Definition:Structure Sheaf of Spectrum of Ring

Let $A$ be a commutative ring with unity. Let $\struct {\Spec A, \tau}$ be its spectrum with Zariski topology $\tau$ === Definition 1 === Note that Principal Open Subsets form Basis of Zariski Topology on Prime Spectrum. We define the structure sheaf of $\Spec A$ to be the sheaf induced by a sheaf on this basis defined as follows: :For $f \in A$, $\map \OO {\map X f}$ is the localization of $A$ at $f$ :For $f, g \in A$ with $\map X f \supset \map X g$, the restriction is the induced homomorphism of $A$-algebras $A_f \to A_g$. === Definition 2 === {{definition wanted|the étalé space point of view}} === Definition 3 === {{definition wanted|using explicit formulas}}


Definition:Irreducible Scheme

Let $\struct {X, \mathcal O_X}$ be a scheme. Then $\struct {X, \mathcal O_X}$ is '''irreducible''' {{iff}} $X$ is an irreducible space.


Definition:Local Ring Homomorphism

Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings. Let $f : A \to B$ be a unital ring homomorphism. === Definition 1 === {{Definition:Local Ring Homomorphism/Definition 1}} === Definition 2 === {{Definition:Local Ring Homomorphism/Definition 2}} === Definition 3 === {{Definition:Local Ring Homomorphism/Definition 3}}


Definition:Quasicompact Scheme

Let $\struct {X, \mathcal O_X}$ be a scheme. Then $X$ is '''quasicompact''' {{iff}} $X$ is a compact space.


Definition:Preadditive Category

A '''preadditive category''' is a monoidally enriched category over the monoidal category of abelian groups $\mathbf {A b}$. That is, a category such that: :its hom sets are abelian groups and where: :composition is bilinear.


Definition:Sequence of Partial Quotients

Let $F$ be a field. Let $n \in \N\cup\{\infty\}$ be an extended natural number. Let $C$ be a continued fraction in $F$ of length $n$. The '''sequence of partial quotients''' of $C$ is just $C$ itself. That is, a continued fraction equals its '''sequence of partial quotients'''.


Definition:Degenerate Connected Set/Topological Space

$T$ is a '''degenerate connected space''' {{iff}} it contains exactly one element.


Definition:Canonical Form

A '''canonical form''' of a mathematical object is a standard way of presenting that object as a mathematical expression.


Definition:Ordered Set of Natural Numbers

The '''ordered set of natural numbers''' is the ordered set $(\N, \leq)$ where: :$\N$ is the set of natural numbers :$\leq$ is the ordering on the natural numbers


Definition:Extended Natural Numbers

Let $\N$ be the set of natural numbers. === Definition 1 === The '''extended natural numbers''' are the elements of the ordered set $\struct {\N \cup \set \infty, \le}$, where :$\infty$ is a new element, called '''infinity''' :$\le$ is the ordering on the extended natural numbers === Definition 2 === The '''extended natural numbers''' are the order completion of the ordered set of natural numbers.


Definition:Length of Continued Fraction

Let $k$ be a field. Let $C$ be a continued fraction in $k$, either finite or infinite. The '''length''' of $C$ is an extended natural number equal to: *$\infty$ if $C$ is an infinite continued fraction. *$n$ if $C$ is a finite continued fraction with domain the integer interval $\left[0 \,.\,.\, n\right]$.


Definition:Nonzero Ring Element

Let $R$ be a ring. A '''nonzero element''' of $R$ is an element that is not its ring zero.


Definition:Divisor-Finite Monoid

Let $(M, *)$ be a monoid. Then $M$ is '''divisor-finite''' {{iff}} for all $m \in M$ the set: :$\{(x, y) \in M^2 : x*y = m \}$ is finite.


Definition:Convolution of Mappings on Divisor-Finite Monoid

Let $\struct {M, \cdot}$ be a divisor-finite monoid. Let $\struct {R, +, \times}$ be a non-associative ring. Let $f, g : M \to R$ be mappings. The '''convolution''' of $f$ and $g$ is the mapping $f * g: M \to R$ defined as: :$\forall m \in M: \map {\paren {f * g} } m := \displaystyle \sum_{x y \mathop = m} \map f x \times \map g y$ where the summation is over the finite set $\set {\tuple {x, y} \in M^2: x y = m}$.


Definition:Big Monoid Ring

Let $(M,\cdot)$ be a divisor-finite monoid. Let $(R, +, \times)$ be an additive semiring. The '''big monoid ring''' of $R$ over $M$ is the ringoid $(R^M, +, *)$ where: :$R^M$ be the set of all mappings $M \to R$ :$+$ is the pointwise operation induced by $+$ :$*$ denotes convolution of mappings


Definition:Generic Point of Topological Space

Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$ be an element of $S$. === Definition 1 === The point $x$ is a '''generic point of $T$''' {{iff}} the closure of the singleton $\left\{{x}\right\}$ is $S$. === Definition 2 === The point $x$ is a '''generic point of $T$''' {{iff}} $x$ is contained in every non-empty open subset of $T$.


Definition:Sober Space

Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is a '''sober space''' {{iff}} each closed irreducible subspace of $T$ has a unique generic point. Category:Definitions/Topology iiihtfrup7ghe0k7t25kuwvjw1yyw1s


Definition:Ring of Formal Power Series

Let $R$ be a commutative ring with unity. === One variable === A '''ring of formal power series''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\left({RX, \iota, X}\right)$ where: :$RX$ is a commutative ring with unity :$\iota : R \to RX$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $RX$, called indeterminate that may be defined as follows: Let $\N$ be the additive monoid of natural numbers. Let $R\N$ be the big monoid ring of $R$ over $\N$. Let $\iota : R \to R\N$ be the embedding. Let $X \in R\N$ be the mapping $X : \N \to R$ defined by $X(n) = 1$ if $n=1$ and $X(n) = 0$ otherwise. The '''ring of formal power series''' over $R$ is the ordered triple $\left({R\N, \iota, X}\right)$ === Multiple variables === {{definition wanted}}


Definition:Ring of Formal Laurent Series

A '''ring of formal Laurent series in one variable''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\tuple {\map R {\paren X}, \iota, X}$ where: :$\map R {\paren X}$ is a commutative ring with unity :$\iota: R \to \map R {\paren X}$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $\map R {\paren X}$, called variable that may be defined as follows: {{explain|Research that double parenthesis notation and investigate its validity and specific meaning}} Let $\tuple {R \sqbrk {\sqbrk X}, \kappa, X}$ be a ring of formal power series in one variable over $R$. Let $\tuple {\map R {\paren X}, \lambda}$ be the localization of $R \sqbrk {\sqbrk X}$ at $X$. The '''ring of formal Laurent series''' over $R$ is the ordered triple $\tuple {\map R {\paren X}, \lambda \circ \kappa, \map \lambda X}$.


Definition:Ring of Formal Dirichlet Series

The '''ring of formal Dirichlet series''' is the topological ring $(\mathcal A, \tau)$ where: :$\mathcal A$ is the ring of arithmetic functions :$\tau$ is the topology on formal dirichlet series Category:Definitions/Dirichlet Series l5z5hd2tvlwjh8bdwmnpwvrr0dx7dw2


Definition:Field of Formal Laurent Series

Let $k$ be a field. The ring of formal Laurent series $k((X))$ over $k$ in a variable $X$ is also called the '''field of formal Laurent series'''.


Definition:Continued Fraction Expansion of Laurent Series

Let $k$ be a field. Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$. === Irrational Laurent series === Let $f \in k((t^{-1}))$ be an irrational formal Laurent series. The '''continued fraction expansion''' of $f$ is the infinite continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as: :$\alpha_n = \displaystyle \begin{cases} f & : n = 0 \\  \dfrac 1 {f - \lfloor f \rfloor} & : n \geq 1 \end{cases}$ where $\lfloor \cdot \rfloor$ denotes the polynomial part. === Rational Laurent series === {{definition wanted}}


Definition:Irreducible Space/Definition 6

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} the closure of every non-empty open set is the whole space: :$\forall U \in \tau: U^- = S$


Definition:Ultraconnected Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''ultraconnected''' {{iff}} no two non-empty closed sets are disjoint.


Definition:Ultraconnected Space/Definition 2

A topological space $T = \struct {S, \tau}$ is '''ultraconnected''' {{iff}} the closures of every distinct pair of elements of $S$ are not disjoint: :$\forall x, y \in S: \set x^- \cap \set y^- \ne \O$


Definition:Irreducible Space/Definition 7

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} every open set of $T$ is connected.


Definition:Locally Connected Space/Definition 4

A topological space $T = \struct {S, \tau}$ is '''locally connected''' {{iff}} the components of the open sets of $T$ are also open in $T$.


Definition:Convergent of Continued Fraction/Definition 1

The '''$k$th convergent''' $C_k$ of $C$ is the value of the finite continued fraction: :$C_k = \left[{a_0, a_1, \ldots, a_k}\right]$


Definition:Convergent of Continued Fraction/Definition 2

The '''$k$th convergent''' $C_k$ of $C$ is the quotient of the $k$th numerator $p_k$ by the $k$th denominator $q_k$: :$C_k = \dfrac{p_k}{q_k}$


Definition:New Element

Let $S$ be a set. A '''new element for $S$''' is a set that is not an element of $S$. By Set does not Contain Itself, it may be defined as $S$ itself.


Definition:Locally Path-Connected Space/Definition 4

A topological space $T = \struct{S, \tau}$ is a '''locally path-connected space''' {{iff}} the path components of open sets of $T$ are also open in $T$.


Definition:Weakly Locally Connected at Point/Definition 1

The space $T$ is '''weakly locally connected at $x$''' {{iff}} $x$ has a neighborhood basis consisting of connected sets.


Definition:Weakly Locally Connected at Point/Definition 2

The space $T$ is '''weakly locally connected at $x$''' {{iff}} every open neighborhood $U$ of $x$ contains an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$.


Definition:Fundamental Group Functor

Let $\mathbf{Top}^*$ be the category of pointed topological spaces. Let $\mathbf{Grp}$ be the category of groups. The '''fundamental group functor''' is the covariant functor $\pi_1 : \mathbf{Top}^* \to \mathbf{Grp}$ with: {{DefineFunctor |ob = A pointed topological space $(X, x_0)$ is sent to its fundamental group. |mor = A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\pi_1(f) = f_*$. }}


Definition:Pointed Topological Space

A '''pointed topological space''' is an ordered pair $\left({T, x_0}\right)$ where: :$T$ is a topological space :$x_0$ is a point of $T$.


Definition:Discrete Category on Set

Let $S$ be a set. The '''discrete category on $S$''' is the discrete category $\mathbf{Dis} \left({S}\right)$ with: {{DefineCategory |ob = elements of $S$ |mor = only identity mappings |comp = composition of mappings }}


Definition:Graded Object

Let $C$ be a category. Let $S$ be a set. Let $\mathbf{Dis}(S)$ be its discrete category. An '''$S$-graded object''' of $C$ is a covariant functor $\mathbf{Dis}(S) \to C$.


Definition:Binary Biproduct

Let $A$ be a category. Let $a_1, a_2$ be objects of $A$. === Definition 1 === A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ such that: :$\tuple {a_1 \oplus a_2, p_1, p_2}$ is a binary product :$\tuple {a_1 \oplus a_2, i_1, i_2}$ is a binary coproduct === Definition 2: for categories with zero morphisms === Let $A$ be a category with zero morphisms. Then $a_1$ and $a_2$ are said to '''have a biproduct''' {{iff}}: :they have a coproduct $\tuple {a_1 \sqcup a_2, j_1, j_2}$ and a product $\tuple {a_1 \times a_2, q_1, q_2}$ :the canonical mapping $r: a_1 \sqcup a_2 \to a_1 \times a_2$ is an isomorphism in which case: :$\tuple {a_1 \sqcup a_2, j_1, j_2, q_1 \circ r, q_2 \circ r}$ :$\tuple {a_1 \times a_2, r \circ j_1, r \circ j_2, q_1, q_2}$ are '''biproducts''' of $a_1$ and $a_2$. === Definition 3: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \oplus a_2$ is an object of $A$ :$i_1 : a_1 \to a_1 \oplus a_2$ :$i_2 : a_1 \to a_1 \oplus a_2$ :$p_1 : a_1 \oplus a_2 \to a_2$ :$p_2 : a_1 \oplus a_2 \to a_2$ are morphisms such that: :$p_1 \circ i_1 = 1_{a_1}$ :$p_2 \circ i_2 = 1_{a_2}$ :$i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1 \oplus a_2}$ where $1$ denotes the identity morphism. === Definition 4: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \oplus a_2$ is an object of $A$ :$i_1 : a_1 \to a_1 \oplus a_2$ :$i_2 : a_1 \to a_1 \oplus a_2$ :$p_1 : a_1 \oplus a_2 \to a_2$ :$p_2 : a_1 \oplus a_2 \to a_2$ are morphisms such that: :$p_1 \circ i_1 = 1_{a_1}$ :$p_2 \circ i_2 = 1_{a_2}$ :$p_1 \circ i_2 = 0_{a_1}$ :$p_2 \circ i_1 = 0_{a_2}$ :$i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1 \oplus a_2}$ where: :$1$ denotes the identity morphism :$0$ denotes the zero morphism.


Definition:Zero Morphism in Preadditive Category

Let $A$ be a preadditive category. Let $a,b$ be objects of $A$. The '''zero morphism''' $0 : a \to b$ is the identity element of their hom set $\operatorname{Hom}(a, b)$.


Definition:Additive Category

Let $A$ be a preadditive category. === Definition 1: binary biproducts === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary biproducts. === Definition 2: binary products === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary products. === Definition 3: binary coproducts === The category $A$ is an '''additive category''' {{iff}}: * it has a zero object * it has all binary coproducts. === Definition 4: finite biproducts === The category $A$ is an '''additive category''' {{iff}} it has all finite biproducts. === Definition 5: finite products === The category $A$ is an '''additive category''' {{iff}} it has all finite products. === Definition 6: finite coproducts === The category $A$ is an '''additive category''' {{iff}} it has all finite coproducts.


Definition:Canonical Mapping from Coproduct to Product

Let $C$ be a category with zero morphisms. Let $a, b$ be objects of $C$. Assume they have a coproduct $(a \sqcup b, i_1, i_2)$ and a product $(a \times b, p_1, p_2)$. === Definition 1 === Let: :$j_1 : a \to a \times b$ be the unique morphism such that: :*$p_1 \circ j_1 = 1 : a \to a$ :*$p_2 \circ j_1 = 0 : a \to b$ :$j_2 : b \to a \times b$ be the unique morphism such that: :*$p_1 \circ j_2 = 0 : b \to a$ :*$p_2 \circ j_2 = 1 : b \to b$ The '''canonical mapping''' from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that: :$r \circ i_1 = j_1$ :$r \circ i_2 = j_2$ === Definition 2 === Let: :$q_1 : a \sqcup b \to a$ be the unique morphism such that: :*$q_1 \circ j_1 = 1 : a \to a$ :*$q_1 \circ j_2 = 0 : b \to a$ :$q_2 : a \sqcup b \to b$ be the unique morphism such that: :*$q_2 \circ j_1 = 0 : a \to a$ :*$q_2 \circ j_2 = 1 : b \to a$ The '''canonical mapping''' from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that: :$p_1 \circ r = q_1$ :$p_2 \circ r = q_2$ {{definition wanted|for arbitrarily many objects}}


Definition:Totally Separated Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''totally separated''' {{iff}}: :For every $x, y \in S: x \ne y$ there exists a separation $U \mid V$ of $T$ such that $x \in U, y \in V$.


Definition:Totally Separated Space/Definition 2

A topological space $T = \struct {S, \tau}$ is '''totally separated''' {{iff}} each of its quasicomponents is a singleton set.


Definition:Pre-Abelian Category

=== Definition 1 === A '''pre-abelian category''' is an additive category in which every morphism has a kernel and a cokernel. === Definition 2 === A '''pre-abelian category''' is an additive category with all finite limits and finite colimits.


Definition:Abelian Category

=== Definition 1 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is a kernel * every epimorphism is a cokernel === Definition 2 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is the kernel of its cokernel * every epimorphism is the cokernel of its kernel === Definition 3 === An '''abelian category''' is a pre-abelian category in which for every morphism $f$, the canonical morphism from its coimage to its image $\operatorname{coim}(f) \to \operatorname{im}(f)$ is an isomorphism.


Definition:Extremally Disconnected Space/Definition 1

A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected''' {{iff}} the closure of every open set of $T$ is open.


Definition:Extremally Disconnected Space/Definition 3

A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected''' {{iff}} the closures of every pair of open sets which are disjoint are also disjoint.


Definition:Extremally Disconnected Space/Definition 2

A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected''' {{iff}} the interior of every closed set of $T$ is closed.


Definition:Isometry (Metric Spaces)/Definition 1

Let $M_1 = \tuple {A_1, d_1}$ and $M_2 = \tuple {A_2, d_2}$ be metric spaces or pseudometric spaces. Let $\phi: A_1 \to A_2$ be a bijection such that: :$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$ Then $\phi$ is called an '''isometry'''. That is, an '''isometry''' is a distance-preserving bijection.


Definition:Isometry (Metric Spaces)/Definition 2

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces or pseudometric spaces. :$M_1$ and $M_2$ are '''isometric''' {{iff}} there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that: ::$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$ :and: ::$\forall u, v \in A_2: \map {d_2} {u, v} = \map {d_1} {\map {\phi^{-1} } u, \map {\phi^{-1} } v}$


Definition:Isometry (Metric Spaces)/Into

Let $\phi: A_1 \to A_2$ be an injection such that: : $\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$ Then $\phi$ is called an '''isometry (from $M_1$) ''into'' $M_2$'''.


Definition:Presheaf on Topological Space/Definition 1

A $\mathbf C$-valued '''presheaf''' on $T$ is a pair $\left({\mathcal F, \operatorname{res} }\right)$ where: : $\mathcal F$ is a mapping on $\tau$ whose image consists of objects of $\mathbf C$ : $\operatorname{res}$ is a mapping on $\left\{ {\left({U, V}\right) \in \tau^2: U \supseteq V}\right\}$ such that for all $U, V, W \in \tau$ with $U \supseteq V \supseteq W$: :: $\operatorname{res}_V^U$ is a morphism from $\mathcal F \left({U}\right)$ to $\mathcal F \left({V}\right)$ :: $\operatorname{res}_U^U = \operatorname{id}_{\mathcal F \left({U}\right)}$, the identity morphism on $\mathcal F \left({U}\right)$ :: $\operatorname{res}_V^U \circ \operatorname{res}_W^V = \operatorname{res}_W^U$, where $\circ$ is the composition in $\mathbf C$


Definition:Presheaf on Topological Space/Definition 2

Let $\tau$ be the category of open sets of $T$. A $\mathbf C$-valued '''presheaf''' on $T$ is a contravariant functor $\tau \to \mathbf C$.


Definition:Section of Étalé Space/Definition 1

A '''section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ on $U$ is a continuous mapping $s: U \to \map {\operatorname {\acute Et} } {\mathcal F}$ such that: :$\pi \circ s = I_U$ where $I_U$ is the identity mapping on $U$.


Definition:Section of Étalé Space/Definition 2

A '''section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ on $U$ is a mapping $s: U \to \map {\operatorname {\acute Et} } {\mathcal F}$ such that: :for all $x \in U$ there exists an open neighborhhood $V$ of $x$ in $U$ such that the restriction of $s$ to $V$ is the section associated to some $t \in \map {\mathcal F} V$.


Definition:Direct Image of Presheaf/Definition 1

The '''direct image''' of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_* \mathcal F$ on $T_2$ with: : $f_* \mathcal F \left({V}\right) = \mathcal F \left({f^{-1} \left({V}\right)}\right)$ for all open $V \subset T_2$ : Restrictions $\operatorname{res}^U_V = \operatorname{res}^{f^{-1} \left({U}\right)}_{f^{-1} \left({V}\right)}$


Definition:Direct Image of Presheaf/Definition 2

The '''direct image''' of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_*\mathcal F$ that is the composition $\mathcal F \circ \operatorname{Open} \left({f}\right)$, where $\operatorname{Open}$ is the open subsets functor.


Definition:Sheaf on Topological Space/Definition 1

A $\mathbf C$-valued '''sheaf''' $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$: :$\left({\mathcal F \left({U}\right), \left({\operatorname{res}^U_{U_i} }\right)_{i \mathop \in I} }\right)$ is the limit of the restriction of $\mathcal F$ to $\left\{ {U}\right\} \cup \left\{ {U_i: i \in I}\right\} \cup \left\{ {U_i \cap U_j : \left({i, j}\right) \in I^2}\right\}$


Definition:Sheaf on Topological Space/Definition 2

Let $\mathbf C$ be a complete category. {{definition wanted|using equalizers}}


Definition:Sheaf on Topological Space/Definition 3

Let $\mathbf C$ be a complete abelian category. A $\mathbf C$-valued '''sheaf''' $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$ the sequence: :$0 \to \mathcal F \left({U}\right) \to \displaystyle \prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \to \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right)$ is exact. {{explain|what the maps are}} {{definition wanted|using Definition:étalé Space of Presheaf, another via Definition:Sheaf of Sets}}


Definition:Edge Cut

Let $G$ be a graph. An '''edge cut''' of $G$ is a set of edges $W \subseteq \map E G$ such that the edge deletion $G \setminus W$ is disconnected.


Definition:Particular Point

Let $T = \struct {S, \tau}$ be a topological space. A '''particular point of $T$''' is an element $p \in S$ which is distinguished in some manner from the other elements of $S$.


Definition:Euclidean Space/Euclidean Topology/Real Number Line

Let $\R$ denote the real number line. Let $d: \R \times \R \to \R$ denote the Euclidean metric on $\R$. Let $\tau_d$ denote the topology on $\R$ induced by $d$. The topology $\tau_d$ induced by $d$ is called the '''Euclidean topology'''. Hence $\struct {\R, \tau_d}$ is referred to as the '''real number line with the Euclidean topology'''.


Definition:Euclidean Space/Euclidean Topology/Real Number Plane

Let $\R^n$ be an $n$-dimensional real vector space. Let $M = \left({\R^2, d}\right)$ be a real Euclidean space of $2$ dimensions. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''. The space $\left({\R^2, \tau_d}\right)$ is known as the '''(real) Euclidean plane'''.


Definition:Fréchet Space (Functional Analysis)

Let $\R^\omega$ denote the countable-dimensional real Cartesian space. Let: :$x := \family {x_i}_{i \mathop \in \N} = \tuple {x_0, x_1, x_2, \ldots}$ and: :$y := \family {y_i}_{i \mathop \in \N} = \tuple {y_0, y_1, y_2, \ldots}$ denote arbitrary elements of $\R^\omega$. Let the distance function $d: \R^\omega \times \R^\omega \to \R$ be applied to $\R^\omega$ as: :$\forall x, y \in \R^\omega: \map d {x, y} = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$ The distance function $d$ is referred to as the '''Fréchet (product) metric'''. The resulting metric space $\struct {\R^\omega, d}$ is then referred to as the '''Fréchet (metric) space'''.


Definition:Hilbert Cube

The '''Hilbert cube''' $\struct {I^\omega, d_2}$ is the subspace of the Hilbert sequence space $I^\omega$ defined as: :$\displaystyle I^\omega = \prod_{k \mathop \in \N} \closedint 0 {\dfrac 1 k}$ under the same metric as that of the Hilbert sequence space: :$\displaystyle \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$


Definition:Convex Set (Order Theory)/Definition 1

A subset $A$ of an ordered set $\left({S, \preceq}\right)$ is '''convex (in $S$)''' {{iff}}: :$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$


Definition:Convex Set (Order Theory)/Definition 2

A subset $A$ of an ordered set $\struct {S, \preceq}$ is '''convex (in $S$)''' {{iff}}: :$\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$


Definition:Interval/Open Interval

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Open Interval on Ordered Set === {{:Definition:Open Interval}} === Open Real Interval === In the context of the real number line $\R$: {{:Definition:Open Real Interval}}


Definition:Interval/Closed Interval

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Closed Interval on Ordered Set === {{:Definition:Closed Interval}} === Closed Real Interval === In the context of the real number line $\R$: {{:Definition:Closed Real Interval}}


Definition:Interval/Half-Open Interval

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Half-Open Real Interval === In the context of the real number line $\R$: {{:Definition:Half-Open Real Interval}}


Definition:Convex Component

Let $\struct {S, \preccurlyeq}$ be an ordered set. The maximally convex sets into which $S$ is partitioned are the '''convex components''' of $S$.


Definition:Interval/Ordered Set/Endpoint

The elements $a, b \in S$ are known as the '''endpoints''' (or '''end points''') of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand end point''' of the interval.


Definition:Ordinal Space

Let $\Gamma$ be a limit ordinal. === Open Ordinal Space === {{:Definition:Ordinal Space/Open}} === Closed Ordinal Space === {{:Definition:Ordinal Space/Closed}}


Definition:Uncountable Ordinal

Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''uncountable''' {{iff}} it is an uncountable set.


Definition:Ordinal Space/Closed

The '''closed ordinal space on $\Gamma$''' is the set $\closedint 0 \Gamma$ of all ordinal numbers less than or equal to $\Gamma$, together with the order topology.


Definition:Ordinal Space/Open

The '''open ordinal space on $\Gamma$''' is the set $\hointr 0 \Gamma$ of all ordinal numbers (strictly) less than $\Gamma$, together with the order topology.


Definition:Ordinal Space/Open/Countable

Let $\Omega$ denote the first uncountable ordinal. The '''countable open ordinal space on $\Gamma$''' is a particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma < \Omega$.


Definition:Ordinal Space/Closed/Uncountable

Let $\Omega$ denote the first uncountable ordinal. The '''uncountable closed ordinal space on $\Omega$''' is the particular case of a closed ordinal space $\closedint 0 \Gamma$ where $\Gamma = \Omega$.


Definition:Ordinal Space/Closed/Countable

Let $\Omega$ denote the first uncountable ordinal. The '''countable closed ordinal space on $\Gamma$''' is a particular case of an closed ordinal space $\closedint 0 \Gamma$ where $\Gamma < \Omega$.


Definition:Ordinal Space/Open/Uncountable

Let $\Omega$ denote the first uncountable ordinal. The '''uncountable open ordinal space on $\Omega$''' is the particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma = \Omega$.


Definition:Characteristic Polynomial of Element of Algebra

Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \in B$. The '''characteristic polynomial''' of $b$ is the characteristic polynomial of the regular representation $\lambda_b : B \to B$ over $A$.


Definition:Trace of Element of Algebra over Ring

Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \in B$. The '''trace''' $\operatorname{Tr}_{B/A}(b)$ of $b$ is the trace of the regular representation $\lambda_b : B \to B$ over $A$.


Definition:Norm of Element of Algebra over Ring

Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module  over $A$. Let $b \in B$. The '''trace''' $N_{B/A}(b)$ of $b$ is the determinant of the regular representation $\lambda_b : B \to B$ over $A$.


Definition:Characteristic Polynomial of Matrix

Let $R$ be a commutative ring with unity. Let $M$ be a square matrix over $R$ of order $n > 0$. Let $I_n$ be the $n\times n$ identity matrix. Let $R[x]$ be the polynomial ring in one variable over $R$. The '''characteristic polynomial''' of $M$ is the determinant of a matrix over $R[x]$: :$p_M (x) = \operatorname{det}(xI - M)$.


Definition:Torsion Element of Module

Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Let $m \in M$. Then $m$ is a '''torsion element''' {{iff}} there exists a regular element $a \in R$ with $a m = 0$.


Definition:Torsion Submodule

Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. The '''torsion submodule''' $T(M)$ of $M$ is the submodule of all torsion elements of $M$.


Definition:Torsion Module

Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Then $M$ is a '''torsion module''' {{iff}} every element of $M$ is of torsion, that is, $M$ equals its torsion submodule $T(M)$.


Definition:Zariski Topology on Maximal Spectrum of Ring

Let $A$ be a commutative ring with unity. Let $\operatorname{MaxSpec}(A)$ be its maximal spectrum. === Definition 1 === The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the topology with as closed sets the maximal zero loci. === Definition 2 === The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the subspace topology induced by the Zariski topology on the spectrum $\operatorname{Spec} A$.


Definition:Torsion Subgroup

Let $G$ be an abelian group. Its '''torsion subgroup''' $\map T G$ is the subgroup of all torsion elements.


Definition:Torsion Group

Let $G$ be a group. Then $G$ is a '''torsion group''' {{iff}} all its elements are torsion elements.


Definition:Purely Inseparable Field Extension

Let $E/F$ be an algebraic field extension. === Definition 1 === The extension $E/F$ is '''purely inseparable''' {{iff}} every element $\alpha \in E \setminus F$ is inseparable. === Definition 2 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} for each $\alpha \in E$ there exists $n \in \N$ such that $\alpha^{p^n} \in F$. === Definition 3 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$.


Definition:Initial Homomorphism from Integers to Ring with Unity

Let $\Z$ be the ring of integers. Let $R$ be a ring with unity. The '''initial homomorphism''' $\Z \to R$ is the unital ring homomorphism that sends $n \in \Z$ to the $n$th power of $1$ in $R$: :$ n \mapsto n \cdot 1$.


Definition:Positive Characteristic

Let $R$ be a ring with unity. Then $R$ has '''positive characteristic''' {{iff}} its characteristic is not $0$.


Definition:Inseparable Element of Field Extension

Let $E/F$ be a field extension. Let $\alpha \in E$ be algebraic over $F$. Then $\alpha$ is '''inseparable''' {{iff}} it is not separable.


Definition:Relative Separable Closure

Let $E/F$ be a field extension. The '''(relative) separable closure''' of $E$ in $F$ is the intermediate field consisting of all separable elements of $E$ over $F$.


Definition:Inseparable Field Extension

Let $E/F$ be an algebraic field extension. Then $E/F$ is '''inseparable''' {{iff}} it is not separable.


Definition:Separable Closure

Let $K$ be a field. A '''separable closure''' of $K$ is a separably closed algebraic field extension of $F$. It may be defined as the relative separable closure of $F$ in its algebraic closure.


Definition:Separably Closed Field

Let $K$ be a field. Then $K$ is '''separably closed''' {{iff}} === Definition 1 === :The only separable field extension of $K$ is $K$ itself. === Definition 2 === :Every separable irreducible polynomial over $K$ has degree $1$. === Definition 3 === :Every separable polynomial over $K$ of strictly positive degree has a root in $K$.


Definition:Separable Degree

Let $E / F$ be a field extension. Let $S \subseteq E$ be the separable closure of $F$ in $E$. The '''separable degree''' $\index E F_{\operatorname {sep} }$ of $E / F$ is the degree $\index S F$.


Definition:Inseparable Degree

Let $E/F$ be a field extension. Let $S \subseteq E$ be the separable closure of $F$ in $E$. The '''inseparable degree''' $[E : F]_i$ of $E/F$ is the degree $[E : S]$.


Definition:Absolute Galois Group

Let $K$ be a field. === Definition 1 === {{:Definition:Absolute Galois Group/Definition 1}} === Definition 2 === {{:Definition:Absolute Galois Group/Definition 2}}


Definition:Profinite Group

=== Definition 1 === A '''profinite group''' is a topological group that is isomorphic to a small inverse limit of finite discrete groups, with the inverse limit topology. === Definition 2 === A '''profinite group''' is a topological group that is compact, Hausdorff and totally disconnected.


Definition:Complete Order Topology

Let $\struct {S, \preceq}$ be a complete ordered set. Let $\tau$ be the order topology on $\struct {S, \preceq}$. Then $\tau$ is a '''complete order topology'''. Hence $\struct {S, \preceq, \tau}$ is a '''complete order space'''.


Definition:Integral Transform

Let $p$ be a variable whose domain is a subset of the set of real numbers $\R$. Let $\closedint a b$ be a closed real interval for some $a, b \in \R: a \le b$. Let $f: \closedint a b \to \R$ be a real function defined on the domain $\closedint a b$. Let $\map K {p, x}$ be a real-valued function defined for all $p$ in its domain and all $x \in \closedint a b$. Let $\map f x \map K {p, x}$ be integrable {{WRT|Integration}} $x$ for all $p$ in its domain and all $x \in \closedint a b$. Consider the real function $\map F p$ defined as: :$\map F p = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$ Then $\map F p$ is an '''integral transform''' of $\map f x$.


Definition:Integral Transform/Kernel

The function $K \left({p, x}\right)$ is the '''kernel''' of $F \left({p}\right)$.


Definition:Integral Transform/Operator

This can be written in the form: :$F = \map T f$ where $T$ is interpreted as the (unitary) operator meaning: :''Multiply this by $\map K {p, x}$ and integrate {{WRT|Integration}} $x$ between the limits $a$ and $b$.'' Thus $T$ transforms the function $\map f x$ into its image $\map F p$, which is itself another real function.


Definition:Integral Transform/Image Space

The domain of $p$ is known as the '''image space''' of $T$.


Definition:Inverse Integral Operator

Let $T: f \to F$ be an integral operator on a function $f$. Let there be a (unitary) operator $T^{-1}: F \to f$ such that for a given $F \left({p}\right)$ there exists a unique $f \left({x}\right)$ such that $f = T \left({f}\right)$. Then $T^{-1}$ is the '''inverse integral operator''' of $T$.


Definition:Inversion Theorem

Let $F \left({p}\right)$ be an integral transform: :$F \left({p}\right) = \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$ Let $T$ be the integral operator associated with $F \left({p}\right)$: :$F = t \left({f}\right)$ An '''inversion theorem''' is a specification for an inverse integral operator $T^{-1}$ of the form $f = T^{-1} \left({F}\right)$ such that: :$f \left({x}\right) = \displaystyle \int_\alpha^\beta F \left({p}\right) H \left({x, p}\right) \, \mathrm d p$ should such an $H \left({x, p}\right)$ exist. It is not necessarily the case that it does exist..


Definition:Trigonometric Series

A '''trigonometric series''' is a series of the type: :$\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ where: :the domain of $x$ is the set of real numbers $\R$ :the coefficients $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are real numbers independent of $x$.


Definition:Trigonometric Series/Complex Form

A '''trigonometric series''' can be expressed in a form using complex functions as follows: :$\map S x = \displaystyle \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$ where: :the domain of $x$ is the set of real numbers $\R$ :the coefficients $\ldots, c_{-n}, \ldots, c_{-2}, c_{-1}, c_0, c_1, c_2, \ldots, c_n, \ldots$ are real numbers independent of $x$ :$c_{-n} = \overline {c_n}$ where $\overline {c_n}$ is the complex conjugate of $c_n$.


Definition:Fourier Series/Fourier Coefficient/Range 2 Pi

The constants: : $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are the '''Fourier coefficients''' of $f$.


Definition:Dirichlet Conditions

{{begin-axiom}} {{axiom | n = \text D 1         | t = $\quad f$ is absolutely integrable }} {{axiom | n = \text D 2         | t = $\quad f$ has a finite number of local maxima and local minima }} {{axiom | n = \text D 3         | t = $\quad f$ has a finite number of discontinuities, all of them finite }} {{end-axiom}}


Definition:Absolutely Integrable Function

Let $f$ be a real function. $\map f x$ is '''absolutely integrable''' on $S \subseteq \R$ {{iff}} its absolute value of $f$ is an integrable function on $S$. That is, {{iff}} the definite integral of the absolute value of $f$ over any interval $\openint \alpha \beta \subseteq S$ is bounded. Category:Definitions/Integral Calculus 35eag25292vx0cl2p32luts6g380kx9


Definition:Fourier Series/Fourier Coefficient


Definition:Spence's Function

'''Spence's function''', also known as the '''dilogarithm''', is a special case of the polylogarithm, defined for $z \in \C$ by the integral:  :$\displaystyle \operatorname {Li}_2 \paren z = -\int_0^z \frac {\Ln \paren {1 - t} } t \rd t$ where: :$\displaystyle \int_0^z$ is an integral across the straight line in the complex plane connecting $0$ and $z$.  :$\Ln$ is the principal branch of the complex natural logarithm.


=== Formulation 1 === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Sine Series/Formulation 2}}


=== Formulation 1 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 2}}


Definition:Fourier Series/Formulation 1

Let $\alpha \in \R$ be a real number. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \cos \frac {n \pi x} \lambda \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x }} {{end-eqn}} Then: :$\displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ is the '''Fourier Series''' for $f$.


Definition:Fourier Series/Formulation 2

Let $a, b \in \R$ be real numbers. Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = A_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{eqn | l = B_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{end-eqn}} Then: :$\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$ is the '''Fourier Series''' for $f$.


Definition:Half-Range Fourier Sine Series/Formulation 2

Let $\map f x$ be a real function defined on the interval $\openint a b$. Then the ''' half-range Fourier sine series''' of $\map f x$ over $\openint a b$ is the series: :$\displaystyle \map f x \sim \sum_{m \mathop = 1}^\infty B_m \sin \frac {m \pi \paren {x - a} } {b - a}$ where for all $n \in \Z_{> 0}$: :$B_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \sin\frac {m \pi \paren {x - a} } {b - a} \rd x$


Definition:Half-Range Fourier Cosine Series/Formulation 2

Let $\map f x$ be a real function defined on the interval $\openint a b$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint a b$ is the series: :$\displaystyle \map f x \sim \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty A_m \cos \frac {m \pi \paren {x - a} } {b - a}$ where for all $m \in \Z_{\ge 0}$: :$A_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \cos \frac {m \pi \paren {x - a} } {b - a} \rd x$


Definition:Orthonormal Set of Real Functions

Let $I$ be an indexing set. Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an indexed family of real functions all of which are integrable over the interval $\openint a b$. Let $S$ have the property that: :$\forall m, n \in I: \displaystyle \int_a^b \map {\phi_m} x \map {\phi_n} x \rd x = \delta_{m n}$ where $\delta_{m n}$ denotes the Kronecker delta. Then $S$ is defined as '''orthonormal'''.


Definition:Complete Orthonormal Set of Real Functions

Let $I$ be an indexing set. Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an orthonormal set of real functions over the interval $\openint a b$. Let $S$ have the property that: :$\forall n \in I: \displaystyle \int_a^b \map \psi x \map {\phi_n} x \rd x = 0 \implies \map \psi x \equiv 0$ for any real function $\psi$ integrable over the interval $\openint a b$. Then $S$ is a '''complete orthonormal set''' of real functions.


Definition:Divisor of Polynomial

Let $D$ be an integral domain. Let $D \sqbrk x$ be the polynomial ring in one variable over $D$. Let $f, g \in D \sqbrk x$ be polynomials. Then: :'''$f$ divides $g$''' :'''$f$ is a divisor of $g$''' :'''$f$ is a factor of $g$''' :'''$g$ is divisible by $f$''' {{iff}}: :$\exists h \in D \sqbrk x : g = f h$ This is denoted: :$f \divides g$ === Notation === {{:Definition:Divisor Notation}}


Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1

The element $y$ is a '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ {{iff}} there exists a mapping $a: I \to \N$ with finite support such that: :$y = \ds \prod_{i \mathop \in I} X_i^{a_i}$ where: :$\prod$ denotes the product with finite support over $I$ :$X_i^{a_i}$ denotes the $a_i$th power of $X_i$.


Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2

The element $y$ is a '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ {{iff}} there exists a finite set $S$ and a mapping $f: S \to \set {X_i : i \in I}$ such that it equals :$y = \ds \prod_{s \mathop \in S} \map f s$ where $\prod$ denotes the product over the finite set $S$.


Definition:Inclusion-Preserving Mapping

Let $A$ and $B$ be sets. Let $f : A \to B$ be a mapping. Then $f$ is '''inclusion-preserving''' {{iff}} for every two sets $a_1, a_2 \in A$: :$a_1 \subseteq a_2 \implies f(a_1) \subseteq f(a_2)$


Definition:Topologically Nilpotent Ring Element

Let $A$ be a Hausdorff topological ring. Let $a \in A$. Then $a$ is '''topologically nilpotent''' {{iff}} the sequence of powers $a^n$ converges to $0$.


Definition:Induced Mapping on Spectra of Rings

Let $A$ and $B$ be commutative rings with unity. Let $f : A \to B$ be a ring homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \operatorname{Spec} B \to \operatorname{Spec} A$ between their spectra with: :$f^* (\mathfrak p) = f^{-1}(\mathfrak p)$, the preimage of a prime ideal $\mathfrak p \in \operatorname{Spec} B$. === Induced morphism of locally ringed spaces === {{definition wanted}}


Definition:Saturation of Ideal by Multiplicatively Closed Subset

Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be an ideal. Let $S \subseteq A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $\mathfrak a$ by $S$ is the ideal: :$\{ a \in A : \exists s \in S : as \in \mathfrak a\}$ === Definition 2 === Let $A \overset \iota \to A_S$ be the localization of $A$ at $S$. The '''saturation''' of $\mathfrak a$ is the preimage of its image under the ring homomorphism $\iota : A \to A_S$: :$\iota^{-1}(\iota(\mathfrak a))$


Definition:Topological Closed Embedding

Let $X$ and $Y$ be topological spaces. Let $f : X \to Y$ be a mapping. Then $f$ is a '''(topological) closed embedding''' {{iff}}: *$f$ is a topological embedding *its image $f(X)$ is closed in $Y$


Definition:Topological Open Embedding

Let $X$ and $Y$ be topological spaces. Let $f : X \to Y$ be a mapping. Then $f$ is a '''(topological) open embedding''' {{iff}}: *$f$ is a topological embedding *its image $f(X)$ is open in $Y$


Definition:Unit Ideal of Ring

Let $R$ be a ring. Its '''unit ideal''' is the ideal $R$.


Definition:Semi-Local Ring

Let $A$ be a commutative ring with unity. Then $A$ is '''semi-local''' {{iff}} $A$ has a finite number of maximal ideals.


Definition:Local Ring/Commutative

Let $A$ be a commutative ring with unity. === Definition 1 === {{Definition:Local Ring/Commutative/Definition 1}} === Definition 2 === {{Definition:Local Ring/Commutative/Definition 2}}


Definition:Localization of Module

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Let $A_S$ be the localization of $A$ at $S$. Let $M$ be an $A$-module. A '''localization of $M$ at $S$''' is a pair $(M_S, \iota)$ where: *$M_S$ is an $A_S$-module *$\iota : M \to M_S$ is the '''localization map''', an $A$-module homomorphism to the restriction of scalars of $M_S$ to $A$ Such that: :For every $A_S$-module $N$ and $A$-module homomorphism $f: M \to \operatorname{res}_A^{A_S}N$ to the restriction of scalars to $A$, there exists a unique $A_S$-module homomorphism $g : M_S \to N$ such that $f = g \circ \iota$. That is, the precomposition mapping between modules of homomorphisms: :$\operatorname{Hom}_{A_S} (M_S, N) \overset {\iota^*} \to \operatorname{Hom}_A(M, N)$ is a bijection.


Definition:Precomposition Mapping

Let $A, B, C$ be sets. Let $\operatorname{Hom}(B, C)$ denote the set of all mappings from $B$ to $C$. Let $f : A \to B$ be a mapping. The '''precomposition mapping''' $f^* : \operatorname{Hom}(B, C) \to \operatorname{Hom}(A, C)$ is the mapping that sends a mapping $g : B \to C$ to the precomposition $g \circ f$ with $f$.


Definition:Vanishing Ideal of Set of Prime Ideals

Let $A$ be a commutative ring with unity. Let $V \subseteq \operatorname{Spec} A$ be a set of prime ideals of $A$. Its '''vanishing ideal''' is its intersection, the set of elements of $A$ that are in each $\mathfrak p \in V$: :$I(V) = \bigcap V$


Definition:Vanishing Ideal of Subset of Affine Space

Let $k$ be a field. Let $n \ge 0$ be a natural number. Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$. Let $S \subseteq \mathbb A^n_k$ be a subset of the standard affine space over $k$. Its '''vanishing ideal''' is the ideal: :$\map I S = \set {f \in k \sqbrk {X_1, \ldots, X_n} : \forall x \in S : \map f x = 0}$


Definition:Divisor (Algebra)/Terminology

Let $x \divides y$ denote that '''$x$ divides $y$'''. Then the following terminology can be used:  : $x$ is a '''divisor''' of $y$ : $y$ is a '''multiple''' of $x$ : $y$ is '''divisible by''' $x$. In the field of Euclidean geometry, in particular: : $x$ '''measures''' $y$. To indicate that $x$ does ''not'' divide $y$, we write $x \nmid y$.


Definition:Divisor (Algebra)/Gaussian Integer

Let $\struct {\Z \left[{i}\right], +, \times}$ be the ring of Gaussian integers. Let $x, y \in \Z \left[{i}\right]$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z \left[{i}\right]: y = t \times x$


Definition:Spectrum of Ring Functor

Let $\mathbf{Ring}$ be the category of commutative rings with unity. === To topological spaces === Let $\mathbf{Top}$ be the category of topological spaces. The '''spectrum functor''' $\Spec {} : \mathbf{Ring} \to \mathbf{Top}$ is the contravariant functor with: {{DefineFunctor |ob = $\Spec A$ is the spectrum of a ring $A$ |mor = If $f : A \to B$ is a ring homomorphism, $\Spec f : \Spec B \to \Spec A$ is the  induced map on spectra }} {{definition wanted|to schemes}} Category:Definitions/Examples of Functors Category:Definitions/Commutative Algebra Category:Definitions/Algebraic Geometry szewf05u9zadxzq1453qaeva2p9bwhz


Definition:Ring Antirepresentation

Let $R$ be a ring. Let $M$ be an abelian group. A '''ring antirepresentation''' of $R$ on $M$ is a ring antihomomorphism to the endomorphism ring of $M$.


Definition:Ring Action Defined by Ring Representation

Let $R$ be a ring. Let $M$ be an abelian group. Let $\rho : R \to \map {\operatorname {End} } M$ be a ring representation. The '''associated (left) ring action''' is the linear ring action: :$R \times M \to M$: :$\tuple {r, m} \mapsto \map {\map \rho r} m$


Definition:Ring Representation/Unital

Let $R$ be a ring with unity. Let $M$ be an abelian group. A '''unital ring representation''' of $R$ on $M$ is a ring representation $R \to \map {\operatorname {End} } M$ which is unital. That is, it is a unital ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$.


Definition:Closed Set Axioms

Let $S$ be a set. The '''closed set axioms''' are the conditions under which a subset $F \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ consists of the closed sets of a topology on $S$: {{begin-axiom}} {{axiom | n = C1         | t = The intersection of an arbitrary subset of $F$ is an element of $F$. }} {{axiom | n = C2         | t = The union of any two elements of $F$ is an element of $F$. }} {{axiom | n = C3         | t = $\varnothing$ is an element of $F$. }} {{end-axiom}}


Definition:Topology Defined by Closed Sets

Let $S$ be a set. Let $F \subseteq \mathcal P(S)$ be a subset of its power set satisfying the closed set axioms. The '''topology defined by $F$''' is the topology whose open sets are the complements of elements of $F$: :$\tau = \{U \subseteq S : S \setminus U \in F\}$


Definition:Two-Sided Linear Combination in Ring

Let $R$ be a ring. Let $\family {x_i}_{i \mathop \in I}$ be a family of elements of $R$. A '''two-sided linear combination''' of the family is an element of the form: :$\displaystyle \sum_{i \mathop \in I} a_i x_i b_i$ where: :$\family {a_i}_{i \mathop \in I}$ and $\family {b_i}_{i \mathop \in I}$ are families in $R$ of finite support :$\sum$ denotes summation with finite support


Definition:Prime Ideal of Number Field

Let $K$ be a number field. Let $\mathcal O_K$ be its ring of integers. Let $\mathfrak p \subseteq \mathcal O_K$ be an ideal. Then $\mathfrak p$ is a '''prime ideal''' {{iff}} it is not the unit ideal $(1)$ and $\mathfrak p$ has no divisors other than $\mathfrak p$ and $(1)$.


Definition:Rational Integer

Let $K$ be a number field. A '''rational integer''' of $K$ is an integer of $K$, that is, an element of $\mathbb Z \subseteq K$.


Definition:Dedekind Domain

=== Definition 1 === {{Definition:Dedekind Domain/Definition 1}} === Definition 2 === {{Definition:Dedekind Domain/Definition 2}} === Definition 3 === {{Definition:Dedekind Domain/Definition 3}} === Definition 4 === {{Definition:Dedekind Domain/Definition 4}} === Definition 5 === {{Definition:Dedekind Domain/Definition 5}} === Definition 6 === {{Definition:Dedekind Domain/Definition 6}}


Definition:Integrally Closed Integral Domain

Let $R$ be an integral domain. Then $R$ is '''integrally closed''' {{iff}} it is integrally closed in its field of fractions.


Definition:Integrally Closed in Ring Extension

Let $\phi : A \hookrightarrow B$ be a ring extension. Let $C$ be the integral closure of $A$ in $B$. Then $A$ is '''integrally closed''' in $B$ {{iff}} $C = \phi(A)$.


Definition:Fractional Ideal

Let $R$ be an integral domain with field of fractions $K$. A '''fractional ideal''' of $R$ is a subset of $K$ that is a product of an ideal $I$ with the inverse $\lambda^{-1}$ of an element of $R$: :$\lambda^{-1} I = \left\{ {\lambda^{-1} a : a \in I}\right\}$


Definition:Dedekind Domain/Definition 1

A '''Dedekind domain''' is an integral domain in which every nonzero proper ideal has a prime ideal factorization that is unique up to permutation of the factors. {{explain|unique up to permutation}}


Definition:Dedekind Domain/Definition 2

A '''Dedekind domain''' is an integral domain of which every nonzero fractional ideal is invertible.


Definition:Dedekind Domain/Definition 3

A '''Dedekind domain''' is a noetherian domain of dimension $1$ that is integrally closed.


Definition:Dedekind Domain/Definition 4

A '''Dedekind domain''' is a noetherian domain of dimension $1$ in which every primary ideal is the power of a prime ideal.


Definition:Dedekind Domain/Definition 5

A '''Dedekind domain''' is a noetherian domain $A$ of dimension $1$ such that for every maximal ideal $\mathfrak p$, the localization $A_{\mathfrak p}$ is a discrete valuation ring.


Definition:Dedekind Domain/Definition 6

A '''Dedekind domain''' is a Krull domain of dimension $1$.


Definition:Polynomial over Ring/Multiple Variables

Let $I$ be a set. A '''polynomial over $I$ in one variable''' is an element of a polynomial ring in $I$ variables over $R$. Thus: :''Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial'' is a short way of saying: : Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.


Definition:Digamma Function

The '''digamma function''', $\psi$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the logarithmic derivative of the gamma function:  :$\map \psi z = \dfrac {\map {\Gamma'} z} {\map \Gamma z}$ where $\Gamma$ is the gamma function, and $\Gamma'$ denotes its derivative.


Definition:Generated Ideal of Ring/Definition 1

The '''ideal generated by $S$''' is the smallest ideal of $R$ containing $S$, that is, the intersection of all ideals containing $S$.


Definition:Generated Ideal of Ring/Definition 2

Let $R$ be a commutative ring with unity.


Definition:Product of Fractional Ideals

Let $R$ be an integral domain with fraction field $K$. Let $I, J \subseteq K$ be fractional ideals of $R$. The '''product of $I$ and $J$''' is the set of summations: :$\left\{ \displaystyle\sum_{i = 1}^r a_i b_i : a_i \in I, b_i \in J, r \in \N \right\}$


Definition:Invertible Fractional Ideal

Let $R$ be an integral domain with fraction field $K$. Let $I\subseteq K$ be a fractional ideal of $R$. Then $I$ is '''invertible''' {{iff}} there exists a fractional ideal $J\subseteq K$ such that their product is the unit ideal of $R$: :$I J = \ideal 1$


Definition:Minimal Condition/Ordered Set

Let $(P, \leq)$ be an ordered set. Then $P$ satisfies the '''minimal condition''' {{iff}} it is '''well-founded''': :Every non-empty subset has a minimal element.


Definition:Coprime Ideals

Let $A$ be a commutative ring with unity. Let $I, J \subseteq A$ be ideals. Then $I$ and $J$ are '''coprime''' {{iff}} their sum is the unit ideal: :$I + J = A$.


Definition:Unit Ideal

Let $A$ be a commutative ring with unity. Its '''unit ideal''' is $A$ itself.


Definition:Saturated Multiplicatively Closed Subset of Ring

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Then $S$ is '''saturated''' {{iff}} it equals its saturation, that is: :$x, y \in A, xy \in S \implies x,y \in S$ Category:Definitions/Localization of Rings eoer31a87qrcjz3ej5uao1tm83mcoov


Definition:Contraction of Ideal

Let $A$ and $B$ be commutative ring with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak b \subseteq B$ be an ideal. The '''contraction''' of $\mathfrak b$ by $f$ is its preimage under $f$: :$\mathfrak b^c = f^{-1}(\mathfrak b)$


Definition:Canonical Homomorphism from Ring to Unital Algebra

Let $A$ be a commutative ring with unity. Let $B$ be an unital algebra over $A$ with unit $1$ whose underlying module is unitary. The '''canonical homomorphism''' $A \to B$ is the algebra homomorphism that sends $a \in A$ to $a \cdot 1$.


Definition:Algebra Homomorphism

Let $R$ be a commutative ring. Let $(A, *)$ and $(B, \times)$ be algebras over $R$. An '''algebra homomorphism''' $f : A \to B$ is a module homomorphism such that: :$\forall a, b \in A : f(a * b) = f(a) \times f(b)$


Definition:Underlying Module of Algebra

Let $R$ be a commutative ring. Let $(A, *)$ be an algebra over $R$. Its '''underlying module''' is the $R$-module $A$.


Definition:Unital Algebra Homomorphism

Let $R$ be a commutative ring. Let $\left({A, *}\right)$ and $\left({B, \times}\right)$ be unital algebras over $R$ with units $1_A$ and $1_B$. A '''unital algebra homomorphism''' $f : A \to B$ is a algebra homomorphism such that $f \left({1_A}\right) = 1_B$. Category:Definitions/Algebras 7qphmf7ukyuspdrvr52lwuc1qes6ebc


Definition:Riemann P-symbol

The '''Riemann P-symbol''', written:  :$\displaystyle \map f z = \operatorname P \set {\begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} }$ denotes the solutions to the hypergeometric differential equation:  {{begin-eqn}} {{eqn | o =        | r = \frac {\d^2 f} {\d z^2} }} {{eqn | o =        | ro= +       | r = \paren {\frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} + \frac {1 - \gamma - \gamma'} {z - c} } \frac {\d f} {\d z} }} {{eqn | o =        | ro= +       | r = \paren {\frac {\alpha \alpha' \paren {a - b} \paren {a - c} } {z - a} + \frac {\beta \beta' \paren {b - c} \paren {b - a} } {z - b} + \frac {\gamma \gamma' \paren {c - a} \paren {c - b} } {z - c} } \frac f {\paren {z - a} \paren {z - b} \paren {z - c} } }} {{eqn | r = 0 }} {{end-eqn}} where: :$\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1$


Definition:Induced Homomorphism between Localizations of Ring

Let $A$ be a commutative ring with unity. Let $S, T \subseteq A$ be multiplicatively closed subsets. Let $S$ be a subset of the saturation of $T$. The '''induced homomorphism''' between localizations $A_S \to A_T$ is the unique $A$-algebra homomorphism between them.


Definition:Principal Open Subset of Spectrum

Let $A$ be a commutative ring with unity. Let $f \in A$. The '''principal open subset''' determined by $f$ of the spectrum $\operatorname{Spec} A$ is the complement of the vanishing set $V \left({f}\right)$: :$D \left({f}\right) = \operatorname{Spec} A - V \left({f}\right)$ That is, it is the set of prime ideals $\mathfrak p \subseteq A$ with $f \notin \mathfrak p$.


Definition:Unital Associative Commutative Algebra Homomorphism

Let $A$ be a commutative ring with unity. Let $B$ and $C$ be $A$-algebras. === Definition 1 === Let $B$ and $C$ be viewed as rings under $A$, say $(B, f)$ and $(C, g)$. An '''$A$-algebra homomorphism''' $h : B \to C$ is a morphism of rings under $A$. That is, a unital ring homomorphism $h$ such that $g = h \circ f$: :$\xymatrix{ A \ar[d]_f \ar[r]^{g} & C\\ B \ar[ru]_{h} }$ === Definition 2 === Let $B$ and $C$ be viewed as unital algebras over $A$. An '''$A$-algebra homomorphism''' $h : B \to C$ is a unital algebra homomorphism.


Definition:Radical of Ideal of Ring/Definition 1

The '''radical of $I$''' is the ideal of elements of which some power is in $I$: :$\map {\operatorname {Rad} } I := \set {a \in A: \exists n \in \N : a^n \in I}$


Definition:Radical of Ideal of Ring/Definition 2

Let $A / I$ be the quotient ring. Let $\Nil {A / I}$ be its nilradical. Let $\pi: A \to A / I$ be the quotient mapping. The '''radical''' of $I$ is the preimage of $\Nil {A / I}$ under $\pi$: :$\map {\operatorname {Rad} } I = \pi^{-1} \sqbrk {\Nil {A / I} }$


Definition:Finitely Generated Algebra

Let $A$ be a commutative ring. Let $B$ be an $A$-algebra. Then $B$ is '''finitely generated''' {{iff}} $B$ has a generator which is finite. Category:Definitions/Algebras qq40jwih7e3gzrirnshtqgb4dep6ajp


Definition:Characteristic Polynomial of Linear Operator

Let $A$ be a commutative ring with unity. Let $M$ be a free module over $A$ of finite rank $n > 0$. Let $\phi : M \to M$ be a linear operator. === Definition 1 === The '''characteristic polynomial''' of $\phi$ is the characteristic polynomial of the relative matrix of $\phi$ with respect to a basis of $M$. === Definition 2 === Let $A[x]$ be the polynomial ring in one variable over $A$. Let $\operatorname{id}$ be the identity mapping on $M$. Let $M \otimes_A A[x]$ be the extension of scalars of $M$ to $A[x]$. The '''characteristic polynomial''' of $\phi$ is the determinant of the linear operator $x\operatorname{id} - \phi$ on $M \otimes_A A[x]$.


Definition:Induced Mapping on Maximal Spectra

Let $k$ be a field. Let $A$ and $B$ be finitely generated $k$-algebras. Let $f : A \to B$ be a $k$-algebra homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \operatorname{Max} B \to \operatorname{Max} A$ between their maximal spectra with: :$f^* (\mathfrak m) = f^{-1}(\mathfrak m)$, the preimage of a maximal ideal $\mathfrak m \in \operatorname{Max} B$. === Induced morphism of locally ringed spaces === {{definition wanted}}


Definition:Finite Algebra over Ring

Let $A$ be a commutative ring with unity. Let $B$ be an $A$-algebra. Then $B$ is '''finite''' over $A$ {{iff}} its underlying module is finitely generated.


Definition:Module Defined by Ring Homomorphism

Let $A$ and $B$ be a rings. Let $f: A \to B$ be a ring homomorphism. === Definition 1 === The '''left $A$-module structure''' of $B$ via $f$ is the module with left ring action: :$A \times B \to B$ :$ \tuple {a, b} \mapsto \map f a \cdot b$ === Definition 2 === The '''left $A$-module structure''' of $B$ via $f$ is the restriction of scalars of the $B$-module structure of $B$. === Definition 3 === Let $\lambda: B \to \map {\operatorname {End} } B$ be its left regular ring representation. The '''left $A$-module structure''' of $B$ via $f$ is the module with ring representation the composition $\lambda \circ f$.


Definition:Weierstrass's Elliptic Function

'''Weierstrass's Elliptic Function''' is an elliptic function, given for all complex $z$ (except for $z \in \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$ where the function has double poles, by Poles of Weierstrass's Elliptic Function) by:  :$\displaystyle \map \wp {z; \omega_1, \omega_2} = \frac 1 {z^2} + {\sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } } \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }$ where $\omega_1$ and $\omega_2$ are non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.


Definition:Right-Hand Derivative/Real Function

Let $f: \R \to \R$ be a real function. The '''right-hand derivative''' of $f$ is defined as the right-hand limit: :$\displaystyle \map {f'_+} x = \lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h$ If the '''right-hand derivative''' exists, then $f$ is said to be '''right-hand differentiable''' at $x$.


Definition:Left-Hand Derivative/Real Function

Let $f: \R \to \R$ be a real function. The '''left-hand derivative''' of $f$ is defined as the left-hand limit: :$\displaystyle \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$ If the '''left-hand derivative''' exists, then $f$ is said to be '''left-hand differentiable''' at $x$.


Definition:Ideal Quotient

Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals of $A$. Their '''ideal quotient''' is the ideal consisting of elements whose product with $\mathfrak b$ is a subset of $\mathfrak a$: :$\ideal {\mathfrak a : \mathfrak b} := \set {x \in A : x \mathfrak b \subseteq \mathfrak a}$


Definition:Annihilator of Ideal of Ring

Let $A$ be a commutative ring with unity. Let $I \subseteq A$ be an ideal. === Definition 1 === The '''annihilator''' of $I$ is the ideal consisting of the elements $a \in A$ such that $a \cdot x = 0$ for all $x \in I$, where $0 \in A$ is its zero. === Definition 2 === The '''annihilator''' of $I$ is the ideal quotient $\ideal {0 : I}$, where $0$ is the zero ideal.


Definition:Extension of Ideal

Let $A$ and $B$ be commutative ring with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak a \subseteq A$ be an ideal. The '''extension''' of $\mathfrak a$ by $f$ is the ideal generated by its image under $f$: :$\mathfrak a^e = \left\langle f(\mathfrak a) \right\rangle$


Definition:Irreducible Polynomial/Definition 1

Let $R$ be an integral domain. An '''irreducible polynomial''' over $R$ is an irreducible element of the polynomial ring $R \left[{X}\right]$.


Definition:Irreducible Polynomial/Definition 2

Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a nonconstant polynomial over $K$ that is not the product of two polynomials of smaller degree.


Definition:Irreducible Polynomial/Definition 3

Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a polynomial over $K$ that is not the product of two nonconstant polynomials.


Definition:Linear Ring Action

Let $R$ be a ring. Let $M$ be an abelian group. === Left Ring Action === {{:Definition:Linear Ring Action/Left}} === Right Ring Action === {{:Definition:Linear Ring Action/Right}}


Definition:Gradation on Abelian Group

Let $G$ be an abelian group. Let $\Delta$ be a set. A '''gradation of type $\Delta$''' on $G$ is a family of subgroups $\family {G_\lambda}_{\lambda \in \Delta}$ of which $G$ is the internal direct sum.


Definition:Graded Abelian Group

Let $\Delta$ be a set. A '''graded abelian group of type $\Delta$''' is a pair $\struct {G, f}$ where: :$G$ is an abelian group :$f$ is a gradation on $G$ indexed by $\Delta$, the '''set of degrees'''.


Definition:N-Graded Ring

Let $\N$ be the set of natural numbers. An '''$\N$-graded ring''' or '''positively $\Z$-graded ring''' is a graded ring of type the additive monoid of natural numbers. That is, it is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a sequence $\sequence {R_n}_{n \mathop \in \N}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\forall x \in R_n, y \in R_m: x y \in R_{m + n}$


Definition:Z-Graded Ring

Let $\Z$ be the set of integers. A '''$\Z$-graded ring''' is a graded ring of type the additive group of integers. That is, it is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a family $\family {R_n}_{n \mathop \in \Z}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\forall x \in R_n, y \in R_m: x y \in R_{m + n}$


Definition:Opposite Ring

Let $\struct {R, +, \times}$ be a ring. Let $* : R \times R \to R$ be the binary operation on $S$ defined by: :$\forall x, y \in S: x * y = y \times x$ The '''opposite ring''' of $R$ is the algebraic structure $\struct {R, +, *}$.


Definition:Ring Representation Defined by Ring Action

Let $R$ be a ring. Let $M$ be an abelian group. Let $\phi : R \times M \to M$ be a left linear ring action. The '''associated ring representation''' is the ring representation $\rho: R \to \map {\operatorname {End} } M$ with: :$\map {\map \rho r} m = \map \phi {r, m}$


Definition:Category of Modules

Let $R$ be a commutative ring. The '''category of $R$-modules''' is the category $\mathbf{R-Mod}$ with: {{DefineCategory |ob = $R$-modules |mor = $R$-module homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of Unitary Modules === {{Definition:Category of Unitary Modules}}


Definition:Underlying Group of Module

Let $R$ be a ring. Let $\struct {M, +, \times}$ be a left module or right module over $R$. Its '''underlying (abelian) group''' is the abelian group $\struct {M, +}$.


Definition:Biadditive Mapping

Let $M, N, P$ be abelian groups. Let $M \times N$ be the cartesian product. A '''biadditive mapping''' $f : M \times N \to P$ is a mapping such that: :$\forall m_1, m_2 \in M : \forall n \in N: \map f {m_1 + m_2, n} = \map f {m_1, n} + \map f {m_2, n}$ :$\forall m \in M: \forall n_1, n_2 \in N: \map f {m, n_1 + n_2} = \map f {m, n_1} + \map f {m, n_2}$


Definition:Cartesian Square of Set

Let $S$ be a set. Its '''cartesian square''' is the cartesian power $S^2$. Category:Definitions/Cartesian Product i2nh881gpp361x4qw0f83uv8um0ek3x


Definition:R-Balanced Mapping

Let $R$ be a ring. Let $M$ be a right $R$-module and $N$ be a left $R$-module. Let $M \times N$ be their cartesian product. Let $P$ be an abelian group. An '''$R$-balanced mapping''' $f : M \times N \to P$ is a biadditive mapping with: :$\forall m \in M: \forall n \in N: \forall r \in R: \map f {m \cdot r, n} = \map f {m, r \cdot n}$


Definition:Tensor Product of Abelian Groups

Let $A$ and $B$ be abelian groups. === Definition 1: by universal property === Their '''tensor product''' is a pair $(A \otimes B, \theta)$ where: * $A \otimes B$ is an abelian group * $\theta : A \times B \to A \otimes B$ is a biadditive mapping such that, for every pair $(C, \omega)$ where: * $C$ is an abelian group * $\omega : A \times B \to C$ is a biadditive mapping there exists a unique group homomorphism $g : A \otimes B \to C$ with $\omega = g \circ \theta$. === Definition 2: construction === Their '''tensor product''' is the pair $(A \otimes B, \theta)$ where: * $A \otimes B$ is the quotient of the free abelian group $\Z^{(A \times B)}$ on the cartesian product $A \times B$ by the subgroup generated by the elements of the form: ** $(a_1 + a_2, b) - (a_1, b) - (a_2, b)$ ** $(a, b_1 + b_2) - (a, b_1) - (a, b_2)$ *:for $a, a_1, a_2 \in A$, $b, b_1, b_2 \in B$, where we denote $(a, b)$ for its image under the canonical mapping $A \times B \to \Z^{(A\times B)}$. * $\theta : A \times B \to A \otimes B$ is the composition of the canonical mapping $A \times B \to \Z^{(A\times B)}$ with the quotient group epimorphism $\Z^{(A\times B)} \to A \otimes B$.


Definition:Right Ring Action Defined by Ring Antirepresentation

Let $R$ be a ring. Let $M$ be an abelian group. Let $\rho: R \to \map {\operatorname {End} } M$ be a ring antirepresentation. The '''associated right ring action''' is the right linear ring action: :$M \times R \to M$: :$\tuple {m, r} \mapsto \map \rho r \paren m$


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: :$\forall a, b \in A: \map f {a b} = \map f a b + a \map f b$ Category:Definitions/Differential Algebra bvtjsn4vemtu8t3g5euveqfulwyq6mi


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''' {{iff}} 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''' {{iff}} for all $a \in A$, the homothety $h_a : M \to M$ is an endomorphism of the $B$-module $M$. That is, {{iff}} 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''' {{iff}} for all $b \in A$, the homothety $h_b : M \to M$ is an endomorphism of the $A$-module $M$. That is, {{iff}} 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 $\mathcal R$''' {{iff}}: :$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 $H$ {{iff}}: : $G = H K$ and $H \cap K = \set e$


Definition:Complement of Subgroup/Definition 2

$K$ is a '''complement''' of $H$ {{iff}}: :$G = K H$ 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 $\sim$ {{iff}}: :$\forall x, y \in X : x \equiv y \implies x \sim y$


Definition:Finer Topology/Definition 1

$\tau_1$ is '''finer''' than $\tau_2$ {{iff}} $\tau_1 \supseteq \tau_2$.


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''' {{iff}} 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''' {{iff}} it is nontrivial and the sum of any two non-units is a non-unit. {{refactor}} * 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''' {{iff}} it has a unique maximal left ideal.


Definition:Local Ring/Noncommutative/Definition 2

$R$ is a '''local ring''' {{iff}} 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''' {{iff}} the quotient ring $R / \operatorname{rad} R$ is a division ring.


Definition:Local Ring Homomorphism/Definition 1

The homomorphism $f$ is '''local''' {{iff}} the image $f(\mathfrak m) \subseteq \mathfrak n$.


Definition:Local Ring Homomorphism/Definition 2

The homomorphism $f$ is '''local''' {{iff}} the preimage $f^{-1}(\mathfrak n) \supseteq \mathfrak m$.


Definition:Local Ring Homomorphism/Definition 3

The homomorphism $f$ is '''local''' {{iff}} the preimage $f^{-1}(\mathfrak n) = \mathfrak m$.


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 $\closedint a b$ {{iff}} 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 $\left\lvert{x}\right\rvert < \pi$


An alternative form of the Euler numbers can often be found. Usually denoted with the symbol $E_n^*$, they are generally not used on {{ProofWiki}}. === 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 $\mathbf V$.


Definition:Gradient Operator

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $f \in \map {\CC^\infty} M$ be a smooth mapping on $M$. The '''gradient''' of $f$ is defined as: {{begin-eqn}} {{eqn | l = \grad f        | o = :=        | r = \nabla f }} {{eqn | r = g^{-1} \d_{\d R} f }} {{end-eqn}} where $\d_{\d R}$ is de Rham differential.


Definition:Beta Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \closedint 0 1$. $X$ is said to have a '''Beta distribution''' {{iff}} it has probability density function:  :$\map {f_X} X = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$ for $\alpha, \beta > 0$, where $\Beta$ denotes the Beta function. This is written:  :$X \sim \BetaDist \alpha \beta$


Definition:Divergence Operator

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $\mathbf X : \map {\CC^\infty} M \to \map {\CC^\infty} M$ be a smooth vector field. The '''divergence''' of $\mathbf X$ is defined as: {{begin-eqn}} {{eqn | l = \operatorname {div} \mathbf X        | o = :=        | r = \nabla \cdot \mathbf X }} {{eqn | r = \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X} }} {{end-eqn}} where: :$\star_g$ is the Hodge star operator of $\struct {M, g}$ :$\d_{\d R}$ is de Rham differential.


Definition:Divergence Operator/Real Cartesian Space

Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$. Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \R^n \to \R^n$ be a vector-valued function on $\R^n$. Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''divergence of $\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \operatorname {div} \mathbf f       | o = :=       | r = \nabla \cdot \mathbf f }} {{eqn | r = \left({\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial u_k} }\right) \cdot \mathbf f       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\partial f_k} {\partial x_k}       | c =  }} {{end-eqn}}


Definition:Curl Operator

Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$. The '''curl''' of $\mathbf f$ is defined as: {{begin-eqn}} {{eqn | l = \curl \mathbf f       | o = :=       | r = \nabla \times \mathbf f       | c = where $\nabla$ denotes the del operator }} {{eqn | r = \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } \times \paren {f_x \mathbf i + f_y \mathbf j + f_z \mathbf k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x & f_y & f_z \end{vmatrix}       | c = {{Defof|Vector Cross Product}} }} {{eqn | r = \paren {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } \mathbf k       | c =  }} {{end-eqn}}