hcju/proofwikips huggingface dataset:
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'''  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'''  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'''  $T$ 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'''  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'''  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'''  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'''  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'''  each point of $S$ has a compact neighborhood. That is,  $T$ is weakly locally compact.
Definition:Locally Compact Hausdorff Space/Definition 2
$T$ is a '''locally compact Hausdorff space'''  each point has a neighborhood basis consisting of compact sets. That is,  $T$ is locally compact (in the general sense).
Definition:Locally Compact Hausdorff Space
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 $B$'''  $A$ is closed in the relative topology of $B$.
Definition:Relatively Closed Set/Definition 2
'''$A$ is relatively closed in $B$'''  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.
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.
Definition:Saturation (Equivalence Relation)
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: :$\displaystyle \overline T = \bigcup_{t \mathop \in T} \eqclass t \sim$
Definition:Saturated Set (Equivalence Relation)
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'''  it equals its saturation: :$T = \overline T$
Definition:Saturated Set (Equivalence Relation)/Definition 3
$T$ is '''saturated'''  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'''  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'''  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'''  its roots are distinct in an algebraic closure of $K$.
Definition:Separable Polynomial/Definition 2
$P$ is '''separable'''  it has no double roots in every field extension of $K$.
Definition:Separable Polynomial/Definition 3
$P$ is '''separable'''  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 $K$'''  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'''  it is normal and separable.
Definition:Galois Extension/Finite/Definition 1
$L/K$ is a '''Galois extension'''  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''' : :$\displaystyle \sum_{p \mathop \divides n} \frac 1 p - \prod_{p \mathop \divides n} \frac 1 p \in \Z_{\ge 0}$
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'''  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'''  :$\mathord\preceq = \mathord\subseteq \cap \left({S \times S}\right)$ 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 $L$.
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'''  :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  $\mathbf h$ :$(3): \quad \displaystyle \size {\mathbf h} = \sum_{i \mathop = 1}^N \size {h_i}_1 = \sum_{i \mathop = 1}^N \max_{a \mathop \le x \mathop \le b} \set {\size {\map {h_i} x} + \size {\map {h_i'} x} }$ :$(4): \quad \epsilon \to 0$ 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: :$\delta^2 J \sqbrk {\mathbf y; \mathbf h}$