hcju/proofwikips huggingface dataset:
Definition:Variational Derivative
Let $\map y x$ be a real function. Let $J = J \sqbrk y$ be a functional dependent on $y$. Let $\map h x$ be a real function, which differs from zero only in the neighbourhood of $x_0$. Consider an increment of functional $\Delta J \sqbrk {y; h}$. Denote the area between $\map y x + \map h x$ and $\map y x$ (or, equivalently, between $\map h x$ and x-axis) as $\Delta \sigma$. Let $\Delta \sigma \to 0$ in such a way, that  $\ds \lim_{\Delta \sigma \mathop \to 0} \map \max {\map h x} = 0$ and the length of interval where $\map h x$ differs from 0 would go to 0. If the ratio $\dfrac {\Delta J \sqbrk {y; h} } {\Delta \sigma}$ converges to a limit as $\Delta \sigma \to 0$, then: $\ds \lim_{\Delta \sigma \mathop \to 0} \frac {\Delta J \sqbrk {y; h} } {\Delta \sigma} = \intlimits {\frac {\delta J} {\delta y} } {x = x_0} {}$ where $\intlimits {\dfrac {\delta J} {\delta y} } {x = x_0} {}$ is called the '''variational derivative''' at the point $x = x_0$ for the function $y = \map y x$. {{Stub|Check if there is a definition not using a special point $x_0$}}
Definition:Euler's Equation for Vanishing Variation
Let $\map y x$ be a real function. Let $\map F {x, y, z}$ be a real function belonging to $C^2$  all its variables. Let $J \sqbrk y$ be a functional of the form: :$\displaystyle \int_a^b \map F {x, y, y'} \rd x$ Then '''Euler's equation for vanishing variation''' is defined a differential equation, resulting from condition: :$\displaystyle \delta \int_a^b \map F {x, y, y'} \rd x = 0$ In other words: :$\displaystyle F_y - \dfrac \d {\d x} F_{y'} = 0$ {{Stub|Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms}} {{NamedforDef|Leonhard Paul Euler|cat = Euler}}
Definition:Directed Suprema Inheriting
Let $L = \left({X, \preceq}\right)$ be an ordered set. Let $S = \left({Y, \precsim}\right)$ be an ordered subset of $L$. Then $S$ '''inherits directed suprema'''  :for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\sup_L A \in Y$
Definition:Ordered Set of Subalgebras
Let $L$ be an ordered set. The '''ordered set of subalgebras''' of $L$ is ordered subset of $\operatorname{ClSystems}\left({L}\right)$ and is defined by :$\operatorname{Subalgeras}\left({L}\right) := \left({X, \precsim}\right)$ where :$X$ is the set of all directed suprema inheriting closure systems on $L$, :$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems on $L$.
Definition:Lattice (Group Theory)/Definition 1
A (point) '''lattice''' is a discrete subgroup of $\R^m$ under addition.
Definition:Lattice (Group Theory)/Definition 2
Let $\R^m$ be the $m$-dimensional real Euclidean space. Let $\set {b_1, b_2, \ldots, b_n}$ be a set of linearly independent vectors of $\R^m$. A '''lattice in $\R^m$''' is the set of all integer linear combinations of such vectors. That is: :$\displaystyle \map \LL {b_1, b_2, \ldots, b_n} = \set {\sum_{i \mathop = 1}^n x_i b_i : x_i \in \Z}$
Definition:Coarser Subset (Order Theory)
Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''coarser (subset)''' than $Y$  :$\forall x \in X: \exists y \in Y: y \preceq x$
Definition:Finer Subset (Order Theory)
Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''finer (subset)''' than $Y$  :$\forall x \in X: \exists y \in Y: x \preceq y$
Definition:Archimedean Polyhedron
An '''Archimedean polyhedron''' is a convex polyhedron with the following properties: :$(1): \quad$ Each of its faces is a regular polygon :$(2): \quad$ It is vertex-transitive :$(3): \quad$ The faces are not all congruent. :$(4): \quad$ It is not a regular prism or a regular antiprism.
Definition:Semiregular Polyhedron
A '''semiregular polyhedron''' is a polyhedron with the following properties: :$(1): \quad$ Each of its faces is a regular polygon :$(2): \quad$ It is vertex-transitive :$(3): \quad$ The faces are not all congruent.
Definition:Inaccessible by Directed Suprema
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''inaccessible by directed suprema'''  :for all directed subsets $D$ of $S$: $\sup D \in X \implies X \cap D \ne \varnothing$
Definition:Closed under Directed Suprema
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''closed under directed suprema'''  :for all directed subsets $D$ of $S$: $D \subseteq X \implies \sup D \in X$
Definition:Small Stellated Dodecahedron
The small stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $5$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}
Definition:Great Stellated Dodecahedron
The great stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $3$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}
Definition:Catalan Polyhedron
A '''Catalan polyhedron''' is a dual polyhedron of an Archimedean polyhedron.
Definition:Contractible Space
=== Definition 1 === {{:Definition:Contractible Space/Definition 1}} === Definition 2 === {{:Definition:Contractible Space/Definition 2}}
Definition:Property (S)
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ has '''property (S)'''  :for all directed subsets $D$ of $S$: $\sup D \in X \implies \exists y \in D:\forall x \in D: y \preceq x \implies x \in X$
Definition:Stellation
'''Stellation''' is the process of extending the extremities of a geometric figure in a specifically structured way, in order to form new figures from those extensions. The figure so generated is itself known as a '''stellation''' or a '''stellated''' original figure.
Definition:Stellation/Polygon
A '''stellation''' of a polygon $P$ is a plane figure formed by extending the sides of $P$ until they meet.
Definition:Stellation/Polyhedron
A '''stellation''' of a polyhedron $H$ is a solid figure formed by extending the faces of $H$ until they meet.
Definition:Quadratic Gauss Sum
Let $p$ be an odd prime. Let $a$ be an integer. The '''quadratic Gauss sum''' is: :$\displaystyle g \left({a, p}\right) = \sum_{n \mathop = 0}^{p - 1} e^{2 \pi i a n^2 / p}$
Definition:Relational Structure with Topology
A triple $\left({S, \preceq, \tau}\right)$ is a '''relational structure with topology''' where :$\preceq$ is a relation on $S$ and :$\tau$ is a topology on $S$.
Definition:Weierstrass Elementary Factor
Let $n \in \Z_{\ge 0}$ be a positive integer. The '''$n$th (Weierstrass) elementary factor''' is the function $E_n: \C \to \C$ defined as: :$E_n \left({z}\right) = \displaystyle \begin{cases} 1 - z & : n = 0 \\ \left({1 - z}\right) \exp \left({z + \dfrac {z^2} 2 + \cdots + \dfrac{z^n} n}\right) & : \text{otherwise}\end{cases}$
Definition:Absolute Convergence of Product/General Definition/Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + a_n}$ is '''absolutely convergent'''  $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + \norm{a_n}}$ is convergent.
Definition:Absolute Convergence of Product/Complex Numbers
Let $\sequence {a_n}$ be a sequence in $\C$. === Definition 1 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 2}} === Definition 3 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 3}}
Definition:Convergent Product/Number Field
Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$. === Nonzero Sequence === {{:Definition:Convergent Product/Number Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Number Field/Arbitrary Sequence}}
Definition:Scott Topology
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Then $T$ has '''Scott topology'''  :$\tau$ is the set of all upper and inaccessible by directed suprema subsets of $S$. {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}
Definition:Convergent Product/Arbitrary Field
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. === Nonzero Sequence === {{:Definition:Convergent Product/Arbitrary Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence}}
Definition:Divergent Product
An infinite product which is not convergent is '''divergent'''. === Divergence to zero === {{:Definition:Divergent Product/Divergence to Zero}}
Definition:Primitive Pythagorean Triangle
A '''primitive Pythagorean triangle''' is a Pythagorean triangle whose sides form a primitive Pythagorean triple.
Definition:O Notation/Big-O Notation/Real/Point
Let $x_0 \in \R$. Let $f$ and $g$ be real-valued or complex-valued functions defined on a punctured neighborhood of $x_0$. The statement: :$f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to x_0$ is equivalent to: :$\exists c \in \R: c \ge 0: \exists \delta \in \R : \delta > 0 : \forall x \in \R : \left({0 < \left\lvert{x - x_0}\right\rvert < \delta \implies \left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert}\right)$ That is: :$\left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert$ for all $x$ in a punctured neighborhood of $x_0$.
Definition:O Notation/Big-O Notation/Complex/Infinity
Let $f$ and $g$ be complex functions defined for all complex numbers whose modulus is sufficiently large. The statement: :$f(z) = \mathcal O \left({g(z)}\right)$ as $|z|\to\infty$ is equivalent to: :$\displaystyle \exists c\in \R: c\ge 0 : \exists r_0 \in \R : \forall z \in \C : (|z| \geq r_0 \implies |f(z)| \leq c \cdot |g(z)|)$ That is: :$|f(z)| \leq c \cdot |g(z)|$ for all $z$ in a neighborhood of infinity in $\CC$.
Definition:O Notation/Big-O Notation/Complex/Point
Let $z_0 \in \C$. Let $f$ and $g$ be complex functions defined on an punctured neighborhood of $z_0$. The statement: :$f(z) = \mathcal O \left({g(z)}\right)$ as $z\to z_0$ is equivalent to: :$\displaystyle \exists c \in \R : c \ge 0 : \exists \delta \in \R : \delta > 0 : \forall z \in \C : (0<|z-z_0|<\delta \implies |f(z)| \leq c \cdot |g(z)|)$ That is: :$|f(z)| \leq c \cdot |g(z)|$ for all $z$ in a punctured neighborhood of $z_0$.
Definition:O Notation/Big-O Notation/Real/Infinity
Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$. The statement: :$\map f x = \map {\mathcal O} {\map g x}$ as $x \to \infty$ is equivalent to: :$\exists c \in \R: c \ge 0: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$ That is: :$\size {\map f x} \le c \cdot \size {\map g x}$ for $x$ sufficiently large. This statement is voiced '''$f$ is big-O of $g$''' or simply '''$f$ is big-O $g$'''.
Definition:Asymptotically Equal/Sequences
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$. === Definition 1 === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Definition 2 === {{:Definition:Asymptotically Equal/Sequences/Definition 2}} === Definition 3 === {{:Definition:Asymptotically Equal/Sequences/Definition 3}} This is denoted: $a_n \sim b_n$.
Definition:Asymptotically Equal/Functions
Let $f$ and $g$ real functions defined on $\R$. Then: : '''$f$ is asymptotically equal to $g$''' : :$\dfrac {f \left({x}\right)} {g \left({x}\right)} \to 1$ as $x \to +\infty$. That is, the larger the $x$, the closer $f$ gets (relatively) to $g$.
Definition:O Notation/Big-O Notation/General Definition/Point
Let $(X,\tau)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$ Let $x_0\in X$. Let $f,g:X\setminus\{x_0\}\to V$ be functions. The statement :$f(x) = \mathcal O \left({g(x)}\right)$ as $x\to x_0$ is equivalent to: :$\displaystyle \exists c\in \R: c\ge 0 : \exists U\in\tau: x_0\in U : \forall x\in U\setminus\{x_0\} : \Vert f(x)\Vert \leq c\cdot\Vert g(x)\Vert$ That is: :$\Vert f(x)\Vert\leq c\cdot\Vert g(x)\Vert$ for all $x$ in a punctured neighborhood of $x_0$.
Definition:Asymptotically Equal/General Definition/Point
Let $T = \left({S, \tau}\right)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\, \cdot \,}\right\Vert$. Let $f, g: S \to V$ be mappings. Let $x_0 \in X$. Then: : $f$ is '''asymptotically equal''' to $g$ as $x \to x_0$ : : $f - g = o \left({g}\right)$ as $x \to x_0$ where $o$ denotes little-O notation.
Definition:O Notation/Little-O Notation/Real Functions
Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 2}} This is denoted: :$f = o \left({g}\right)\qquad(x\to\infty)$  This statement is voiced '''$f$ is little-o of $g$''' or simply '''$f$ is little-o $g$'''.
Definition:Sequence of Partial Products
The sequence $\sequence {p_N}$ defined as: :$\displaystyle p_N: = \prod_{n \mathop = 1}^N a_n = a_1 \cdot a_2 \cdots a_N$ is the '''sequence of partial products''' of the infinite product $\displaystyle p = \prod_{n \mathop = 1}^\infty a_n$.
Definition:O Notation/Little-O Notation/General Definition/Point
Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$ Let $f,g:X\to V$ be functions. Let $x_0\in X$. The statement :$f(x) = o \left({g(x)}\right)$ as $x\to x_0$ is equivalent to the statement: :For all $\epsilon>0$, there exists a neighborhood $U$ of $x_0$ such that $\Vert f(x)\Vert\leq \epsilon\cdot\Vert g(x)\Vert$ for all $x\in U$
Definition:Topological Lattice
A '''topological lattice''' is a relational structure with topology $\left({S, \preceq, \tau}\right)$ where $\left({S, \preceq}\right)$ is a lattice and $\left({S, \tau}\right)$ is a topological space.
Definition:Generator for Primitive Pythagorean Triple
The '''generator''' for a primitive Pythagorean triple is an ordered pair: :$G = \left({m, n}\right)$ where $m, n \in \Z$ such that: : $m, n \in \Z_{>0}$ are (strictly) positive integers : $m \perp n$, that is, $m$ and $n$ are coprime : $m$ and $n$ are of opposite parity : $m > n$. The primitive Pythagorean triple which has been '''generated''' by $G$ is: :$\left({2 m n, m^2 - n^2, m^2 + n^2}\right)$
Definition:Generator for Pythagorean Triple
Let $P = \left\{ {a, b, c}\right\}$ be a Pythagorean triple  A '''generator''' for $P$ is an ordered pair: :$G = \left({m, n}\right)$ where $m, n \in \Z$ such that: : $a = 2 m n$ : $b = m^2 - n^2$ : $c = m^2 + n^2$
Definition:Almost Isosceles Pythagorean Triangle
Let $P$ be a Pythagorean triangle whose legs are $a$ and $b$. Then $P$ is '''almost isosceles'''  $\size {a - b} = 1$ That is, if the legs of $P$ differ by $1$. === Sequence === {{:Generator for Almost Isosceles Pythagorean Triangle/Sequence}}
Definition:Neighborhood of Infinity (Real Analysis)
=== Neighborhood of Positive Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity}} === Neighborhood of Negative Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity}}
Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity
A '''neighborhood of $+\infty$''' is a subset of the set of real numbers $\R$ wich contains an interval $(a,+\infty)$ for some $a\in\R$. That is, a subset which contains all sufficiently large real numbers.
Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity
A '''neighborhood of $-\infty$''' is a subset of the set of real numbers $\R$ wich contains an interval $(-\infty,a)$ for some $a\in\R$. That is, a subset which contains all sufficiently large negative real numbers.
Definition:Nearest Integer Function
The '''nearest integer function''' is defined as: :$\forall x \in \R: \nint x = \begin {cases} \floor {x + \dfrac 1 2} & : x \notin 2 \Z + \dfrac 1 2 \\ x - \dfrac 1 2 & : x \in 2 \Z + \dfrac 1 2 \end{cases}$ where $\floor x$ is the floor function.
Definition:Pythagorean Prime
A '''Pythagorean prime''' is a prime number of the form: :$p = 4 n + 1$ where $n \in \Z_{\ge 0}$ is a positive integer. === Sequence === {{:Definition:Pythagorean Prime/Sequence}}
Definition:Non-Pythagorean Prime
A '''non-Pythagorean prime''' is an odd prime number of the form: :$p = 4 n + 3$ where $n \in \Z_{\ge 0}$ is a positive integer. === Sequence === {{:Definition:Non-Pythagorean Prime/Sequence}}
Definition:O Notation/Big-O Notation/Sequence
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. '''$a_n$ is big-O of $b_n$'''  :$\exists c \in \R: c \ge 0 : \exists n_0 \in \N : \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$ That is: :$\size {a_n} \le c \cdot \size {b_n}$ for all sufficiently large $n$.
Definition:O Notation/Big-O Notation/General Definition
=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/General Definition/Point}}
Definition:Neighborhood of Infinity (Topology)
Let $X$ be a non-empty topological space. A '''neighborhood of infinity''' is a subset of $X$ which contains the complement of a closed and compact subset of $X$.
Definition:Hamiltonian
Let $J \sqbrk {\dotsm y_i \dotsm}$ be a functional of the form: :$\displaystyle J \sqbrk {\dotsm y_i \dotsm} = \intlimits {\int_{x_0}^{x_1} \map F {x, \cdots y_i \dotsm, \dotsm y_i \dotsm} \rd x} {i \mathop = 1} {i \mathop = n}$ Then the '''Hamilonian''' $H$ corresponding to $J \sqbrk {\dotsm y_i \dotsm}$ is defined as :$\displaystyle H = -F + \sum_{i \mathop = 1}^n y_i' F_{y_i'}$ {{NamedforDef|William Rowan Hamilton|cat = Hamilton}}
Definition:Canonical Variable
Let $\map {\mathbf y} x : \R \to \R^n$, $n \in \N $ be a vector-valued function. Let $F: \R^{2 n + 1} \to \R$ be a differentiable mapping. Let $J \sqbrk {\mathbf y}$ be a functional of the form: :$\d J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ Consider the (real) variables $x, \mathbf y, \mathbf y'$, and the mapping $F$. Make the following transformation: :$F_{y_i'} = p_i$ where $F_x$ denotes the partial derivative of $F$  $x$. Let $H$ be the Hamiltonian corresponding to $J \sqbrk {\mathbf y}$. Then the new variables $x, \mathbf y, \mathbf p$, and the mapping $H$ corresponding to $J \sqbrk {\mathbf y}$ are called '''the canonical variables'''.
Definition:Time/Unit/Week
:$1$ '''week''' $=7$ days.
Definition:Time/Unit/Week/Symbol
The symbol for the '''week''' is $\mathrm {week}$ or $\mathrm {wk}$.
Definition:Time/Unit/Fortnight
:$1$ '''fortnight''' $=2$ weeks $=14$ days.
Definition:O Notation/Big-O Notation/Parameter
Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$ Let $x_0 \in X \cup \set \infty$. Let $A$ be a set. Let $X_\alpha$ be a subset of $X$ for every $\alpha \in A$, which, if $x_0 \ne \infty$, contains $x_0$. Let $f_\alpha : X_\alpha \setminus \set {x_0} \to V$ be a mapping for every $\alpha \in A$. Let $g : X \to V$ be a mapping. The statement: :$f_\alpha = \map {\OO_\alpha} g$ as $x \to x_0$ is equivalent to: :$\forall \alpha \in A : f_\alpha = \map \OO g$ as $x \to x_0$ The $\OO$-estimate is said to be '''independent''' of $\alpha \in A$ : :there exists a neighborhood $U$ of $x_0$ in $X$ such that: ::$\exists c \in \R: c \ge 0 : \forall \alpha \in A : \forall x \in \paren {U \setminus \set {x_0} } \cap X_\alpha : \norm {\map {f_\alpha} x} \le c \cdot \norm {\map g x}$ That is, if the implied constant and implied neighborhood can be chosen the same for all $\alpha \in A$.
Definition:O Notation/Big-O Notation/Real
=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Real/Point}}
Definition:O Notation/Big-O Notation/Complex
=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Complex/Point}}
Definition:O Notation/Big-O Notation/General Definition/Infinity
Let $\left({X, \tau}\right)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$. Let $f, g : X \to V$ be functions. The statement: :$f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to \infty$ is equivalent to: :There exists a neighborhood of infinity $U \subset X$ such that: ::$\exists c \in \R: c \ge 0: \forall x \in U: \left\Vert{f \left({x}\right)}\right\Vert \le c \cdot \left\Vert{g \left({x}\right)}\right\Vert$ That is: :$\Vert f \left({x}\right) \Vert \le c \cdot \Vert g \left({x}\right) \Vert$ for all $x$ in a neighborhood of infinity.
Definition:Time/Unit/Month
The '''month''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''month''' is between $28$ and $31$ days, depending on which '''month''' it is.
Definition:Time/Unit/Season
The '''season''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''season''' $= 3$ months, although exactly which months fall into which '''season''' depends on the locale and the culture.
Definition:Neighborhood of Infinity (Complex Analysis)
A '''neighborhood of $\infty$''' in $\C$ is a subset of the set of complex numbers $\C$ wich contains a set of the form $\{z \in \C : |z| > r\}$ for some $r\in\R$. That is, a subset which contains all complex numbers whose modulus is sufficiently large.
Definition:Covolume of Lattice
Let $L$ be a lattice in $\R^n$. Let $\left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$. Let $v_i = \left({v_{i 1}, \ldots , v_{i n} }\right)$ for $i \in \left\{ {1, \ldots, n}\right\}$. The '''covolume''' of $L$ is the determinant of the matrix: :$\begin{bmatrix} v_{1 1} & v_{1 2} & \cdots & v_{1 n} \\ v_{2 1} & v_{2 2} & \cdots & v_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n 1} & v_{n 2} & \cdots & v_{n n} \\ \end{bmatrix}$
Definition:Integer Lattice
Let $n$ be a positive integer. The '''integer lattice''' in $\R^n$ is the lattice $\Z^n$.
Definition:Fundamental Domain (Lattice)
Let $L \subset \R^n$ be an integral lattice. Let $B = \left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$. The '''fundamental domain''' of $L$ associated to $B$ is the set: :$\displaystyle \left\{ {\sum_{i \mathop = 1}^n \lambda_i v_i: \forall i: 0 \le \lambda_i < 1}\right\}$
Definition:Integral Lattice
Let $n$ be a positive integer. An '''integral lattice''' in $\R^n$ is a subgroup that is isomorphic to the integer lattice $\Z^n$.
Definition:Square Pyramidal Number
'''Square pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a square pyramid.
Definition:Discrete Group
A '''discrete group''' is a topological group whose topology is discrete.
Definition:Discrete Subgroup
Let $G$ be a topological group. Let $H$ be a subgroup of $G$. Then $H$ is called a '''discrete subgroup''' if it is a discrete group for the induced topology. === Real Numbers === {{:Definition:Discrete Subgroup/Real Numbers}}
Definition:Limit Inferior of Moore-Smith Sequence
Let $\left({S, \preceq}\right)$ be a directed set. Let $L = \left({T, \precsim}\right)$ be a complete lattice. Let $N: S \to T$ be a Moore-Smith sequence in $T$. Then '''limit inferior''' of $N$ is defined as follows: :$\liminf N := \sup_L \left\{ {\inf_L \left({N \left[{\preceq \left({j}\right)}\right]}\right): j \in S}\right\}$ where :$\preceq \left({j}\right)$ denotes the image of $j$ by $\preceq$, :$N \left[{\preceq \left({j}\right)}\right]$ denotes the image of $\preceq \left({j}\right)$ under $N$.