hcju/proofwikips huggingface dataset:
Definition:Discrete Subgroup/Real Numbers
Let $G$ be a subgroup of the additive group of real numbers. Then $G$ is '''discrete''' : :$\forall g \in G : \exists \epsilon > 0: \openint {g - \epsilon} {g + \epsilon} \cap G = \set g$ That is, there exists a neighborhood of $g$ which contains no other elements of $G$.
Definition:Everywhere Dense/Real Numbers
Let $S$ be a subset of the real numbers. Then $S$ is '''(everywhere) dense''' in $\R$ : :$\forall x \in \R : \forall \epsilon \in \R : \epsilon > 0: \exists s \in S: x - \epsilon < s < x + \epsilon$. That is,  in every neighborhood of every real number lies an element of $S$.
Definition:Poisson Bracket
Let  :$\map A {x, \mathbf y, \mathbf p}: \R^{2n+1} \to \R$, :$\map B {x, \mathbf y, \mathbf p}: \R^{2n+1} \to \R$, $n \in \N$  be real functions, dependent on canonical variables. The '''Poisson Bracket''' of functions $A$ and $B$ is defined as: :$\displaystyle \sqbrk {A, B} := \sum_{i \mathop = 1}^n \paren {\frac {\partial A} {\partial y_i} \frac {\partial B} {\partial p_i} - \frac {\partial B} {\partial y_i} \frac {\partial A} {\partial p_i} }$ where the notation $\dfrac {\partial A} {\partial y_i}$ denotes partial differentiation. {{NamedforDef|Siméon-Denis Poisson|cat = Poisson}}
Definition:First Integral of System of Differential Equations
Let $S$ be a system of differential equations. Let $g$ be a function which satisfies $S$. Let $f$ be a function. {{explain|Establish the domain and codomain of these functions}} Let $f$ depend on variables (denoted by ordered tuples here) of $S$ independently as well as through $g$ and its derivatives: :$f = \map f {\sequence {x_i} _{1 \mathop \le i \mathop \le n}, \sequence {g^{\paren j} \paren {\sequence {x_i}_{1 \mathop \le i \mathop \le n} }_{0 \mathop \le j \mathop \le k} } }, \quad {n, k} \in \N$ Suppose there exists $g$ such that $f$ is a constant. Then $f$ is the first integral of $S$.
Definition:Proper Mapping
Let $X$ and $Y$ be topological spaces. A mapping $f: X \to Y$ is '''proper'''  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'''  $\phi$ 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'''  $\phi$ 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 $T$)''' : :$\map \gamma 0 = \map \gamma 1$ === 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.<br/>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<br/>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'''  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'''  $K$ 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''' : :$\dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i},\quad\dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i}$ 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)'''  :$A$ 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 : :$J \sqbrk \Gamma = J \sqbrk \gamma$ That is, : :$\displaystyle \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{X_0}^{X_1} \map F {X, \mathbf Y, \mathbf Y'} \rd X$
Definition:Dense (Order Theory)/Element
Let $x \in S$. Then $x$ is '''dense'''  :$\forall y \in S: y \ne \bot \implies x \wedge y \ne \bot$ where $\bot$ denotes the smallest element in $L$.
Definition:Dense (Order Theory)/Subset
Let $A$ be a subset of $S$. Then $A$ is '''dense'''  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}}
Definition:Almost Perfect Number/Definition 1
Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''almost perfect'''  $A \left({n}\right) = -1$.
Definition:Almost Perfect Number/Definition 2
$n$ is '''almost perfect''' : :$\sigma \left({n}\right) = 2 n - 1$ where $\sigma \left({n}\right)$ denotes the $\sigma$ function of $n$.
Definition:Almost Perfect Number/Definition 3
$n$ is '''almost perfect'''  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''' : :$\sigma \left({n}\right) = 2 n + 1$ where $\sigma \left({n}\right)$ denotes the $\sigma$ function of $n$.
Definition:Quasiperfect Number/Definition 3
$n$ is '''quasiperfect'''  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)'''  :$\exists x \in I: x$ is upper bound for $I$
Definition:Principal Ideal of Ordered Set/Definition 2
Then $I$ is '''principal (ideal)'''  :$\exists x \in S: I = x^\preceq$ where $x^\preceq$ denotes the lower closure of $x$.
Definition:Principal Ideal of Ordered Set
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 $L$  :$S$ 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  $y_1$ for fixed $y_2$, and linear  $y_2$ 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  $h$ :$\epsilon \to 0$ 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.
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  $y$, 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:Generalized Pentagonal Number
=== 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''' : :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.