temperature scale only in the US and Belize, although can still be seen on occasion in the contexts of weather reporting and health monitoring in the UK. The [http://en.wikipedia.org/wiki/Rankine_scale Rankine scale] is used in a few specialist engineering applications in the US and Canada.


Definition:Multiplicatively Perfect Number

Let $n \in \Z_{>0}$ be a positive integer. $n$ is '''multiplicatively perfect''' {{iff}} the product of the divisors of $n$ is equal to $n^2$.


Definition:Order of Group Element/Definition 3

The '''order of $x$ (in $G$)''', denoted $\left\vert{x}\right\vert$, is the largest $k \in \Z_{\gt 0}$ such that: :$\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$


Definition:Inclusion Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set. $\left({S, \preceq}\right)$ is '''inclusion ordered set''' {{iff}} :$\mathord\preceq = \mathord\subseteq \cap \left({S \times S}\right)$ That means, :$\forall x, y \in S: x \preceq y \iff x \subseteq y$


Definition:Unit Square

A '''unit square''' is a square of side of unit length.


Definition:Unit Length

A '''unit length''' is a straight line segment of arbitrary length can defined as having a length of $1$ ('''one'''). During the course of the exposition arising from the definition a '''unit length''' $L$, all subsequent lengths can then be expressed as multiples of $L$. Category:Definitions/Geometry 4z1h1wptocuc7qcfs0wp4yp6laxaf55


Definition:Polyomino

A '''polyomino''' is a configuration of $n$ unit squares, for a given (strictly) positive integer $n$, which are placed side by side with vertices touching, to form a plane figure.


Definition:Monomino

A '''monomino''' is a $1$-omino: :100px That is, it is a polyomino consisting of exactly $1$ unit square.


Definition:Domino

A '''domino''' is a $2$-omino: :200px That is, it is a polyomino consisting of exactly $2$ unit squares.


Definition:Tromino

A '''tromino''' is a $3$-omino: :500px That is, it is a polyomino consisting of exactly $3$ unit squares.


Definition:Pascal's Triangle/Column

Each of the vertical lines of numbers headed by a given $\dbinom n m$ is known as the '''$m$th column''' of Pascal's triangle. Hence the leftmost '''column''', containing all $1$s, is identified as the '''zeroth column''', or '''column $0$'''.


Definition:Pascal's Triangle/Row

Each of the horizontal lines of numbers corresponding to a given $n$ is known as the '''$n$th row''' of Pascal's triangle. Hence the top '''row''', containing a single $1$, is identified as the '''zeroth row''', or '''row $0$'''.


Definition:Pascal's Triangle/Diagonal

The $n$th '''diagonal''' of Pascal's triangle consists of the entries $\dbinom {n + m} m$ for $m \ge 0$: :$\dbinom n 0, \dbinom {n + 1} 1, \dbinom {n + 2} 2, \dbinom {n + 3} 3, \ldots$ Hence the '''diagonal''' leading down and to the right from $\dbinom 0 0$, containing all $1$s, is identified as the '''zeroth diagonal''', or '''diagonal $0$'''.


Definition:Pascal's Triangle/Lesser Diagonal

The $n$th '''lesser diagonal''' of Pascal's triangle consists of the entries $\dbinom {n - m} m$ for $m \ge 0$, leading up and to the right from the entry in row $n$ and column $0$: :$\dbinom n 0, \dbinom {n - 1} 1, \dbinom {n - 2} 2, \dbinom {n - 3} 3, \ldots$


Definition:Pascal's Triangle/Order of Numbers

The entries in column $n$ can be referred to as '''numbers of the $n$th order (of Pascal's triangle)''', or '''$n$th order numbers'''.


Definition:Pascal's Triangle/Edge Cells

The numbers in column $0$ and in diagonal $0$, containing all $1$s, are referred to as the '''edge cells'''.


Definition:Simplicial Polytopic Number

The '''simplicial polytopic numbers''' are those denumerating a collection of objects which can be arranged in the form of a simplex.


Definition:Harmonic Sequence

A '''harmonic sequence''' is a sequence $\sequence {a_k}$ in $\R$ defined as: :$h_k = \dfrac 1 {a + k d}$ where: :$k \in \set {0, 1, 2, \ldots}$ :$-\dfrac a d \notin \set {0, 1, 2, \ldots}$ Thus its general form is: :$\dfrac 1 a, \dfrac 1 {a + d}, \dfrac 1 {a + 2 d}, \dfrac 1 {a + 3 d}, \ldots$


Definition:Leibniz Harmonic Triangle

:$\begin{array}{r|rrrrrr} n & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0  & \frac 1 1 \\ 1  & \frac 1 2 & \frac 1 2 \\ 2  & \frac 1 3 & \frac 1 6 &  \frac 1 3 \\ 3  & \frac 1 4 & \frac 1 {12} & \frac 1 {12} & \frac 1 4 \\ 4  & \frac 1 5 & \frac 1 {20} & \frac 1 {30} & \frac 1 {20} & \frac 1 5 \\ 5  & \frac 1 6 & \frac 1 {30} & \frac 1 {60} & \frac 1 {60} & \frac 1 {30} & \frac 1 6 \\ \end{array}$


Definition:Leibniz Harmonic Triangle/Row

Each of the horizontal lines of numbers corresponding to a given $n$ is known as the '''$n$th row''' of Leibniz harmonic triangle. Hence the top '''row''', containing a single $1$, is identified as the '''zeroth row''', or '''row $0$'''.


Definition:Leibniz Harmonic Triangle/Column

Each of the vertical lines of numbers is known as the '''$m$th column''' of Leibniz harmonic triangle. The leftmost '''column''', containing the reciprocals of the non-negative integers, is identified as the '''zeroth column''', or '''column $0$'''.


Definition:Leibniz Harmonic Triangle/Diagonal

The $n$th '''diagonal''' of Leibniz harmonic triangle consists of the entries in row $n + m$ and column $m$ for $m \ge 0$: :$\left({n, 0}\right), \left({n + 1, 1}\right), \left({n + 2, 2}\right), \ldots$ Hence the '''diagonal''' leading down and to the right from $\left({0, 0}\right)$, containing  the reciprocals of the non-negative integers, is identified as the '''zeroth diagonal''', or '''diagonal $0$'''.


Definition:Knot (Knot Theory)/Sphere Knot

A '''knotted $n$-sphere''' is a knotted embedding: : $\phi: \Bbb S^n \to \R^{n + 2}$


Definition:Knot (Knot Theory)/Elementary Knot

Circle knots can often be quite wild and unwieldy - most of modern knot theory concerns itself with a specific kind of knot. These knots are described as a finite set of points in $\R^3$ called $\left\{{ x_1, x_2, \dots, x_n }\right\}$, together with line segments from $x_i$ to $x_{i+1}$ and a line segment from $x_n$ to $x_1$. The union of all these line segments is clearly a circle knot, or an '''unknot''', an embedding of the circle which is homotopic to a circle. {{explain|Rewrite in a rigorous dictionary style so as to make it understandable independently of the parent page.}}


Definition:Centered Hexagonal Number/Linguistic Note

A '''centered hexagonal number''' is rendered in British English as '''centred hexagonal number''', on account of the British spelling of '''centre'''.


Definition:Scott Sigma

Let $L = \left({S, \preceq}\right)$ be an ordered set. '''Scott sigma''' $\sigma\left({L}\right)$ is a set of subsets of $S$ and equals to :$\left\{ {U \subseteq S: U}\right.$ is upper and has property (S)$\left.\right\}$ {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}


Definition:Jointly Scott Continuous

Let $\struct{S, \preceq}$ be an ordered set. Let $f:S \times S \to S$ be a mapping. $f$ is '''jointly Scott continuous''' {{iff}} :for every relational structure with Scott topology $\struct{S, \preceq, \tau}$ :for every topological space $T = \struct{S, \tau}$: $f$ is continuous as a mapping from $T \times T$ into $T$. where $T \times T$ denotes the product space. {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}


Definition:Euler Lucky Number/Definition 1

The '''Euler lucky numbers''' are the prime numbers $p$ such that: :$n^2 + n + p$ is prime for $0 \le n < p - 1$.


Definition:Euler Lucky Number/Definition 2

The '''Euler lucky numbers''' are the prime numbers $p$ such that: :$n^2 - n + p$ is prime for $1 \le n < p$.


Definition:Twice Differentiable/Functional/Dependent on N functions

Let $\Delta J \sqbrk {\mathbf y; \mathbf h}$ be an increment of a functional, where $\mathbf y = \paren {\sequence {y_i}_{1 \mathop \le i \mathop \le N} }$ is a vector. Let: :$\Delta J \sqbrk {\mathbf y; \mathbf h} = \phi_1 \sqbrk {\mathbf y; \mathbf h} + \phi_2 \sqbrk {\mathbf y; \mathbf h} + \epsilon \size {\mathbf h}^2$ where: :$(1): \quad \phi_1 \sqbrk {\mathbf y; \mathbf h}$ is a linear functional :$(2): \quad \phi_2 \sqbrk {\mathbf y; \mathbf h}$ is a quadratic functional {{WRT}} $\mathbf h$ :$(3): \quad \displaystyle \size {\mathbf h} = \sum_{i \mathop = 1}^N \size {h_i}_1 = \sum_{i \mathop = 1}^N \max_{a \mathop \le x \mathop \le b} \set {\size {\map {h_i} x} + \size {\map {h_i'} x} }$ :$(4): \quad \epsilon \to 0$ as $\size {\mathbf h} \to 0$. Then the functional $J \sqbrk {\mathbf y}$ is '''twice differentiable'''. {{refactor|Extract the following two into their own definition pages, appropriately named and transcluded.}} The linear part $\phi_1$ is the '''first variation''', denoted: :$\delta J \sqbrk {\mathbf y; \mathbf h}$ $\phi_2$ is called the '''second variation''' (or '''differential''') of a functional, and is denoted: :$\delta^2 J \sqbrk {\mathbf y; \mathbf h}$


Definition:Class Mapping

Let $S$ and $T$ be classes. A '''class mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a class relation $f = \mathcal R \subseteq S \times T$ such that: :$\forall x \in S: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$ and :$\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$


Definition:Class Injection

Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class injection''' if and only if: :$\forall x_1, x_2 \in A: f \left({x_1}\right) = f \left({x_2}\right) \implies x_1 = x_2$


Definition:Class Surjection

Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class surjection''' if and only if: :$\forall y \in B: \exists x \in A: f \left({x}\right) = y$


Definition:Class Bijection

Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class bijection''' if and only if: :$f$ is both a class injection and a class surjection. Category:Definitions/Class Mappings pddqb0wn23p9f221zl7n4nj8v1wwasw


Definition:Greatest Common Divisor/Polynomial Ring over Field

Let $F$ be a field. Let $P, Q, R \in F \left[{X}\right]$ be polynomials. Then $R$ is '''the greatest common divisor''' of $P$ and $Q$ {{iff}} it is a monic greatest common divisor. This is denoted $\gcd \left({P, Q}\right) = R$.


Definition:Euclidean Number Theory

'''Euclidean number theory''' concerns aspects of number theory and real analysis as presented in {{ElementsLink}}. Particular attention is paid to certain properties of the irrational numbers.


Definition:Göbel's Sequence

'''Göbel's sequence''' is the sequence defined recursively as: :$x_n = \begin{cases} 1 & : n = 0 \\ \displaystyle \left({1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2}\right) / n & : n > 0 \end{cases}$


Definition:Göbel's Sequence/Sequence

'''Göbel's sequence''' begins: :$1, 2, 3, 5, 10, 28, 154, 3520, 1 \, 551 \, 880, 267 \, 593 \, 772 \, 160, \ldots$


Definition:Göbel's Sequence/Historical Note

Some sources link Göbel's sequence with the name of {{AuthorRef|Michael Somos}}, but it appears that the latter has a different sequence named for him.


Definition:Göbel's Sequence/General/Examples/3

The '''$3$-Göbel sequence''' begins: :$1, 2, 5, 45, 22 \, 815, 2 \, 375 \, 152 \, 056 \, 927, \ldots$


Definition:Class Relation

Let $S$ and $T$ be classes. Let $S \times T$ be the Cartesian Product of $S$ and $T$. Then a '''relation''' on $S \times T$ is defined as a subclass of $S \times T$: :$\mathcal R \subseteq S \times T$ Category:Definitions/Class Theory igju4mpaz3swp08ql307vsp0po3to1b


Definition:Conjugate Point/Dependent on N Functions

Let $K$ be a functional such that: :$\ds K \sqbrk h = \int_a^b \paren {\mathbf h'\mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ Consider Euler's equation related to the functional $K$: :$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = 0$ where $\mathbf P$ and $\mathbf Q$ are symmetric matrices. Let the general solution to this equation be: :$\set {\mathbf h^{\paren i} = \paren {\sequence {h_{ij} } }: i,j \in \N_{\le N} }$ Let: :$\exists j: \forall k \ne j: \paren {\map {\mathbf h^{\paren j} } a = 0} \land \paren {\map {h_{j j}'} a = 1, h'_{j k} = 0}$ Let the determinant, built from $h_{ij}$, be such that: :$\size {h_{i j} } \paren {\tilde a} = 0$ Here $i$ denotes rows, and $j$ denotes columns. Then $\tilde a$ is said to be '''conjugate''' to point $a$ {{WRT}} the functional $K$.


Definition:Derangement/Historical Note

The number of a derangements of a finite set was first investigated by {{AuthorRef|Nicolaus I Bernoulli}} and {{AuthorRef|Pierre Raymond de Montmort}} between about $1708$ and $1713$. They solved it at around the same time. The question is often couched in the idea of counting the number of ways of placing letters at random in envelopes such that nobody receives the correct letter.


Definition:Kaprekar Triple

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Suppose that $n^3$, when expressed in number base $b$, can be split into three parts that add up to $n$. Then $n$ is a '''Kaprekar triple''' for base $b$.


Definition:Kaprekar Triple/Sequence

The sequence of Kaprekar triples begins: :$1, 8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, \ldots$


Definition:Hexamorphic Number

A '''hexamorphic number''' is a hexagonal number $H_n$ whose decimal representation ends in $n$.


Definition:Hexamorphic Number/Sequence

The sequence of hexamorphic numbers, for $n \in \Z_{\ge 0}$, begins: {{begin-eqn}} {{eqn | l = H_1       | r = 1 }} {{eqn | l = H_5       | r = 45 }} {{eqn | l = H_6       | r = 66 }} {{eqn | l = H_{25}       | r = 1225 }} {{eqn | l = H_{26}       | r = 1326 }} {{eqn | l = H_{50}       | r = 4950 }} {{end-eqn}}


Definition:Uniform Absolute Convergence

Let $S$ be a set. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges uniformly absolutely''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges uniformly.


Definition:Locally Uniform Convergence

=== General Definition === {{:Definition:Locally Uniform Convergence/General Definition}} === Complex Functions === {{:Definition:Locally Uniform Convergence/Complex Functions}} === Series === {{:Definition:Locally Uniform Convergence/Series}}


Definition:Locally Uniform Convergence/General Definition

Let $X$ be a topological space. Let $M$ be a metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n: X \to M$. Then $f_n$ '''converges locally uniformly''' to $f: X \to M$ if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.


Definition:Compact Convergence

Let $X$ be a topological space. Let $M$ be a metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n : X \to M$. Let $f: X \to M$ be a mapping. Then $f_n$ '''converges compactly''' to $f$ {{iff}} $f_n$ converges uniformly to $f$ on every compact subset of $X$.


Definition:Amicable Pair

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Pair/Definition 1}} === Definition 2 === {{:Definition:Amicable Pair/Definition 2}} === Definition 3 === {{:Definition:Amicable Pair/Definition 3}}


Definition:Quasiamicable Numbers

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Quasiamicable Numbers/Definition 1}} === Definition 2 === {{:Definition:Quasiamicable Numbers/Definition 2}}


Definition:Uniform Convergence of Product/Compact Space

Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a complete valued field. Let $\sequence{f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges uniformly''' {{Iff}} there exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly and is nonzero.


Definition:Locally Uniform Absolute Convergence

Let $X$ be a topological space. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: X \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly absolutely''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges locally uniformly.


Definition:Compact Absolute Convergence

Let $X$ be a topological space. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: X \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges compactly (uniformly) absolutely''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges compactly.


Definition:Local Uniform Absolute Convergence of Product

Let $X$ be a topological space. Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to \mathbb K$. Then the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly absolutely''' {{iff}} every point of $X$ has a neighborhood on which it converges uniformly absolutely.


Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\sequence {a_n}$ be a sequence in $\mathbb K$. === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}


Definition:Absolute Convergence of Product/General Definition/Definition 2

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + a_n}$ is '''absolutely convergent''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.


Definition:Quasiamicable Numbers/Definition 1

$m$ and $n$ are '''quasiamicable numbers''' {{iff}}: :the sum of the proper divisors of $m$ is equal to $n$ and: :the sum of the proper divisors of $n$ is equal to $m$.


Definition:Quasiamicable Numbers/Definition 2

$m$ and $n$ are '''quasiamicable numbers''' {{iff}}: :$\sigma \left({m}\right) = \sigma \left({n}\right) = m + n + 1$ where $\sigma \left({m}\right)$ denotes the $\sigma$ function.


Definition:Absolute Convergence of Product/Complex Numbers/Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is '''absolutely convergent''' {{iff}} $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\vert{a_n}\right\vert}\right)$ is convergent.


Definition:Absolute Convergence of Product/Complex Numbers/Definition 2

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is '''absolutely convergent''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.


Definition:Absolute Convergence of Product/Complex Numbers/Definition 3

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is '''absolutely convergent''' {{iff}} there exists $n_0 \in \N$ such that: :$a_n \ne -1$ for $n > n_0$ :The series $\displaystyle \sum_{n \mathop = n_0 + 1}^\infty \log \paren {1 + a_n}$ is absolutely convergent where $\log$ denotes the complex logarithm.


Definition:Trimorphic Number

An '''trimorphic number''' is a positive integer whose cube ends in that number.


Definition:Divergent Product/Divergence to Zero

If either: :there exist infinitely many $n \in \N$ with $a_n = 0$ :there exists $n_0 \in \N$ with $a_n \ne 0$ for all $n > n_0$ and the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$ the product '''diverges to $0$''', and we assign the value: :$\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$


Definition:Cauchy's Criterion for Products

Let $\struct {\mathbb K, \norm {\,\cdot\,}}$ be a valued field. Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ satisfies '''Cauchy's criterion''' {{Iff}}: :$\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m\geq n \ge N: \norm {\prod_{k \mathop = n}^m a_n - 1} < \epsilon$


Definition:Cauchy Sequence/Cauchy Criterion

The '''Cauchy criterion''' is the condition: :For any (strictly) positive real number $\epsilon \in \R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$. Informally: :For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked. Or to put it another way, the terms get arbitrarily close together the farther out you go.


Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \to \mathbb K$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2}}


Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely''' {{iff}} the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty(1+ \norm{f_n})$ converges uniformly.


Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely''' {{iff}} the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly absolutely.


Definition:Uniform Absolute Convergence of Product/Compact Space

{{refactor|integrate this in the other lists of definitions}} Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \to \mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely''' {{iff}} $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \norm{f_n}}\right)$ converges uniformly.


Let $X$ be a set. Let $\left \langle {f_n} \right \rangle$ be a sequence of bounded mappings $f_n: X \to \C$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2}} === Definition 3 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3}}


Let $T = \left({S, \tau}\right)$ be a weakly locally compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left \langle {f_n} \right \rangle$ be a sequence of locally bounded mappings $f_n: S \to \mathbb K$. === Definition 1 === {{:Definition:Locally Uniform Convergence of Product/Definition 1}} === Definition 2 === {{:Definition:Locally Uniform Convergence of Product/Definition 2}}


Definition:Locally Uniform Convergence of Product/Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly''' {{iff}} every point of $T$ has a compact neighborhood on which it converges uniformly.


Definition:Locally Uniform Convergence of Product/Definition 2

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly''' {{iff}} it converges uniformly on every compact subspace of $T$.


Definition:Normed Division Ring

Let $\struct {R, +, \circ}$ be a division ring. Let $\norm {\,\cdot\,}$ be a norm on $R$. Then $\struct {R, \norm{\,\cdot\,} }$ is a '''normed division ring'''.


Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely''' {{iff}} the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\vert{f_n}\right\vert}\right)$ converges uniformly.


Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely''' {{iff}} there exists $n_0 \in \N$ such that: :$(1): \quad f_n \left({x}\right) \ne -1$ for $n \ge n_0$ and $x \in X$ and: :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 + f_n}\right)$ is uniformly absolutely convergent.


Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2


Definition:Locally Uniform Convergence/Series

Let $X$ be a topological space. Let $V$ be a normed vector space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n:X\to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly''' {{iff}} the sequence of partial sums converges locally uniformly.


Definition:Order of Entire Function/Definition 2

Let $f$ be not identically zero. The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which: :$\displaystyle \map \ln {\max_{\size z \mathop \le R} \size {\map f z} } = \map \OO {R^\beta}$ or $\infty$ if no such $\beta$ exists, where $\OO$ denotes big-O notation The '''order''' of $0$ is $0$.


Definition:Order of Entire Function/Definition 3

Let $f$ be non-constant. The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the limit superior: :$\displaystyle \limsup_{R \mathop \to \infty} \frac {\displaystyle \ln \ln \max_{\cmod z \le R} \cmod f} {\ln R}$ The '''order''' of a constant function is $0$.


Definition:Order of Entire Function/Definition 1

The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which: :$\map f z = \map \OO {\map \exp {\size z^\beta} }$ or $\infty$ if no such $\beta$ exists, where $\OO$ denotes big-O notation.


Definition:Jacobi's Equation of Functional/Dependent on N Functions

Let: :$\ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ be a (real) functional, where $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$. Let: :$\ds \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ be a quadratic functional, where: :$P_{ij} = \dfrac 1 2 F_{y_i'y_j'}$ :$Q_{ij} = \dfrac 1 2 \paren {F_{y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$ Then the Euler's equation of the latter functional: :$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = \mathbf 0$ is called '''Jacobi's Equation''' of the former functional. {{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}


Definition:Exponent of Convergence

Let $\sequence {a_n}$ be a sequence of nonzero complex numbers. The '''exponent of convergence''' of $\sequence {a_n}$ is the infimum of $\tau \ge 0$ for which the series $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}^{-\tau}$ converges. The '''exponent of convergence''' of a finite sequence is $0$.


Definition:Rank of Entire Function

Let $f: \C \to \C$ be an entire function. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''rank''' of $f$ is the smallest integer $p\geq0$ for which the series $\displaystyle\sum_{n\mathop=1}^\infty |a_n|^{-p-1}$ converges, or $\infty$ if there is no such integer. If $f$ has finitely many zeroes, its '''rank''' is $0$.


Definition:Hadamard's Canonical Factorization

Let $f: \C \to \C$ be a nonzero entire function of finite rank $p \in \N$. Let $0$ be a zero of $f$ of multiplicity $m\geq0$. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''canonical representation''' of $f$ is: :$\displaystyle f \left({z}\right) = z^m e^{g \left({z}\right)} \prod_{n \mathop = 1}^\infty E_p\left({\frac z {a_n} }\right)$ where: :$g: \C \to \C$ is an entire function :$E_p$ denotes the $p$th Weierstrass elementary factor. If $f$ has finitely many zeroes, the product is understood to be finite. {{NamedforDef|Jacques Salomon Hadamard|cat = Hadamard}}


Definition:Pentagonal Pyramidal Number

'''Pentagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a pentagonal pyramid.


Definition:Hexagonal Pyramidal Number

'''Hexagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a hexagonal pyramid.


A '''cubic recurring digital invariant''' is a recurring digital invariant of order $3$. === Sequence === {{:Definition:Cubic Recurring Digital Invariant/Sequence}}


Definition:Mutually Consistent Boundary Conditions

Let $\map {\mathbf y} x$, $\map {\boldsymbol \psi} {\mathbf y}$ be an N-dimensional vectors. Consider the system of differential equations: :$(1): \quad \mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$ Let derivatives of $\mathbf y$ satisfy: :$\bigintlimits {\mathbf y'} {x \mathop = x_1} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1} {}$ :$\bigintlimits {\mathbf y'} {x \mathop = x_2} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2} {}$ If every solution of $(1)$ satisfying conditions at $x = x_1$ automatically satisfies conditions at $x = x_2$ (or vice versa), then these boundary conditions are called '''mutually consistent'''.


Definition:Recurring Digital Invariant

Let $k \in \Z_{>0}$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$\forall m \in \Z_{>0}: f \left({m}\right) = $ the sum of the $k$th powers of the digits of $n$. Let $n_0 \in \Z_{>0}$ be a positive integer. Consider the sequence: :$s_n = \begin{cases} n_0 & : n = 0 \\ f \left({s_{n - 1} }\right) & : n > 0 \end{cases}$ If: :$\exists r \in \N_{>0}: s_r = n_0$ then the smallest of the terms $n_0, n_1, \ldots, n_r$ is a '''recurring digital invariant of order $k$'''.


Definition:Separated Sets/Definition 2

$A$ and $B$ are '''separated (in $T$)''' {{iff}} there exist $U,V\in\tau$ with: :$A\subset U$ and $U\cap B = \varnothing$ :$B\subset V$ and $V\cap A = \varnothing$ where $\varnothing$ denotes the empty set.


Definition:Separated Sets/Definition 1

$A$ and $B$ are '''separated (in $T$)''' {{iff}}: :$A^- \cap B = A \cap B^- = \O$ where: :$A^-$ denotes the closure of $A$ in $T$ :$\O$ denotes the empty set.


Definition:Logarithmic Mean Value

Let $f: \N \to \C$ be an arithmetic function. The '''logarithmic mean value''' of $f$ is the limit: :$L \left({f}\right) = \displaystyle \lim_{x \mathop \to \infty} \frac 1 {\ln x} \sum_{n \mathop \le x} \frac {f \left({n}\right)} n$ if it exists.


Definition:Ordinary Mean Value

Let $f: \N \to \C$ be an arithmetic function. The '''(ordinary) mean value''' of $f$ is the limit: :$M \left({f}\right) = \displaystyle \lim_{x \mathop \to \infty} \frac 1 x \sum_{n \mathop \le x} f \left({n}\right)$ if it exists.


Definition:Summatory Function

Let $a: \N \to \C$ be an arithmetic function. Its '''summatory function''' is the mapping: :$A \left({x}\right) = \displaystyle\sum_{n \mathop \le x} a \left({n}\right)$ defined for $x \in \Z_{\ge 0}$.


Definition:Dirichlet Convolution/Definition 1

The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\displaystyle \left({f * g}\right) \left({n}\right) := \sum_{d \mathop \backslash n} f \left({d}\right) g \left({\frac n d}\right)$ where the summation runs over the set of positive divisors $d$ of $n$.


Definition:Dirichlet Convolution/Definition 2

The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\displaystyle \left({f * g}\right) \left({n}\right) := \sum_{a b \mathop = n} f \left({a}\right) g \left({b}\right)$ where the summation runs over all pairs of positive integers $\left({a, b}\right)$ with $a b = n$.


Definition:Pentatope Number

'''Pentatope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular pentatope.


Definition:Polytope Number

'''Polytope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular polytope.


Definition:Polytope Number/Examples

=== Pentatope Number === {{:Definition:Pentatope Number}}


Definition:Convergent Product/Number Field/Nonzero Sequence

Let $\sequence{a_n}$ be a sequence of nonzero elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}} its sequence of partial products converges to a nonzero limit $a\in\mathbb K\setminus\{0\}$.


Definition:Convergent Product/Number Field/Arbitrary Sequence

Let $\sequence{a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}}: * There exists $n_0\in\N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b\in\mathbb K\setminus\{0\}$. The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a\in\mathbb K$.


Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence

Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}}: : There exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b \in \mathbb K \setminus \set 0$. The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a \in \mathbb K$.


Definition:Convergent Product/Arbitrary Field/Nonzero Sequence

Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}} its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.


Definition:Group of Permutation Matrices

The '''group of permutation matrices''' of order $n$ is the set of all $n \times n$ permutation matrices with multiplication of matrices. {{Disambiguate|Definition:Order}}


Definition:Change of Basis Matrix/Definition 1

The '''matrix of change of basis from $A$ to $B$''' is the matrix whose columns are the coordinate vectors of the elements of the '''new basis''' $\left \langle {b_n} \right \rangle$ relative to the '''original basis''' $\left \langle {a_n} \right \rangle$.


Definition:Change of Basis Matrix/Definition 2

Let $I_G$ be the identity linear operator on $G$. Let $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ be the matrix of $I_G$ relative to $\left \langle {b_n} \right \rangle$ and $\left \langle {a_n} \right \rangle$. Then $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ is called the '''matrix corresponding to the change of basis from $\left \langle {a_n} \right \rangle$ to $\left \langle {b_n} \right \rangle$'''.


Definition:Basis of Module

Let $R$ be a ring with unity. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Definition 1 === {{:Definition:Basis of Module/Definition 1}} === Definition 2 === {{:Definition:Basis of Module/Definition 2}}


Definition:Basis of Vector Space/Definition 1

A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$.


Definition:Basis of Vector Space/Definition 2

A '''basis''' is a maximal linearly independent subset of $G$.


Definition:Basis of Vector Space

Let $K$ be a division ring. Let $\struct {G, +_G, \circ}_R$ be a vector space over $K$. === Definition 1 === {{:Definition:Basis of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Basis of Vector Space/Definition 2}}


Definition:Field of Directions

Consider the following system of differential equations: :$(1): \quad \mathbf y' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$ where $\mathbf y$ is an $n$-dimensional vector. Let the boundary conditions be prescribed $\forall x \in \closedint a b$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let these boundary conditions be consistent for all $x_1, x_2 \in \closedint a b$. Then the family of mutually consistent boundary conditions is called a '''field of directions''' for the given system $(1)$. That is, the first-order system is valid in an interval instead of a countable set of points.


Definition:Angle/Unit/Minute

The '''minute (of arc)''' is a measurement of plane angles, symbolized by $'$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''minute''' }} {{eqn | r = 60       | c = seconds }} {{eqn | r = \dfrac 1 {60}       | c = degree of arc (by definition) }} {{end-eqn}}


Definition:Angle/Unit/Second

The '''second (of arc)''' is a measurement of plane angles, symbolized by $''$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''second''' }} {{eqn | r = \dfrac 1 {60}       | c = minute of arc (by definition) }} {{eqn | r = \dfrac 1 {60 \times 60} = \dfrac 1 {3600}       | c = degree of arc }} {{end-eqn}}


Definition:Second

=== Second of Time === {{:Definition:Second of Time}} === Second of Arc === {{:Definition:Second of Arc}}


Definition:Minute

=== Minute of Time === {{:Definition:Minute of Time}} === Minute of Arc === {{:Definition:Minute of Arc}}


Definition:Splitting Polynomial

Let $K$ be a field. Let $P \in K \sqbrk X$ be a polynomial. === Definition 1 === The polynomial $P$ '''splits (completely)''' in $K$ {{iff}} it is a product of polynomials of degree $1$. === Definition 2 === The polynomial $P$ '''splits (completely)''' in $K$ {{iff}} every irreducible factor of $P$ has degree $1$.


Definition:Self-Adjoint Boundary Conditions

Consider the functional $J \sqbrk {\mathbf y}$, such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let the momenta of $J$ be: :$\mathbf p = \nabla_{\mathbf y'} \map F {x, \mathbf y, \mathbf y'}$ Let the following boundary conditions hold: :$\map {\mathbf y'} a = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$ If: :$\exists \map g {x, \mathbf y}: \bigvalueat {\map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {x \mathop = a} = \bigvalueat {\nabla_{\mathbf y'} \map g {x, \mathbf y} } {x \mathop = a}$ then the boundary conditions are called '''self-adjoint'''.


Definition:Unitary Group

Let $n$ be a positive integer. The '''unitary group''' $\operatorname{U}_n$ is the group of all $n\times n$ unitary matrices. === Unitary Group of Hilbert Space === Let $H$ be a Hilbert space. Its '''unitary group''' is the group consisting of its unitary operators.


Definition:Special Unitary Group

Let $n$ be a positive integer. The '''special unitary group''' $\operatorname{SU}_n$ is the group of all $n\times n$ unitary matrices with determinant $1$: :$\operatorname{SU}_n = \operatorname{U}_n \cap \operatorname{SL}_n(\C)$


Definition:Standard Basis Matrix

Let $R$ be a ring with unity. Let $m, n \ge 1$ be positive integers. Let $i, j \in \set {1, \ldots, m} \times \set {1, \ldots, n}$. The $\tuple {i, j}$th '''standard basis matrix''' is the $m \times n$ matrix which is $0$ everywhere except a $1$ at the $\tuple {i, j}$th indices.


Definition:Coordinate Vector

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of dimension $n$. Let $B = \left \langle {b_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $M$. Let $x\in M$. If $\lambda_1,\ldots,\lambda_n\in R$ are such that $x = \displaystyle\sum_{i=1}^n \lambda_i b_i$, then $(\lambda_1,\ldots,\lambda_n)^\intercal \in R^n$ is the '''coordinate vector''' of $x$ with respect to $B$. This can be denoted: $[x]_B$.


Definition:Highly Composite Number

Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''highly composite''' {{iff}}: :$\forall m \in \Z_{>0}, m < n: \map \tau m < \map \tau n$ where $\map \tau n$ is the $\tau$ function of $n$.


Definition:Highly Composite Number/Sequence

The sequence of highly composite numbers begins: :$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, \ldots$


Let $R$ be a ring. Let $M$ be an $R$-module. Let $(M_i)_{i\in I}$ be a family of submodules. === Definition 1 === {{:Definition:Internal Direct Sum of Modules/Definition 1}} === Definition 2 === {{:Definition:Internal Direct Sum of Modules/Definition 2}} === Definition 3 === {{:Definition:Internal Direct Sum of Modules/Definition 3}}


Definition:Seventh Power

A '''seventh power''' is an integer which can be expressed as the $7$th power of an integer.


Definition:Internal Direct Sum of Modules/Definition 1

$M$ is the '''internal direct sum''' of $(M_i)_{i\in I}$ {{Iff}} every $m\in M$ can be written uniquely as a sum $\sum m_i$ with each $m_i\in M_i$.


Definition:Internal Direct Sum of Modules/Definition 2

$M$ is the '''internal direct sum''' of $(M_i)_{i\in I}$ {{Iff}}: * $\displaystyle\bigcup_{i\in I}M_i$ generates $M$ * For all $i\in I$, $M_i\cap \displaystyle\sum_{j\neq i} M_j = \{0\}$


Definition:Internal Direct Sum of Modules/Definition 3

Let $\displaystyle \bigoplus_{i \mathop \in I} M_i$ be the external direct sum of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$. $M$ is the '''internal direct sum''' of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$ {{iff}} the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$.


Definition:Basis of Module/Definition 1

A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$.


Definition:Basis of Module/Definition 2

Let $\mathcal B = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$. Let $\Psi: R^{\paren I} \to M$ be the homomorphism given by Universal Property of Free Module on Set. Then $\mathcal B$ is a '''basis''' {{Iff}} $\Psi$ is an isomorphism.


Definition:Relative Matrix of Bilinear Form

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal B = \left \langle {b_m} \right \rangle$ be an ordered basis of $M$. Let $f : M\times M\to R$ be a bilinear form. The '''matrix of $f$ relative to $\mathcal B$''' is the $n\times n$ matrix $\mathbf M_{f, \mathcal B}$ where: :$\displaystyle \forall \left({i, j}\right) \in \left[{1 \,.\,.\, n}\right] \times \left[{1 \,.\,.\, n}\right] : (\mathbf M_{f, \mathcal B})_{ij} = f \left({b_i, b_j}\right)$


Definition:Mutually Consistent Boundary Conditions/wrt Functional

Let $J$ be a (real) functional, such that: :$\displaystyle J = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where its Euler's equations are: :$\nabla_{\mathbf y'} F - \dfrac \d {\d x} \nabla_{\mathbf y} F = 0$ Consider the following boundary conditions: :$\bigvalueat {\mathbf y} {x \mathop = x_1} = \bigvalueat {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1}$ :$\bigvalueat {\mathbf y} {x \mathop = x_2} = \bigvalueat {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2}$ If they are consistent {{WRT}} the Euler equations, then these boundary conditions are called '''mutually consistent''' {{WRT}} the functional $J$.


Definition:Endomorphism Ring of Module

Let $R$ be a ring. Let $M$ be an $R$-module. The '''endomorphism ring''' $\operatorname{End}_R(M)$ is the ring of all endomorphisms of $M$ where: : Multiplication is composition of mappings : Addition is point-wise


Definition:Module Endomorphism

Let $R$ be a ring. Let $M$ be an $R$-module. An '''module endomorphism''' of $M$ is a module homomorphism from $M$ to itself.


Definition:Associated Quadratic Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b : V\times V \to \mathbb K$ be a bilinear form. The '''quadratic form associated to $b$''' is the quadratic form: :$q : V \to \mathbb K : x \mapsto b(x, x)$


Definition:Associated Bilinear Form

Let $\mathbb K$ be a field of characteristic $\operatorname{char} \mathbb K\neq2$. Let $V$ be a vector space over $\mathbb K$. Let $q : V \to \mathbb K$ be a quadratic form. The '''bilinear form associated to $q$''' is the bilinear form: :$b : V \times V \to \mathbb K : (v,w) \mapsto \frac12\left( q(v+w) - q(v) - q(w) \right)$


Definition:Matrix Congruence

Let $R$ be a commutative ring with unity. Let $n$ be a positive integer. Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$ over $R$. Then $\mathbf A$ and $\mathbf B$ are '''congruent''' {{Iff}} there exists an invertible matrix $\mathbf P\in R^{n\times n}$ such that $\mathbf B = \mathbf P^\intercal \mathbf A \mathbf P$.


Definition:Symmetric Bilinear Form

Let $R$ be a ring Let $M$ be an $R$-module. Let $b: M \times M \to R$ be a bilinear form. Then $b$ is '''symmetric''' {{iff}}: :$\forall v, w, \in M: b \left({v, w}\right) = b \left({w, v}\right)$


Definition:Represented by Quadratic Form

Let $R$ be an integral domain. Let $M$ be an $R$-module. Let $q : M \to R$ be a quadratic form. Let $a\in R$. Then '''$q$ represents $a$ (over $R$)''' {{Iff}} there exists $x\in M\setminus\{0\}$ with $q(x) = a$.


Definition:Isotropic Quadratic Form

Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq 2$. Let $V$ be a vector space over $\mathbb K$. Let $q : V\times V \mapsto \mathbb K$ be a quadratic form. Then $q$ is '''isotropic''' {{Iff}} it represents $0$. That is: $q(v) = 0$ for some $v\in V\setminus\{0\}$. === Anisotropic Quadratic Form === A quadratic form that is not isotropic is said to be '''anisotropic'''.


Definition:Alternating Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''alternating''' {{iff}}: : $\forall v \in V: b \left({v, v}\right) = 0$


Definition:Reflexive Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''reflexive''' {{iff}}: : $\forall v, w \in V: b \left({v, w}\right) = 0 \implies b \left({w, v}\right) = 0$


Definition:Apéry's Constant

'''Apéry's constant''' is the value of the infinite sum: :$\map \zeta 3 = \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$ where $\zeta$ denotes the Riemann zeta function. Its approximate value is given by: :$\map \zeta 3 \approx 1 \cdotp 20205 \, 69031 \, 59594 \, 28539 \, 97381 \, 61511 \, 44999 \, 07649 \, 86292 \ldots$


Definition:Discriminant of Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$. Let $b : V\times V \to \mathbb K$ be a bilinear form on $V$. Let $A$ be the matrix of $b$ relative to an ordered basis of $V$. If $b$ is nondegenerate, its '''discriminant''' is the equivalence class of the determinant $\det A$ in the quotient group $\mathbb K^\times / (\mathbb K^\times)^2$. If $b$ is degenerate, its '''discriminant''' is $0$.


Definition:Degenerate Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b : V\times V \to \mathbb K$ be a bilinear form on $V$. Then $b$ is '''degenerate''' {{Iff}} there exists $v\in V\setminus\{0\}$ such that $b(v,u) = 0$ for all $u\in V$. === Nondegenerate Bilinear Form === {{:Definition:Nondegenerate Bilinear Form}}


Definition:Nondegenerate Bilinear Form

A bilinear form on $V$ which is not degenerate is '''nondegenerate'''.


Definition:Quadratic Space

Let $\mathbb K$ be a field. A '''quadratic space''' over $\mathbb K$ is a pair $\left({V, q}\right)$ where: :$V$ is a vector space over $\mathbb K$ of finite dimension $n>0$ :$q$ is a quadratic form on $V$.


Definition:Orthogonal (Bilinear Form)

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b: V \times V \to \mathbb K$ be a reflexive bilinear form on $V$. Let $v,w\in V$. Then $v$ and $w$ are '''orthogonal (with respect to $b$)''' {{iff}} $b \left({v, w}\right) = b \left({w, v}\right) = 0$ This is denoted: $v \perp w$. === Orthogonal Subsets === {{:Definition:Orthogonal (Bilinear Form)/Subsets}} === Orthogonal Complement === {{:Definition:Orthogonal (Bilinear Form)/Orthogonal Complement}} === Radical === {{:Definition:Orthogonal (Bilinear Form)/Radical}}


Definition:Orthogonal (Bilinear Form)/Radical

The '''radical''' of $V$ is the orthogonal complement of $V$: :$\operatorname{rad}(V) = V^\perp$


Definition:Orthogonal (Bilinear Form)/Orthogonal Complement

Let $S\subset V$ be a subset. The '''orthogonal complement of $S$ (with respect to $b$)''' is the set of all $v \in V$ which are orthogonal to all $s \in S$. This is denoted: $S^\perp$. If $S = \left\{ {v}\right\}$ is a singleton, we also write $v^\perp$.


Definition:Relative Matrix of Quadratic Form

Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq2$. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$. Let $\mathcal B$ be an ordered basis of $V$. Let $q$ be a quadratic form on $V$. Its '''matrix  relative to $\mathcal B$''' is the matrix of its associated bilinear form relative to $\mathcal B$, denoted $\mathbf M_{q, \mathcal B}$. Category:Definitions/Quadratic Forms 4nmj6syc0tmb5ovh892qw2ag3haw907


Definition:Symplectic Basis

Let $\mathbb K$ be a field. Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $2 n > 0$. Let $\BB = \tuple {b_1, c_1, \ldots, b_n, c_n}$ be an ordered basis of $V$. Then $\BB$ is '''symplectic''' {{iff}}: :$\map f {b_i, b_j} = \map f {c_i, c_j} = 0$ for all $i, j$ :$\map f {b_i, c_j} = \delta_{i j}$ for all $i, j$ where $\delta$ denotes Kronecker delta. That is, {{iff}} the matrix of $f$ relative to $\BB$ has the form: :$\begin{pmatrix}  0 & 1 \\ -1 & 0 \\  & &  0 & 1 \\  & & -1 & 0 \\  & & & & \ddots \\  & & & & &  0 & 1 \\   & & & & & -1 & 0 \end{pmatrix}$


Definition:Bilinear Space

Let $\mathbb K$ be a field. A '''bilinear space''' over $\mathbb K$ is a pair $\left({V, f}\right)$ where: :$V$ be a vector space over $\mathbb K$ of finite dimension $n > 0$ :$f$ is a bilinear form on $V$.


Definition:Orthogonal Basis

=== Orthogonal Basis of Vector Space === {{:Definition:Orthogonal Basis of Vector Space}} === Orthogonal Basis of Inner Product Space === {{:Definition:Orthogonal Basis of Inner Product Space}} === Orthogonal Basis of Bilinear Space === {{:Definition:Orthogonal Basis of Bilinear Space}} Category:Definitions/Linear Algebra 5ccrihgddx0hdkbw28ybw4mk0efqf71


Definition:Orthogonal Basis/Bilinear Space

Let $\mathbb K$ be a field. Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $n > 0$. Let $\BB = \tuple {b_1, \ldots, b_n}$ be an ordered basis of $V$. Then $\BB$ is '''orthogonal''' {{iff}}: :$\map f {b_i, b_j} = 0$ for $i \ne j$. That is, {{iff}} the matrix of $f$ relative to $\mathcal B$ is diagonal.


Definition:Internal Orthogonal Sum (Bilinear Space)

Let $\mathbb K$ be a field. Let $\left({V, f}\right)$ be a reflexive bilinear space over $\mathbb K$. Let $U, W \subset V$ be subspaces of $V$. Then $V$ is the '''internal orthogonal (direct) sum''' of $U$ and $W$ {{iff}}: :$V = U \oplus W$, that is, $V$ is the internal direct sum of $U$ and $W$ :$U \perp W$, that is, $U$ and $W$ are orthogonal. This is denoted: $V = U\oplus W$.


Definition:Reflexive Bilinear Space

Let $\mathbb K$ be a field. Let $\left({V, f}\right)$ be a bilinear space over $\mathbb K$. Then $\left({V, f}\right)$ is '''reflexive''' {{iff}} $f$ is a reflexive bilinear form. Category:Definitions/Bilinear Forms hk94i1mcmdwfx8os8bp7i5ldr18shcq


Definition:Orthogonal (Bilinear Form)/Subsets

Let $S, T \subset V$ be subsets. Then $S$ and $T$ are '''orthogonal''' {{iff}} for all $s\in S$ and $t\in T$, $s$ and $t$ are orthogonal: $s \perp t$.


Definition:Field of Quotients/Definition 2

A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ such that: :$(1): \quad F$ is a field :$(2): \quad \iota: D \to F$ is a ring monomorphism  :$(3): \quad$ If $K$ is a field with $\iota \sqbrk D \subset K \subset F$, then $K = F$. That is, the '''field of quotients''' of an integral domain $D$ is the smallest field containing $D$ as a subring.


Definition:Field of Quotients/Definition 3

A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ where: :$(1): \quad$ $F$ is a field :$(2): \quad$ $\iota : D \to F$ is a ring monomorphism :$(3): \quad$ it satisfies the following universal property: ::::For every field $E$ and for every ring monomorphism $\varphi: D \to E$, there exists a unique field homomorphism $\bar \varphi: F \to E$ such that $\varphi = \bar \varphi \circ \iota$


Definition:Field of Quotients/Definition 1

A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ where: :$(1): \quad$ $F$ is a field :$(2): \quad$ $\iota : D \to F$ is a ring monomorphism :$(3): \quad \forall z \in F: \exists x \in D, y \in D_{\ne 0}: z = \dfrac {\map \iota x} {\map \iota y}$


Definition:Field of Directions/Functional

Let $\mathbf y$ be an $N$-dimensional vector. Let the functional $J$ be such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x $ Let the following be a family of boundary conditions, presribed $\forall x \in \closedint a b$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let these boundary conditions be self-adjoint and consistent $\forall x_1, x_2 \in \closedint a b$. Then these boundary conditions are called '''field of directions''' of the functional $J$.


Definition:Pluperfect Digital Invariant

Let $n \in \Z_{>0}$ be a positive integer. $n$ is a '''pluperfect digital invariant''' {{iff}} when expressed in decimal notation the $n$th powers of its digits add up to $n$.


Definition:Inconsummate Number

Let $m \in \Z_{>0}$ be a positive integer. Let $s_{10}$ denote the digit sum base $10$  . $m$ is an '''inconsummate number''' {{iff}}: :$\nexists n \in \Z_{>0}: n = m \times s_{10} \left({n}\right)$ That is, {{iff}} there exists no positive integer $n \in \Z_{>0}$ such that $n$ equals $m$ multiplied by the digit sum of $n$.


Definition:Kaprekar Mapping

The '''Kaprekar mapping''' is the arithmetic function $K: \Z_{>0} \to \Z_{>0}$ defined on the positive integers  as follows: Let $n \in \Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by arranging the digits of $n$ into descending order of size. Let $n''$ be the positive integer created by arranging the digits of $n$ into ascending order of size. Then: :$K \left({n}\right) = n' - n''$ making sure to retain any leading zeroes to ensure that $K \left({n}\right)$ has the same number of digits as $n$.


Definition:Kaprekar's Process

'''Kaprekar's process''' is the repeated application of the '''Kaprekar mapping''' to a given positive integer.


Definition:Sixth Power

A '''sixth power''' is an integer which can be expressed as the $6$th power of an integer.


Definition:Epsilon Relation/Restriction

Let $S$ be a set. The '''restriction of the epsilon relation''' on $S$ is defined as the endorelation $\Epsilon {\restriction_S} = \left({S, S, \in_S}\right)$, where: :$\in_S \; := \left\{{\left({x, y}\right) \in S \times S: x \in y}\right\}$


Definition:Weird Number

A '''weird number''' is an abundant number which is not at the same time semiperfect.


Definition:Weird Number/Sequence

The sequence of weird numbers begins: :$70, 836, 4030, 5830, 7192, 7912, 9272, \ldots$


Definition:Heptagonal Number

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


Definition:Heptagonal Number/Definition 3

:$\forall n \in \N: H_n = P \left({7, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({7, n - 1}\right) + 5 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Definition:Heptagonal Number/Definition 1

:$H_n = \begin{cases} 0 & : n = 0 \\ H_{n-1} + 5 n - 4 & : n > 0 \end{cases}$


Definition:Heptagonal Number/Definition 2

:$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({5 i - 4}\right) = 1 + 6 + \cdots + \left({5 \left({n - 1}\right) - 4}\right) + \left({5 n - 4}\right)$


Definition:Heptagonal Number/Sequence

The sequence of heptagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, \ldots$


Definition:Zero Digit

Let $x \in \R$ be a number. Let $b \in \Z$ such that $b > 1$ be a number base in which $x$ is represented. By the Basis Representation Theorem, $x$ can be expressed uniquely in the form: :$\displaystyle x = \sum_{j \mathop \in \Z}^m r_j b^j$ Any instance of $r_j$ being equal to $0$ is known as a '''zero (digit)''' of $n$.


Definition:Prime Gap

A '''prime gap''' is a difference between two consecutive prime numbers.


Definition:Sophie Germain Prime/Sequence

The sequence of Sophie Germain primes begins: :$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, \ldots$


Definition:Reverse-and-Add

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $r: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :For all $n \in \Z_{>0}$: reverse the digits of $m$ and add the result to $m$. The mapping $r$ is the '''reverse-and-add operation'''.


Definition:Central Field

Let $R$ be a simply connected region in $\paren {n + 1}$-dimensional space. Let $\tuple {x, \mathbf y}$ be a point in $R$. Let $c = \tuple {\sequence {c_i}_{0 \mathop \le i \mathop \le n} }$ be a point lying outside of $R$. Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ whose extremals $\mathbf y$ are curves in $\paren{n+1}$-dimensional space. {{explain|Extremum or extremal, or are they the same thing? If they are, then an "also known as" section needs to be placed on the page Definition:Extremum of Functional. If not, then the terminology needs to be specifically amended and/or corrected.}} Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\tuple {x, \mathbf y}$, thereby for every point in $R$ defining a relation: :$(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$. Then the field of directions $(1)$ is called a '''central field'''.


Definition:Time/Unit/Quarter-Year

The '''quarter-year''' is defined as: :$91$ days or :$13$ weeks of $7$ days each.


Definition:Fermat Pseudoprime/Base 3

Let $q$ be a composite number such that $3^q \equiv 3 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $3$'''.


Definition:Celsius

'''Celsius''' is a temperature scale. Its two reference points are: : $0 \cels$, which is set at the melting point of water. : $100 \cels$, which is set at the boiling point of water, as defined at sea level and standard atmospheric pressure. A temperature measured in '''Celsius''' is often referred to as so many '''degrees Celsius'''. === Symbol === {{:Definition:Celsius/Symbol}}


Definition:Celsius/Symbol

The symbol for the '''degree Celsius''' is $\cels$.


Definition:Array/Row

Let $\mathbf A$ be an array. The '''rows''' of $\mathbf A$ are the lines of elements reading '''across''' the page.


Definition:Array/Element/Index

Let $\mathbf A$ be an $m \times n$ array. Let $a_{i j}$ be an element of $\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$.


Definition:Array/Element

Let $\mathbf A$ be an Array. The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the '''elements''' of $\mathbf L$. The '''element''' at row $i$ and column $j$ is called '''element $\tuple {i, j}$ of $\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \left({i, j}\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the array where the '''element''' is positioned.


Definition:Array/Square

An array whose dimensions are equal is called a '''square array'''. That is, a '''square array''' is an array which has the same number of rows as it has columns. A '''square array''' $\left[{a}\right]_{n n}$ is usually denoted $\left[{a}\right]_n$.