hcju/proofwikips huggingface dataset:
Definition:Tensor Product of Modules as Abelian Group
Let $R$ be a ring with unity. Let $M$ be a unitary right module and $N$ a unitary left module over $R$. === Definition 1: by universal property === Their '''tensor product''' is a pair $(M \otimes_R N, \theta)$ where: * $M \otimes_R N$ is an abelian group * $\theta : M \times N \to M \otimes_R N$ is an $R$-balanced mapping satisfying the following '''universal property''': :For every pair $(P, \omega)$ of an abelian group and an $R$-balanced mapping $\omega : M \times N \to P$, there exists a unique group homomorphism $f : M \otimes_R N \to P$ with $\omega = f \circ \theta$. === Definition 2: direct construction === Their '''tensor product''' is the pair $(M \otimes_R N, \theta)$, where: *$M \otimes_R N$ is the quotient group of the free $R$-module $R^{(M\times N)}$ on the direct product $M \times N$, by the subgroup generated by the set of elements of the form: ** $(m_1 + m_2, n) - \lambda (m_1, n) - (m_2, n)$ ** $(m, n_1 + n_2) - \lambda (m, n_1) - (m, n_2)$ ** $(m \cdot \lambda, n) - (m, \lambda \cdot n)$ *:for $m, m_1, m_2 \in M$, $n, n_1, n_2 \in N$ and $\lambda \in R$, where we denote $(m, n)$ for its image under the canonical mapping $M \times N \to R^{(M\times N)}$. * $\theta : M \times N \to M \otimes_R N$ is the composition of the canonical mapping $M \times N \to R^{(M\times N)}$ with the quotient module epimorphism $R^{(M\times N)} \to M \otimes_R N$. === Definition 3: construction via abelian groups === Let $(M \otimes_\Z N, u)$ be the tensor product of their underlying abelian groups. The '''tensor product''' of $M$ and $N$ is the the pair $(M \otimes_R N, \theta)$, where: * $M \otimes_R N$ is the quotient of $M \otimes_\Z N$ by the subgroup generated by the elements of the form: *:$u(m \cdot \lambda, n) - u(m, \lambda \cdot n)$ * $\theta : M \times N \to M \otimes_R N$ is the composition of $u$ with the quotient group epimorphism $M \otimes_\Z N \to M \otimes_R N$.
Definition:Free Abelian Group on Set
Let $\Z$ be the additive group of integers. Let $S$ be a set. The '''free abelian group on $S$''' is the pair $(\Z^{(S)}, \iota)$ where: * $\Z^{(S)}$ is the direct sum of $S$ copies of $\Z$. That is, of the indexed family $S \to \{\Z\}$ * $\iota : S \to \Z^{(S)}$ is the '''canonical mapping''', which sends $s$ to the mapping $\delta_{st} \in \Z^{(S)}$, where $\delta$ denotes Kronecker delta.
Definition:Direct Sum of Groups
Let $I$ be a indexing set. Let $(G_i)_{i \in I}$ be a family of groups. The '''direct sum''' of $(G_i)_{i \in I}$ is the subgroup of their direct product consisting of mappings of finite support.
Definition:Category of Right Modules
Let $R$ be a ring. The '''category of right $R$-modules''' is the category $\mathbf {Mod-R}$ with: {{DefineCategory | ob = right modules over $R$ | mor = right $R$-module homomorphisms | id = identity mappings | comp = composition of mappings }}
Definition:Category of Left Modules
Let $R$ be a ring. The '''category of left $R$-modules''' is the category $\mathbf {R-Mod}$ with: {{DefineCategory | ob = left modules over $R$ | mor = left $R$-module homomorphisms | id = identity mappings | comp = composition of mappings }}
Definition:Category of Unitary Modules
Let $R$ be a commutative ring with unity. The '''category of unitary $R$-modules''' is the category $\mathbf{R-Mod}$ with: {{DefineCategory |ob = unitary $R$-modules |mor = $R$-module homomorphisms |id = identity mappings |comp = composition of mappings }}
Definition:Forgetful Functor from Modules to Abelian Groups
Let $R$ be a ring. Let $\mathbf C$ be the category of left modules or category of left modules over $R$. Let $\mathbf{Ab}$ be the category of abelian groups. The '''forgetful functor''' $\mathbf C \to \mathbf{Ab}$ is the covariant functor with {{DefineFunctor | ob = sends a left module or right module to its underlying abelian group. | mor = sends a mapping to itself }}
Definition:Associated Z-Module Functor
Let $\mathbf{Ab}$ be the category of abelian groups. Let $\Z$ be the ring of integers and $\mathbf{Z-Mod}$ the category of unitary modules over $\Z$. The '''associated Z-module functor''' $\mathbf{Ab} \to \mathbf{Z-Mod}$ is the covariant functor with {{DefineFunctor |ob = sends an abelian group to its associated Z-module |mor = sends a mapping to itself }}
Definition:Jacobi Theta Function
Let $\tau$ be a complex constant with a positive imaginary part.  Let $q = e^{i \pi \tau}$.  Then the '''Jacobi Theta functions''' are defined for all complex $z$ by: === First Type === {{:Definition:Jacobi Theta Function/First Type}} === Second Type === {{:Definition:Jacobi Theta Function/Second Type}} === Third Type === {{:Definition:Jacobi Theta Function/Third Type}} === Fourth Type === {{:Definition:Jacobi Theta Function/Fourth Type}}
Definition:Jacobi Theta Function/First Type
:$\displaystyle \vartheta_1 \left({z, q}\right) = 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{\left({n + \frac 1 2}\right)^2} \sin \left({2 n + 1}\right) z$
Definition:Jacobi Theta Function/Second Type
:$\displaystyle \vartheta_2 \left({z, q}\right) = 2 \sum_{n \mathop = 0}^\infty q^{\left({n + \frac 1 2}\right)^2} \cos \left({2 n + 1}\right) z$
Definition:Jacobi Theta Function/Third Type
:$\displaystyle \vartheta_3 \left({z, q}\right) = 1 + 2 \sum_{n \mathop = 0}^\infty q^{n^2} \cos 2 n z$
Definition:Jacobi Theta Function/Fourth Type
:$\displaystyle \vartheta_4 \left({z, q}\right) = 1 + 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{n^2} \cos 2 n z$
Definition:Leibniz Law
Let $R$ be a ring. Let $A$ be a $R$-algebra. Then a function $f : A \to A$ is said to be satisfying '''Leibniz Law''' if the following holds: :$\forall a, b \in A: \map f {a b} = \map f a b + a \map f b$
Definition:Compatible Module Structures
Let $A$ and $B$ be rings. Let $\struct {M, +}$ be an abelian group. Let $* : A \times M \to M$ and $\circledast: B \times M \to M$ be left or right linear ring actions so that: :$(1): \quad \struct {M, +, *}$ is a left or right module over $A$ :$(2): \quad \struct {M, +, \circledast}$ is a left or right module over $B$ === Definition 1 === The module structures are '''compatible'''  for all $a \in A$, $b \in B$, the homotheties $h_a$ and $h_b$ commute. That is, for all $m \in M$, $a \in A$, $b \in B$: :$a * \paren {b \circledast m} = b \circledast \paren {a * m}$ === Definition 2 === The module structures are '''compatible'''  for all $a \in A$, the homothety $h_a : M \to M$ is an endomorphism of the $B$-module $M$. That is,  the image of the ring representation $A \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_B } M$. === Definition 3 === The module structures are '''compatible'''  for all $b \in A$, the homothety $h_b : M \to M$ is an endomorphism of the $A$-module $M$. That is,  the image of the ring representation $B \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_A} M$.
Definition:Invariant Mapping Under Equivalence Relation
Let $S$ and $T$ be sets. Let $\mathcal R$ be an equivalence relation on $S$. Let $f: S \to T$ be a mapping. Then $f$ is '''invariant under $\mathcal R$''' : :$x \mathrel {\mathcal R} y \implies f \left({x}\right) = f \left({y}\right) $
Definition:Complement of Subgroup/Definition 1
$K$ is a '''complement''' of $H$ : : $G = H K$ and $H \cap K = \set e$
Definition:Complement of Subgroup/Definition 2
$K$ is a '''complement''' of $H$ : :$G = K H$ and $H \cap K = \set e$
Definition:Finer Equivalence Relation
Let $X$ be a set. Let $\equiv$ and $\sim$ be equivalence relations on $X$. Then $\equiv$ is '''finer''' than $\sim$ : :$\forall x, y \in X : x \equiv y \implies x \sim y$
Definition:Finer Topology/Definition 1
$\tau_1$ is '''finer''' than $\tau_2$  $\tau_1 \supseteq \tau_2$.
Definition:Gaussian Hypergeometric Function
The '''Gaussian Hypergeometric Function''' is a hypergeometric function, given for $\size z < 1$ by: :$\displaystyle {}_2 \map {F_1} {a, b; c; z} = \sum_{n \mathop = 0}^\infty \dfrac { a^{\overline n} b^{\overline n} } { c^{\overline n} } \dfrac {z^n} {n!}$ where $x^{\overline n}$ denotes the $n$th rising factorial power of $x$.
Definition:Local Ring/Commutative/Definition 1
The ring $A$ is '''local'''  it has a unique maximal ideal.
Definition:Local Ring/Noncommutative
Let $\struct {R, +, \circ}$ be a ring with unity. === Definition 1 === {{Definition:Local Ring/Noncommutative/Definition 1}} === Definition 2 === {{Definition:Local Ring/Noncommutative/Definition 2}} === Definition 3 === {{Definition:Local Ring/Noncommutative/Definition 3}} === Definition 4 === $R$ is a '''local ring'''  it is nontrivial and the sum of any two non-units is a non-unit.  * The zero does not equal the unity, and for all $a \in R$, either $a$ or $1 + \paren {-a}$ is a unit. * If the summation $\displaystyle \sum_{i \mathop = 1}^n a_i$ is a unit, then some of the $a_i$ are also units (in particular the empty sum is not a unit).
Definition:Local Ring/Noncommutative/Definition 1
$R$ is a '''local ring'''  it has a unique maximal left ideal.
Definition:Local Ring/Noncommutative/Definition 2
$R$ is a '''local ring'''  it has a unique maximal right ideal.
Definition:Local Ring/Noncommutative/Definition 3
Let $\operatorname {rad} R$ be its Jacobson radical. Then $R$ is a '''local ring'''  the quotient ring $R / \operatorname{rad} R$ is a division ring.
Definition:Local Ring Homomorphism/Definition 1
The homomorphism $f$ is '''local'''  the image $f(\mathfrak m) \subseteq \mathfrak n$.
Definition:Local Ring Homomorphism/Definition 2
The homomorphism $f$ is '''local'''  the preimage $f^{-1}(\mathfrak n) \supseteq \mathfrak m$.
Definition:Local Ring Homomorphism/Definition 3
The homomorphism $f$ is '''local'''  the preimage $f^{-1}(\mathfrak n) = \mathfrak m$.
Definition:Tensor Product of Abelian Groups/Family
Let $I$ be an indexing set. Let $\family {G_i}_{i \mathop \in I}$ be a family of abelian groups. Let $G = \displaystyle \prod_{i \mathop \in I} G_i$ be their direct product. === Definition 1: by universal property === {{Definition:Tensor Product of Abelian Groups/Family/Definition 1}} === Definition 2: construction === {{Definition:Tensor Product of Abelian Groups/Family/Definition 2}}
Definition:Free Module on Set/Canonical Mapping
The '''canonical mapping''' $I \to R^{(I)}$ is the mapping that sends $i \in I$ to the $i$th standard basis element $e_i$.
Definition:Tensor Product of Abelian Groups/Family/Definition 1
Their '''tensor product''' is a pair $\left( \displaystyle \bigotimes_{i \in I} G_i, \theta \right)$ where: * $\displaystyle \bigotimes_{i \in I} G_i$ is an abelian group * $\theta : G \to \displaystyle\bigotimes_{i \in I} G_i$ is a multiadditive mapping such that, for every pair $(C, \omega)$ where: * $C$ is an abelian group * $\omega : G \to C$ is a multiadditive mapping there exists a unique group homomorphism $g : \displaystyle \bigotimes_{i \in I} G_i \to C$ with $\omega = g \circ \theta$. :$\xymatrix{ G \ar[d]_\theta \ar[r]^\omega & C\\ \displaystyle \bigotimes_{i \in I} G_i \ar@{.>}[ru]_g }$
Definition:Tensor Product of Abelian Groups/Family/Definition 2
Their '''tensor product''' is the pair: :$\tuple {\displaystyle \bigotimes_{i \mathop \in I} G_i, \theta}$ where: *$\displaystyle \bigotimes_{i \in I} G_i$ is the quotient of the free abelian group $\Z \sqbrk G$ on $G$, by the subgroup generated by the elements of the form $\tuple {x + y, \family {z_i}_{i \mathop \ne j} } - \tuple {x, \family {z_i}_{i \mathop \ne j} } - \tuple {y, \family {z_i}_{i \mathop \ne j} }$ *:for $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \displaystyle \prod_{i \mathop \ne j} G_i$, where we denote $\tuple {x, \family {z_i}_{i \mathop \ne j} }$ for: *:*the family in $G$ whose $j$th term is $x$ and whose $i$th term is $z_i$, for $i \ne j$ *:*its image under the canonical mapping $G \to \Z \sqbrk G$. * $\theta : G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$ is the composition of the canonical mapping $G \to \Z \sqbrk G$ with the quotient group epimorphism $\Z \sqbrk G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$: *:$G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \displaystyle \bigotimes_{i \mathop \in I} G_i$
Definition:Hurwitz Zeta Function
The '''Hurwitz zeta function''' is a generalisation of the Riemann zeta function. It is defined for $\map \Re s > 1$ and $\map \Re q > 0$, by:  :$\displaystyle \map \zeta {s, q} = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {n + q}^s}$
Definition:Ordered Tuple/Empty
Let $S$ be a set. The '''empty ordered tuple on $S$''' is the empty mapping: :$\varnothing \to S$ from the empty set $\varnothing$ to $S$.
Definition:Ordered Tuple/On Set
Let $S$ be a set. Let $s: \N^*_n \to S$ be an ordered tuple. Then $s$ is called an '''ordered tuple on $S$''', its codomain.
Definition:Weierstrass E-Function/Definition 1
The following mapping is known as the '''Weierstrass E-Function''' of $J \sqbrk {\mathbf y}$: :$\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} F_{\mathbf y'} \paren {x, \mathbf y, \mathbf z}$
Definition:Weierstrass E-Function/Definition 2
Let $\theta \in \R: 0 < \theta < 1$. The following mapping is known as the '''Weierstrass E-Function of $J \sqbrk {\mathbf y}$''': :$\ds \map E {x, \mathbf y, \mathbf z, \mathbf w} = \frac 1 2 \sum_{i, k \mathop = 1}^n \paren {w_i - z_i} \paren {w_k - z_k} F_{y_i' y_k'} \paren {x, \mathbf y, \mathbf z + \theta \paren {\mathbf w - \mathbf z} }$
Definition:Euler Numbers/Sequence
The sequence of Euler numbers begins: {{begin-eqn}} {{eqn | l = E_0       | r = 1       | c =  }} {{eqn | l = E_2       | r = -1       | c =  }} {{eqn | l = E_4       | r = 5       | c =  }} {{eqn | l = E_6       | r = -61       | c =  }} {{eqn | l = E_8       | r = 1385       | c =  }} {{eqn | l = E_{10}       | r = -50 \, 521       | c =  }} {{eqn | l = E_{12}       | r = 2 \, 702 \, 765       | c =  }} {{eqn | l = E_{14}       | r = -199 \, 360 \, 981       | c =  }} {{eqn | l = E_{16}       | r = 19 \, 391 \, 512 \, 145       | c =  }} {{eqn | l = E_{18}       | r = -2 \, 404 \, 879 \, 675 \, 441       | c =  }} {{eqn | l = E_{20}       | r = 370 \, 371 \, 188 \, 237 \, 525       | c =  }} {{eqn | l = E_{22}       | r = -69 \, 348 \, 874 \, 393 \, 137 \, 901       | c =  }} {{eqn | l = E_{24}       | r = 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905       | c =  }} {{end-eqn}} Odd index Euler numbers are all $0$.
Definition:Million
'''Million''' is a name for $10^6$. :$1 \, 000 \, 000 = 1000 \times 1000$
Definition:Variance
=== Discrete Random Variable === {{:Definition:Variance of Discrete Random Variable}} === Continuous Random Variable === {{:Definition:Variance of Continuous Random Variable}}
Definition:Variance/Continuous
Let $X$ be a continuous random variable.  Then the '''variance of $X$''', written $\var X$, is a measure of how much the values of $X$ varies from the expectation $\expect X$, and is defined as: :$\var X := \expect {\paren {X - \expect X}^2}$ That is, the expectation of the squares of the deviations from the expectation.
Definition:Taylor Series/Two Variables
Let $f: \R^2 \to \R$ be a real-valued function of $2$ variables which is smooth on the open rectangle $\left({a \,.\,.\, b}\right) \times \left({c \,.\,.\, d}\right)$. Let $\left({\xi, \zeta}\right) \in \left({a \,.\,.\, b}\right) \times \left({c \,.\,.\, d}\right)$. Then the '''Taylor series expansion of $f$''' about $\left({\xi, \zeta}\right)$ is: {{begin-eqn}} {{eqn | l = f \left({x, y}\right)       | r = f \left({\xi, \zeta}\right) + \left({x - \xi}\right) f_x \left({\xi, \zeta}\right) + \left({y - \zeta}\right) f_y \left({\xi, \zeta}\right)       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {2!} \left({\left({x - \xi}\right)^2 f_{xx} \left({\xi, \zeta}\right) + 2 \left({x - \xi}\right) \left({y - \zeta}\right) f_{xy} \left({\xi, \zeta}\right) + \left({y - \zeta}\right)^2 f_{yy} \left({\xi, \zeta}\right) }\right)       | c =  }} {{eqn | o =        | ro= +       | r = \cdots       | c =  }} {{end-eqn}} where $f_x \left({\xi, \zeta}\right)$, $f_y \left({\xi, \zeta}\right)$ denote partial derivatives {{WRT|Differentiation}} $x, y, \ldots$ evaluated at $x = \xi$, $y = \zeta$.
Definition:Uniform Distribution/Continuous
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $a, b \in \R$ such that $a < b$. $X$ is said to be '''uniformly distributed''' on the closed real interval $\closedint a b$  it has probability density function:  :$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$  This is written:  :$X \sim \ContinuousUniform a b$
Definition:Median of Continuous Random Variable
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have probability density function $f_X$.  A '''median''' of $X$ is defined as a real number $m_X$ such that:  :$\displaystyle \map \Pr {X < m_X} = \int_{-\infty}^{m_X} \map {f_X} x \rd x = \frac 1 2$
Definition:Mode of Continuous Random Variable
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have probability density function $f_X$.  We call $M$ a '''mode''' of $X$ if $f_X$ attains its (global) maximum at $M$.
Definition:Gamma Distribution
Let $X$ be a continuous random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Let $\operatorname{Im} \left({X}\right) = \R_{\ge 0}$. $X$ is said to have a '''Gamma distribution''' if it has probability density function:  :$\displaystyle f_X\left({x}\right) = \frac{ \beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\Gamma \left({\alpha}\right)}$ for $\alpha, \beta > 0$, where $\Gamma$ is the Gamma function. This is written:  :$X \sim \Gamma \left({\alpha, \beta}\right)$
Definition:Bernoulli Numbers/Archaic Form/Definition 1
{{begin-eqn}} {{eqn | l = \frac x {e^x - 1}       | r = 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = 1 - \frac x 2 + \frac{B_1^* x^2} {2!} - \frac{B_2^* x^4} {4!} + \frac{B_3^* x^6} {6!} - \cdots       | c =  }} {{end-eqn}} for $x \in \R$ such that $\left\lvert{x}\right\rvert < 2 \pi$
Definition:Bernoulli Numbers/Archaic Form/Sequence
The sequence of old style Bernoulli numbers begins: {{begin-eqn}} {{eqn | l = B_1^*       | r = \dfrac 1 6       | c =  }} {{eqn | l = B_2^*       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_3^*       | r = \dfrac 1 {42}       | c =  }} {{eqn | l = B_4^*       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_5^*       | r = \dfrac 5 {66}       | c =  }} {{eqn | l = B_6^*       | r = \dfrac {691} {2730}       | c =  }} {{eqn | l = B_7^*       | r = \dfrac 7 6       | c =  }} {{eqn | l = B_8^*       | r = \dfrac {3617} {510}       | c =  }} {{eqn | l = B_9^*       | r = \dfrac {43 \, 867} {798}       | c =  }} {{eqn | l = B_{10}^*       | r = \dfrac {174 \, 611} {330}       | c =  }} {{eqn | l = B_{11}^*       | r = \dfrac {854 \, 513} {138}       | c =  }} {{eqn | l = B_{12}^*       | r = \dfrac {236 \, 364 \, 091} {2730}       | c =  }} {{end-eqn}}
Definition:Bernoulli Numbers/Archaic Form/Definition 2
:$\displaystyle 1 - \frac x 2 \cot \frac x 2 = \sum_{n \mathop = 1}^\infty \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}$ {{begin-eqn}} {{eqn | l = 1 - \frac x 2 \cot \frac x 2       | r = \sum_{n \mathop = 1}^\infty \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac{B_1^* x^2} {2!} + \frac{B_2^* x^4} {4!} + \frac{B_3^* x^6} {6!} + \cdots       | c =  }} {{end-eqn}} for $x \in \R$ such that $\left\lvert{x}\right\rvert < \pi$
Definition:Euler Numbers/Alternative Form
An alternative form of the Euler numbers can often be found. Usually denoted with the symbol $E_n^*$, they are generally not used on . === Definition 1 === {{:Definition:Euler Numbers/Alternative Form/Definition 1}} === Definition 2 === {{:Definition:Euler Numbers/Alternative Form/Definition 2}}
Definition:Euler Numbers/Alternative Form/Definition 1
{{begin-eqn}} {{eqn | l = \sech x       | r = 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {E_n^* x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} - \frac {E_3^* x^6} {6!} + \cdots       | c =  }} {{end-eqn}} where $\size x < \dfrac \pi 2 $.
Definition:Euler Numbers/Alternative Form/Definition 2
{{begin-eqn}} {{eqn | l = \sec x       | r = 1 + \sum_{n \mathop = 1}^\infty \frac {E^*_n x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 + \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} + \frac {E_3^* x^6} {6!} + \cdots       | c =  }} {{end-eqn}} where $\size x < \dfrac \pi 2$.
Definition:Euler Numbers/Alternative Form/Sequence
The sequence of the alternative form of Euler numbers begins: {{begin-eqn}} {{eqn | l = E_1^*       | r = 1       | c =  }} {{eqn | l = E_2^*       | r = 5       | c =  }} {{eqn | l = E_3^*       | r = 61       | c =  }} {{eqn | l = E_4^*       | r = 1385       | c =  }} {{eqn | l = E_5^*       | r = 50 \, 521       | c =  }} {{eqn | l = E_6^*       | r = 2 \, 702 \, 765       | c =  }} {{eqn | l = E_7^*       | r = 199 \, 360 \, 981       | c =  }} {{eqn | l = E_8^*       | r = 19 \, 391 \, 512 \, 145       | c =  }} {{eqn | l = E_9^*       | r = 2 \, 404 \, 879 \, 675 \, 441       | c =  }} {{eqn | l = E_{10}^*       | r = 370 \, 371 \, 188 \, 237 \, 525       | c =  }} {{eqn | l = E_{11}^*       | r = 69 \, 348 \, 874 \, 393 \, 137 \, 901       | c =  }} {{eqn | l = E_{12}^*       | r = 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905       | c =  }} {{end-eqn}}
Definition:Moment Generating Function
Let $X$ be a random variable.  Then the '''moment generating function''' of $X$, $M_X$, is defined as:  :$\map {M_X} t = \expect {e^{t X} }$ for all $t$ such that this expectation exists.
Definition:Velocity/Dimension
The dimension of measurement of '''velocity''' is $\mathsf {L T}^{-1}$.
Definition:Velocity/Units
=== SI === {{:Definition:Velocity/Units/SI}} === CGS === {{:Definition:Velocity/Units/CGS}} === FPS === {{:Definition:Velocity/Units/FPS}}
Definition:Vector Quantity/Component
Let $\mathbf a$ be a vector quantity embedded in an $n$-dimensional Cartesian coordinate system $C_n$. Let $\mathbf a$ be represented with its initial point at the origin of $C_n$. Let $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ be the unit vectors in the positive direction of the coordinate axes of $C_n$. Then: :$\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$ where: :$a_1 \mathbf e_1, a_2 \mathbf e_2, \ldots, a_3 \mathbf e_n$ are the '''component vectors''' of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$  :$a_1, a_2, \ldots, a_3$ are the '''components''' of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$. The number of '''components''' in $\mathbf a$ is determined by the number of dimensions in the Cartesian coordinate system of its frame of reference.
Definition:Differential Entropy
'''Differential entropy''' extends the concept of entropy to continuous random variables. Let $X$ be a continuous random variable. Let $X$ have probability density function $f_X$.  Then the '''differential entropy''' of $X$, $\map h X$ measured in nats, is given by:  :$\ds \map h X = -\int_{-\infty}^\infty \map {f_X} x \ln \map {f_X} x \rd x$ Where $\map {f_X} x = 0$, we take $\map {f_X} x \ln \map {f_X} x = 0$ by convention.
Definition:Vector Calculus
'''Vector calculus''' is the branch of linear algebra concerned with the application of the techniques of calculus to vector spaces.
Definition:Del Operator
Let $\mathbf V$ be a vector space of $n$ dimensions. Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$. The '''del operator''' is a unary operator on $\mathbf V$ defined as: :$\nabla := \displaystyle \sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k}$ where $\mathbf v = \displaystyle \sum_{k \mathop = 0}^n x_k \mathbf e_k$ is an arbitrary vector of $\mathbf V$.