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|---|---|---|
1074_(GTM232)An Introduction to Number Theory | Definition 4.12 |
Definition 4.12. A nonzero ideal \( I \neq R \) in a commutative ring \( R \) is called maximal if for any ideal \( J, J \mid I \) implies that \( J = I \) . An ideal \( P \) is prime if \( P \mid {IJ} \) implies that \( P \mid I \) or \( P \mid J \) .
Exercise 4.12. In a commutative ring \( R \), let \( M \) and \(... |
1139_(GTM44)Elementary Algebraic Geometry | Definition 3.5 |
Definition 3.5. If \( R \) is the coordinate ring of an irreducible variety \( V \subset {\mathbb{C}}^{n} \) , and if \( \mathfrak{p} = \mathrm{J}\left( W\right) \) is the prime ideal of an irreducible subvariety \( W \) of \( V \) , then the local ring \( {R}_{\mathrm{p}} \) is called the localization of \( V \) at ... |
18_Algebra Chapter 0 | Definition 1.8 |
Definition 1.8. An element \( a \) in a ring \( R \) is a left-zero-divisor if there exist elements \( b \neq 0 \) in \( R \) for which \( {ab} = 0 \) .
The reader will have no difficulty figuring out what a right-zero-divisor should be. The element 0 is a zero-divisor in all nonzero rings \( R \) ; the zero ring is... |
108_The Joys of Haar Measure | Definition 7.1.7 |
Definition 7.1.7. Let \( E \) and \( {E}^{\prime } \) be two elliptic curves with identity elements \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) respectively. An isogeny \( \phi \) from \( E \) to \( {E}^{\prime } \) is a morphism of algebraic curves from \( E \) to \( {E}^{\prime } \) such that \( \phi \left(... |
1359_[陈省身] Lectures on Differential Geometry | Definition 2.1 |
Definition 2.1. Suppose \( C : {u}^{i} = {u}^{i}\left( t\right) \) is a parametrized curve on \( M \), and \( X\left( t\right) \) is a tangent vector field defined on \( C \) given by
\[
X\left( t\right) = {x}^{i}\left( t\right) {\left( \frac{\partial }{\partial {u}^{i}}\right) }_{C\left( t\right) }.
\]
(2.17)
We ... |
1139_(GTM44)Elementary Algebraic Geometry | Definition 8.9 |
Definition 8.9. Let \( \left( {A, M,\pi }\right) \) be a near cover (Definition 5.2). A connected open set \( \mathcal{O} \subset M \) is said to be liftable to \( A \) if there is an open set \( \mathcal{Q} \subset A \) such that \( \pi \mid \mathcal{Q} \) is a homeomorphism from \( \mathcal{Q} \) to \( \mathcal{O};... |
18_Algebra Chapter 0 | Definition 2.7 |
Definition 2.7. A Euclidean valuation on an integral domain \( R \) is a valuation satisfying the following property 8 : for all \( a \in R \) and all nonzero \( b \in R \) there exist \( q, r \in R \) such that
\[
a = {qb} + r
\]
with either \( r = 0 \) or \( v\left( r\right) < v\left( b\right) \) . An integral do... |
1009_(GTM175)An Introduction to Knot Theory | Definition 3.4 |
Definition 3.4. The writhe \( w\left( D\right) \) of a diagram \( D \) of an oriented link is the sum of the signs of the crossings of \( D \), where each crossing has sign +1 or -1 as defined (by convention) in Figure 1.11.
Note that this definition of \( w\left( D\right) \) uses the orientation of the plane and th... |
1116_(GTM270)Fundamentals of Algebraic Topology | Definition 6.2.25 |
Definition 6.2.25. Let \( \left\{ {\bar{\varphi }}_{y}\right\} \) be an orientation of int \( \left( M\right) \), i.e., a compatible system of local orientations. The induced orientation of \( \partial M \) is the compatible system of local orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) obtained as follows: ... |
1088_(GTM245)Complex Analysis | Definition 9.10 |
Definition 9.10. A harmonic conjugate of a real-valued harmonic function \( u \) is any real-valued function \( v \) such that \( u + {uv} \) is holomorphic.
Harmonic conjugates always exist locally, and globally on simply connected domains. They are unique up to additive real constants. In fact, it is easy to see t... |
1042_(GTM203)The Symmetric Group | Definition 4.4.1 |
Definition 4.4.1 Given a partition \( \lambda \), the associated Schur function is
\[
{s}_{\lambda }\left( \mathbf{x}\right) = \mathop{\sum }\limits_{T}{\mathbf{x}}^{T}
\]
where the sum is over all semistandard \( \lambda \) -tableaux \( T \) .
By way of illustration, if \( \lambda = \left( {2,1}\right) \), then s... |
1074_(GTM232)An Introduction to Number Theory | Definition 2.2 |
Definition 2.2. A commutative ring \( R \) is Euclidean if there is a function
\[
N : R \smallsetminus \{ 0\} \rightarrow \mathbb{N}
\]
with the following properties:
(1) \( N\left( {ab}\right) = N\left( a\right) N\left( b\right) \) for all \( a, b \in R \), and
(2) for all \( a, b \in R \), if \( b \neq 0 \), th... |
1359_[陈省身] Lectures on Differential Geometry | Definition 2.2 |
Definition 2.2. Suppose \( M \) is a connected Riemannian manifold, and \( p, q \) are two arbitrary points in \( M \) . Let
\[
\rho \left( {p, q}\right) = \inf \overset{⏜}{pq}
\]
\( \left( {2.47}\right) \)
where \( \overset{⏜}{pq} \) denotes the arc length of a curve connecting \( p \) and \( q \) with measurable... |
1088_(GTM245)Complex Analysis | Definition 9.32 |
Definition 9.32. Let \( D \) be a domain in \( \mathbb{C} \) . A Perron family \( \mathcal{F} \) in \( D \) is a nonempty collection of subharmonic functions in \( D \) such that
(a) If \( u, v \) are in \( \mathcal{F} \), then so is \( \max \{ u, v\} \) .
(b) If \( u \) is in \( \mathcal{F} \), then so is \( {u}_{... |
113_Topological Groups | Definition 8.22 |
Definition 8.22. Let \( \mathcal{P} = \left( {n, c, P}\right) \) be a sentential language. Members of \( {}^{P}2 \) are called models of \( \mathcal{P} \) . (Intuitively,0 means falsity,1 means truth, and a function \( f \in {}^{P}2 \) is just an assignment of a truth value to each sentence of P.) Using the recursion... |
1074_(GTM232)An Introduction to Number Theory | Definition 10.7 |
Definition 10.7. Let \( G \) be a finite Abelian group. A character of \( G \) is a homomorphism
\[
\chi : G \rightarrow \left( {{\mathbb{C}}^{ * }, \cdot }\right)
\]
The multiplicative group \( {\mathbb{C}}^{ * } \) is \( \mathbb{C} \smallsetminus \{ 0\} \) equipped with the usual multiplication. By convention, we... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 23.21 |
Definition 23.21 A smooth, complex-valued function \( f \) on \( N \) is quantizable with respect to \( P \) if \( {Q}_{\text{pre }}\left( f\right) \) preserves the space of smooth sections that are polarized with respect to \( P \) .
The following definition will provide a natural geometric condition guaranteeing q... |
1143_(GTM48)General Relativity for Mathematicians | Definition 6.4.4 |
Definition 6.4.4. A simple cosmological model is a cosmological model \( \left( {M,\mathcal{M}, z}\right) \) such that: (a) \( \left( {M, g}\right) \) is a simple cosmological spacetime,(b) \( M = {\mathbb{R}}^{3} \times \left( {0,\infty }\right) \) and \( R \rightarrow 0 \) as \( {u}^{4} \rightarrow 0 \) ; (c) \( {u... |
1056_(GTM216)Matrices | Definition 5.2 |
Definition 5.2 A square matrix \( M \in {\mathbf{M}}_{n}\left( \mathbb{R}\right) \) is
- Symmetric if \( {M}^{T} = M \)
- Skew-symmetric if \( {M}^{T} = - M \)
- Orthogonal if \( {M}^{T} = {M}^{-1} \)
We denote by \( {\mathbf{H}}_{n} \) the set of Hermitian matrices in \( {\mathbf{M}}_{n}\left( \mathbb{C}\right) ... |
1167_(GTM73)Algebra | Definition 6.1 |
Definition 6.1. An algebra A with identity over a field \( \mathbf{K} \) is said to be central simple if A is a simple \( \mathbf{K} \) -algebra and the center of \( \mathbf{A} \) is precisely \( \mathbf{K} \) .
EXAMPLE. Let \( D \) be a division ring and let \( K \) be the center of \( D \) . It is easy to verify t... |
1065_(GTM224)Metric Structures in Differential Geometry | Definition 1.1 |
Definition 1.1. A function \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is said to be symmetric if for any permutation \( \sigma \) of \( \{ 1,\ldots, n\}, f\left( {{\lambda }_{\sigma \left( 1\right) },\ldots ,{\lambda }_{\sigma \left( n\right) }}\right) = f\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \... |
1083_(GTM240)Number Theory II | Definition 11.3.13 |
Definition 11.3.13. For \( k \in \mathbb{Z} \smallsetminus \{ 0\} \) we define the p-adic \( \chi \) -Bernoulli numbers and \( \chi \) -Euler constant by
\[
{B}_{k, p}\left( \chi \right) = \mathop{\lim }\limits_{{r \rightarrow \infty }}{B}_{\phi \left( {p}^{r}\right) + k}\left( \chi \right) = - k{L}_{p}\left( {\chi ... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 10.5 |
Definition 10.5 The affine GIT quotient of \( {\mathbb{C}}^{n} \) modulo \( G \) is the affine toric variety \( \operatorname{Spec}\left( {S}^{G}\right) \) whose coordinate ring is the invariant ring \( {S}^{G} \) :
\[
{\mathbb{C}}^{n}//G \mathrel{\text{:=}} \operatorname{Spec}\left( {S}^{G}\right) = \operatorname{S... |
1282_[张恭庆] Methods in Nonlinear Analysis | Definition 4.8.6 |
Definition 4.8.6 Let \( X \) be a Banach space. Let \( Q \subset X \) be a compact manifold with boundary \( \partial Q \) and let \( S \subset X \) be a closed subset of \( X \) . \( \partial Q \) is said linking with \( S \), if
1. \( \partial Q \cap S = \varnothing \) ,
2. \( \forall \varphi : Q \rightarrow X \)... |
1088_(GTM245)Complex Analysis | Definition 8.25 |
Definition 8.25. We define the hyperbolic length of a piecewise differentiable curve \( \gamma \) in \( D \) by
\[
{l}_{D}\left( \gamma \right) = {\int }_{\gamma }{\lambda }_{D}\left( z\right) \left| {\mathrm{d}z}\right|
\]
and if \( {z}_{1} \) and \( {z}_{2} \) are any two points in \( D \), the hyperbolic (or Poin... |
1189_(GTM95)Probability-1 | Definition 2 |
Definition 2. A sequence of probability measures \( \left\{ {\mathrm{P}}_{n}\right\} \) converges weakly to the probability measure \( \mathrm{P} \) (notation: \( {\mathrm{P}}_{n}\overset{w}{ \rightarrow }\mathrm{P} \) ) if
\[
{\int }_{E}f\left( x\right) {\mathrm{P}}_{n}\left( {dx}\right) \rightarrow {\int }_{E}f\le... |
1075_(GTM233)Topics in Banach Space Theory | Definition 1.1.1 |
Definition 1.1.1. A sequence of elements \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in an infinite-dimensional Banach space \( X \) is said to be a basis of \( X \) if for each \( x \in X \) there is a unique sequence of scalars \( {\left( {a}_{n}\right) }_{n = 1}^{\infty } \) such that
\[
x = \mathop{\sum }\l... |
18_Algebra Chapter 0 | Definition 1.12 |
Definition 1.12. If \( G \) is finite as a set, its order \( \left| G\right| \) is the number of its elements; we write \( \left| G\right| = \infty \) if \( G \) is infinite.
Cancellation implies that \( \left| g\right| \leq \left| G\right| \) for all \( g \in G \) . Indeed, this is vacuously true if \( \left| G\r... |
1329_[肖梁] Abstract Algebra (2022F) | Definition 3.3.1 |
Definition 3.3.1. A (finite or infinite) group \( G \) is called simple if \( \left| G\right| > 1 \) and the only normal subgroups of \( G \) are \( \{ 1\} \) and \( G \) .
Example 3.3.2. (1) \( {\mathbf{Z}}_{p} \) for a prime number \( p \) . (This is all abelian simple groups.)
(2) Alternating group \( {A}_{n} \)... |
1329_[肖梁] Abstract Algebra (2022F) | Definition 12.3.1 |
Definition 12.3.1. A partial order on a nonempty set \( A \) is a relation \( \preccurlyeq \) on \( A \) satisfying for all \( x, y, z \in A \) ,
(1) (reflexive) \( x \preccurlyeq x \) ;
(2) (antisymmetric) if \( x \preccurlyeq y \) and \( y \preccurlyeq x \), then \( x = y \) ;
(3) (transitive) if \( x \preccurly... |
1167_(GTM73)Algebra | Definition 8.3 |
Definition 8.3. Let \( \alpha \) and \( \beta \) be cardinal numbers. The \( \operatorname{sum}\alpha + \beta \) is defined to be the cardinal number \( \left| {\mathrm{A} \cup \mathrm{B}}\right| \), where \( \mathrm{A} \) and \( \mathrm{B} \) are disjoint sets such that \( \left| \mathrm{A}\right| = \alpha \) and \(... |
1329_[肖梁] Abstract Algebra (2022F) | Definition 16.3.5 |
Definition 16.3.5. Let \( K \) be an extension of \( F \), and let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \in K \) .
(1) The field extension of \( F \) generated by \( {\alpha }_{1},\ldots ,{\alpha }_{n} \), denoted by \( F\left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) \), is the smallest subfield of \( K \) co... |
113_Topological Groups | Definition 24.8 |
Definition 24.8. Let \( \mathbf{K} \) be a class of \( \mathcal{L} \) -structures, where \( \mathcal{L} \) is not necessarily algebraic. We set
\( \mathbf{{HK}} = \{ \mathfrak{A} : \mathfrak{A} \) is a homomorphic image of some \( \mathfrak{B} \in \mathbf{K}\} ; \)
\( \mathbf{{UpK}} = \{ \mathfrak{A} : \mathfrak{A}... |
1063_(GTM222)Lie Groups, Lie Algebras, and Representations | Definition 1.16 |
Definition 1.16. The real projective space of dimension \( n \), denoted \( \mathbb{R}{P}^{n} \), is the set of lines through the origin in \( {\mathbb{R}}^{n + 1} \) . Since each line through the origin intersects the unit sphere exactly twice, we may think of \( \mathbb{R}{P}^{n} \) as the unit sphere \( {S}^{n} \)... |
1167_(GTM73)Algebra | Definition 1.3 |
Definition 1.3. A module A is said to satisfy the maximum condition [resp. minimum condition] on submodules if every nonempty set of submodules of A contains a maximal [resp. minimal] element (with respect to set theoretic inclusion).
Theorem 1.4. A module A satisfies the ascending [resp. descending] chain condition... |
1088_(GTM245)Complex Analysis | Definition 8.12 |
Definition 8.12. A circle in \( \widehat{\mathbb{C}} \) is either an Euclidean (ordinary) circle in \( \mathbb{C} \), or a straight line in \( \mathbb{C} \) together with \( \infty \) (this is a circle passing through \( \infty \) ). See Exercise 3.21 for a justification for the name.
![a50267de-c956-4a7f-8c2e-850ad... |
1098_(GTM254)Algebraic Function Fields and Codes | Definition 4.2.2 |
Definition 4.2.2. (a) A valued field \( T \) is said to be complete if every Cauchy sequence in \( T \) is convergent.
(b) Suppose that \( \left( {T, v}\right) \) is a valued field. A completion of \( T \) is a valued field \( \left( {\widehat{T},\widehat{v}}\right) \) with the following properties:
(1) \( T \subse... |
1096_(GTM252)Distributions and Operators | Definition 3.10 |
Definition 3.10. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) .
\( {1}^{ \circ } \) We say that \( u \) is 0 on the open subset \( \omega \subset \Omega \) when
\[
\langle u,\varphi \rangle = 0\;\text{ for all }\varphi \in {C}_{0}^{\infty }\left( \omega \right) .
\]
(3.32)
\( {2}^{ \circ } \) The ... |
1116_(GTM270)Fundamentals of Algebraic Topology | Definition 2.2.1 |
Definition 2.2.1. Let \( X \) and \( Y \) be arbitrary spaces, and let \( p : Y \rightarrow X \) be a map. Then \( Y \) is a covering space of \( X \), and \( p \) is a covering projection, if every \( x \in X \) has an open neighborhood \( U \) with \( {p}^{-1}\left( U\right) = \left\{ {V}_{i}\right\} \) a union of ... |
1234_[丁一文] Number Theory 1 | Definition 4.3.7 |
Definition 4.3.7. (1) A function \( f : {\left\lbrack 0,1\right\rbrack }^{n - 1} \rightarrow {\mathbb{R}}^{n} \) is called Lipschitz, if the ratio \( \frac{\left| f\left( x\right) - f\left( y\right) \right| }{\left| x - y\right| } \) is uniformly bounded for \( x \neq y \in {\left\lbrack 0,1\right\rbrack }^{n - 1} \)... |
1167_(GTM73)Algebra | Definition 7.7 |
Definition 7.7. Let \( \mathrm{F} \) be an object in a concrete category \( \mathrm{e},\mathrm{X} \) a nonempty set, and \( \mathrm{i} : \mathrm{X} \rightarrow \mathrm{F} \) a map (of sets). \( \mathrm{F} \) is free on the set \( \mathrm{X} \) provided that for any object \( \mathrm{A} \) of \( \mathrm{C} \) and map ... |
109_The rising sea Foundations of Algebraic Geometry | Definition 3.64 |
Definition 3.64. A simplicial complex \( \sum \) is called a Coxeter complex if it is isomorphic to \( \sum \left( {W, S}\right) \) for some Coxeter system \( \left( {W, S}\right) \) . It is called a spherical Coxeter complex if it is finite.
This differs from our previous use of the term "Coxeter complex" in that w... |
1069_(GTM228)A First Course in Modular Forms | Definition 8.2.1 |
Definition 8.2.1. Let \( C \) be a projective curve over \( {\overline{\mathbb{F}}}_{p} \) . The Frobenius map on \( C \) is
\[
{\sigma }_{p} : C \rightarrow {C}^{{\sigma }_{p}},\;\left\lbrack {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right\rbrack \mapsto \left\lbrack {{x}_{0}^{p},{x}_{1}^{p},\ldots ,{x}_{n}^{p}}\right\rbra... |
113_Topological Groups | Definition 1.11 |
Definition 1.11. If \( a \) is any set and \( m \in \omega \), let \( {a}^{\left( m\right) } \) be the unique element of \( {}^{m}\{ a\} \) . Thus \( {a}^{\left( m\right) } \) is an \( m \) -termed sequence of \( a \) ’s, \( {a}^{\left( m\right) } = \langle a, a,\ldots, a\rangle (m \) times). If \( x \) and \( y \) a... |
1077_(GTM235)Compact Lie Groups | Definition 1.36 |
Definition 1.36. (1) Let \( {\mathcal{C}}_{n}^{ + }\left( \mathbb{R}\right) \) be the subalgebra of \( {\mathcal{C}}_{n}\left( \mathbb{R}\right) \) spanned by all products of an even number of elements of \( {\mathbb{R}}^{n} \) .
(2) Let \( {\mathcal{C}}_{n}^{ - }\left( \mathbb{R}\right) \) be the subspace of \( {\m... |
113_Topological Groups | Definition 10.6 |
Definition 10.6. We introduce some operations on expressions \( \varphi ,\psi \) :
(i) \( \neg \varphi = \left\langle {L}_{0}\right\rangle \varphi \) ,
(ii) \( \varphi \vee \psi = \left\langle {L}_{1}\right\rangle {\varphi \psi } \)
(iii) \( \varphi \land \psi = \left\langle {L}_{2}\right\rangle {\varphi \psi } \)... |
109_The rising sea Foundations of Algebraic Geometry | Definition 11.3 |
Definition 11.3. A metric space \( X \) is called a \( \operatorname{CAT}\left( 0\right) \) space if for any \( x, y \) in \( X \) there is a geodesic \( \left\lbrack {x, y}\right\rbrack \) with the following property: For all \( p \in \left\lbrack {x, y}\right\rbrack \) and all \( z \in X \), we have
\[
{d}_{X}\lef... |
1167_(GTM73)Algebra | Definition 3.1 |
Definition 3.1. A nonzero element a of a commutative ring \( \mathrm{R} \) is said to divide an element \( \mathrm{b}\varepsilon \mathrm{R} \) (notation: \( \mathrm{a} \mid \mathrm{b} \) ) if there exists \( \mathrm{x}\varepsilon \mathrm{R} \) such that \( \mathrm{{ax}} = \mathrm{b} \) . Elements \( \mathrm{a},\mathr... |
1029_(GTM192)Elements of Functional Analysis | Definition 2.4 |
Definition 2.4 We say that a family of closed subsets \( {F}_{1},\ldots ,{F}_{n} \) of \( {\mathbb{R}}^{d} \) satisfies condition (C) if, for every compact subset \( K \) of \( {\mathbb{R}}^{d} \), the set
\[
\left\{ {\left( {{x}^{1},\ldots ,{x}^{n}}\right) \in {F}_{1} \times \cdots \times {F}_{n} : {x}^{1} + \cdots... |
18_Algebra Chapter 0 | Definition 1.1 |
Definition 1.1. A field extension \( k \subseteq F \) is finite, of degree \( n \), if \( F \) has (finite) dimension \( \dim F = n \) as a vector space over \( k \) . The extension is infinite otherwise.
The degree of a finite extension \( k \subseteq F \) is denoted by \( \left\lbrack {F : k}\right\rbrack \) (and ... |
1042_(GTM203)The Symmetric Group | Definition 3.10.1 |
Definition 3.10.1 If \( v = \left( {i, j}\right) \) is a node in the diagram of \( \lambda \), then it has hook
\[
{H}_{v} = {H}_{i, j} = \left\{ {\left( {i,{j}^{\prime }}\right) : {j}^{\prime } \geq j}\right\} \cup \left\{ {\left( {{i}^{\prime }, j}\right) : {i}^{\prime } \geq i}\right\}
\]
with corresponding hook... |
1116_(GTM270)Fundamentals of Algebraic Topology | Definition 5.1.2 |
Definition 5.1.2. Let \( {I}^{n} \) be the standard \( n \) -cube. Its boundary is given by
\[
\partial {I}^{0} = 0
\]
and
\[
\partial {I}^{0} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i}\left( {{A}_{i} - {B}_{i}}\right)
\]
for \( n > 0 \) .
\( \diamond \)
In this definition, \( \partial {I}^{n}... |
18_Algebra Chapter 0 | Definition 1.2 |
Definition 1.2. A field extension \( k \subseteq F \) is simple if there exists an element \( \alpha \in F \) such that \( F = k\left( \alpha \right) \) .
The extensions recalled above are of this kind: if \( K = k\left\lbrack t\right\rbrack /\left( {f\left( t\right) }\right) \) and \( \alpha \) denotes the coset of... |
1063_(GTM222)Lie Groups, Lie Algebras, and Representations | Definition 10.35 |
Definition 10.35. Let \( \Lambda \) denote the set of integral elements. If \( \mu \) is a dominant integral element, define a set \( {S}_{\mu } \subset \Lambda \) as follows:
\[
{S}_{\mu } = \{ \eta \in \Lambda \mid \langle \eta + \delta ,\eta + \delta \rangle = \langle \mu + \delta ,\mu + \delta \rangle \} .
\]
N... |
1042_(GTM203)The Symmetric Group | Definition 3.6.4 |
Definition 3.6.4 The ith skeleton of \( \pi \in {\mathcal{S}}_{n},{\pi }^{\left( i\right) } \), is defined inductively by \( {\pi }^{\left( 1\right) } = \pi \) and
\[
{\pi }^{\left( i\right) } = \begin{array}{llll} {k}_{1} & {k}_{2} & \cdots & {k}_{m} \\ {l}_{1} & {l}_{2} & \cdots & {l}_{m} \end{array}
\]
where \( ... |
1088_(GTM245)Complex Analysis | Definition 8.2 |
Definition 8.2. Let \( D \) be a domain in \( \widehat{\mathbb{C}} \) . Aut \( D \) is defined as the group (under composition) of conformal automorphisms (or conformal bijections) of \( D \) ; that is, it consists of the conformal maps from \( D \) onto itself.
There are two naturally related problems:
Problem I. ... |
1059_(GTM219)The Arithmetic of Hyperbolic 3-Manifolds | Definition 2.1.3 |
Definition 2.1.3 Let \( {A}_{0} \) be the subspace of \( A \) spanned by the vectors \( i, j \) and \( k \) . Then the elements of \( {A}_{0} \) are the pure quaternions in \( A \) .
This definition does not depend on the choice of basis. For, let \( x = {a}_{0} + \) \( {a}_{1}i + {a}_{2}j + {a}_{3}k \) . Then
\[
{... |
1065_(GTM224)Metric Structures in Differential Geometry | Definition 11.2 |
Definition 11.2. A tensor field of type \( \left( {r, s}\right) \) on \( M \) is a (smooth) map \( T : M \rightarrow {T}_{r, s}\left( M\right) \) such that \( \pi \circ T = {1}_{M} \) . A differential \( k \) -form on \( M \) is a map \( \alpha : M \rightarrow {\Lambda }_{k}^{ * }\left( M\right) \) such that \( \pi \... |
1089_(GTM246)A Course in Commutative Banach Algebras | Definition 2.10.12 |
Definition 2.10.12. The topological group \( G \) is said to be almost periodic if the homomorphism \( \phi : G \rightarrow \Delta \left( {{AP}\left( G\right) }\right) \) is injective. Even though in general \( \phi \) need not be injective, \( \Delta \left( {{AP}\left( G\right) }\right) \) is called the Bohr or almo... |
1059_(GTM219)The Arithmetic of Hyperbolic 3-Manifolds | Definition 0.3.5 |
Definition 0.3.5 If \( I \) is a non-zero ideal of \( {R}_{k} \), define the norm of \( I \) by
\[
N\left( I\right) = \left| {{R}_{k}/I}\right|
\]
The unique factorisation enables the determination of the norm of ideals to be reduced to the determination of norms of prime ideals. This reduction firstly requires th... |
1343_[鄂维南&李铁军&Vanden-Eijnden] Applied Stochastic Analysis -GSM199 | Definition 1.11 |
Definition 1.11. The distribution of the random variable \( X \) is a probability measure \( \mu \) on \( \mathbb{R} \), defined for any set \( B \in \mathcal{R} \) by
(1.11)
\[
\mu \left( B\right) = \mathbb{P}\left( {X \in B}\right) = \mathbb{P} \circ {X}^{-1}\left( B\right) .
\]
In particular, we define the dist... |
1569_混合相依变量的极限理论(陆传荣) | Definition 6.2.1 |
Definition 6.2.1. The random field \( \left\{ {{\xi }_{n,\mathbf{j}},\mathbf{j} \in {\mathcal{J}}_{n}}\right\} \) is said to be \( \alpha \) -mixing if \( \alpha \left( {nx}\right) \rightarrow 0 \) as \( n \rightarrow \infty \), where
\[
\alpha \left( {nx}\right) = \mathop{\sup }\limits_{\substack{{I, J \subset {\ma... |
1167_(GTM73)Algebra | Definition 7.3 |
Definition 7.3. Let \( \mathrm{K} \) be a commutative ring with identity and \( \mathrm{A},\mathrm{B}\mathrm{K} \) -algebras.
(i) \( A \) subalgebra of \( \mathrm{A} \) is a subring of \( \mathrm{A} \) that is also a \( \mathrm{K} \) -submodule of \( \mathrm{A} \) .
(ii) \( A \) (left, right, two-sided) algebra ide... |
109_The rising sea Foundations of Algebraic Geometry | Definition 4.116 |
Definition 4.116. Given an arbitrary collection of simplices containing at least one chamber, their convex hull is the intersection of all convex chamber subcomplexes containing them.
The convex hull is itself a convex chamber subcomplex containing the given simplices, so it is the smallest one. We will also call th... |
1099_(GTM255)Symmetry, Representations, and Invariants | Definition 9.3.16 |
Definition 9.3.16. A semistandard skew tableau of shape \( \lambda /\mu \) and weight \( v \in {\mathbb{N}}^{n} \) is an assignment of positive integers to the boxes of the skew Ferrers diagram \( \lambda /\mu \) such that
1. if \( {\mathbf{v}}_{j} \neq 0 \) then the integer \( j \) occurs in \( {\mathbf{v}}_{j} \) ... |
109_The rising sea Foundations of Algebraic Geometry | Definition 3.86 |
Definition 3.86. Let \( \left( {W, S}\right) \) be a Coxeter system with Coxeter matrix \( M \) . We say that a Coxeter complex \( \sum \) is of type \( \left( {W, S}\right) \) (or of type \( M \) ) if \( \sum \) comes equipped with a type function having values in \( S \) such that the Coxeter matrix of \( \sum \) i... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 11.18 |
Definition 11.18 An (injective) monomial matrix is a matrix of constants \( {\lambda }_{qp} \in \mathbb{k} \) such that:
1. Each row is labeled by a vector \( {\mathbf{a}}_{q} \in {\mathbb{Z}}^{d} \) and a face \( {F}^{q} \) of \( Q \) .
2. Each column is labeled by a vector \( {\mathbf{a}}_{p} \in {\mathbb{Z}}^{d}... |
108_The Joys of Haar Measure | Definition 3.3.1 |
Definition 3.3.1. Let \( A \in \mathbb{Z}\left\lbrack X\right\rbrack \) be a nonzero polynomial. We define the content of \( A \) and denote by \( c\left( A\right) \) the GCD of all the coefficients of \( A \) .
Proposition 3.3.2 (Gauss’s lemma). If \( A \) and \( B \) are two nonzero polynomials in \( \mathbb{Z}\le... |
113_Topological Groups | Definition 24.12 |
Definition 24.12. Let \( \mathcal{L} \) be an algebraic language. We construct an \( \mathcal{L} \) - structure \( {\mathfrak{{Fr}}}_{\mathcal{L}} \) which will be called the absolutely free \( \mathcal{L} \) -algebra. Its universe is \( {\operatorname{Trm}}_{\mathcal{L}} \), and for any operation symbol \( \mathbf{O... |
1059_(GTM219)The Arithmetic of Hyperbolic 3-Manifolds | Definition 2.7.1 |
Definition 2.7.1 If \( A \) is a quaternion algebra over the number field \( k \), let \( {A}_{v} \) (resp. \( {A}_{\mathcal{P}} \) ) denote the quaternion algebra \( A{ \otimes }_{k}{k}_{v} \) (resp. \( A{ \otimes }_{k}{k}_{\mathcal{P}} \) ) over \( {k}_{v} \) (resp \( {k}_{\mathcal{P}} \) ). Then \( A \) is said to... |
1088_(GTM245)Complex Analysis | Definition 4.6 |
Definition 4.6. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be a continuous path. We say that \( \gamma \) is a piecewise differentiable path (henceforth abbreviated \( {pdp} \) ) if there exists a partition of \( \left\lbrack {a, b}\right\rbrack \) of the form given in (4.1) such that ... |
1288_[张芷芬&丁同仁&黄文灶&董镇喜] Qualitative Theory of Differential Equations | Definition 4.1 |
Definition 4.1. Let \( l \subset {R}^{2} \) be a smooth Jordan curve, and let \( G \) be the region enclosed by \( l \) . Suppose at the point \( P \in l \), the tangent to \( l \) coincides with the field vector determined by (4.1). If there exists a sufficiently small orbit arc \( r\left( P\right) \) for equation (... |
110_The Schwarz Function and Its Generalization to Higher Dimensions | Definition 9.2 |
Definition 9.2. Let \( {x}^{ * } \) be a feasible point of \( \left( P\right) \) in (9.1). A tangent direction of \( \mathcal{F}\left( P\right) \) at \( {x}^{ * } \) is called a feasible direction of \( \left( P\right) \) at \( {x}^{ * } \) . We denote the set of feasible directions of \( \left( P\right) \) at \( {x}... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 3.10 |
Definition 3.10 A planar map is a graph \( G \) together with an embedding of \( G \) into a surface homeomorphic to the plane \( {\mathbb{R}}^{2} \) .
That being said, we refer to the planar map simply as \( G \) if its embedding is given. We require the surface to be homeomorphic rather than equal to \( {\mathbb{R... |
1105_(GTM260)Monomial.Ideals,Jurgen.Herzog(2010) | Definition 6.1.1 |
Definition 6.1.1. The numerical function
\[
H\left( {M, - }\right) : \mathbb{Z} \rightarrow \mathbb{Z},\;i \mapsto H\left( {M, i}\right) \mathrel{\text{:=}} {\dim }_{K}{M}_{i}
\]
is called the Hilbert function of \( M \) . The formal Laurent series \( {H}_{M}\left( t\right) = \) \( \mathop{\sum }\limits_{{i \in \m... |
1185_(GTM91)The Geometry of Discrete Groups | Definition 7.2.3 |
Definition 7.2.3. Let \( E \) be a subset of the hyperbolic plane. Then
(i) \( \widetilde{E} \) denotes the closure of \( E \) relative to the hyperbolic plane;
(ii) \( \bar{E} \) denotes the closure of \( E \) relative to the closed hyperbolic plane.
Of course, \( \bar{E} \) is also the closure of \( E \) in \( \... |
113_Topological Groups | Definition 6.26 |
Definition 6.26. A set \( A \subseteq \omega \) is simple if \( A \) is r.e., \( \omega \sim A \) is infinite, and \( B \cap A \neq 0 \) whenever \( B \) is an infinite r.e. set.
Theorem 6.27. A simple set is neither recursive nor creative.
Proof. If \( A \) is simple and recursive, then \( \omega \sim A \) is an i... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 10.19 |
Definition 10.19 A bounded operator \( A \) on \( \mathbf{H} \) is normal if \( A \) commutes with its adjoint: \( A{A}^{ * } = {A}^{ * }A \) .
Every bounded self-adjoint operator is obviously normal. Other examples of normal operators are skew-self-adjoint operators \( \left( {{A}^{ * } = - A}\right) \) and unitary... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 9.6 |
Definition 9.6 An unbounded operator \( A \) on \( \mathbf{H} \) is said to be closed if the graph of \( A \) is a closed subset of \( \overline{\mathbf{H}} \times \mathbf{H} \) . An unbounded operator \( A \) on \( \mathbf{H} \) is said to be closable if the closure in \( \mathbf{H} \times \mathbf{H} \) of the graph... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 21.3 |
Definition 21.3 If \( f \) and \( g \) are smooth functions on \( N \), define the Poisson bracket \( \{ f, g\} \) of \( f \) and \( g \) by
\[
\{ f, g\} = - {\omega }^{-1}\left( {{df},{dg}}\right)
\]
In particular, if \( \mathbf{1} \) denotes the constant function on \( N \), then \( \{ \mathbf{1}, f\} = \) \( \{ ... |
111_Three Dimensional Navier-Stokes Equations-James_C._Robinson,_Jos_L._Rodrigo,_Witold_Sadows(z-lib.org | Definition 6.50 |
Definition 6.50. The multimaps \( {F}_{i} : X \rightrightarrows {Y}_{i} \) are said to be cooperative (resp. coordinated) at \( \left( {\bar{x},{\bar{y}}_{1},\ldots ,{\bar{y}}_{k}}\right) \) if the graphs of the multimaps \( {M}_{i} : \bar{X} \rightrightarrows Y \mathrel{\text{:=}} {Y}_{1} \times \) \( \cdots \times ... |
1129_(GTM35)Several Complex Variables and Banach Algebras | Definition 4.6 |
Definition 4.6. A \( k \) -form \( {\omega }^{k} \) on \( \Omega \) is a map \( {\omega }^{k} \) assigning to each \( x \) in \( \Omega \) an element of \( { \land }^{k}\left( {T}_{x}\right) \) .
\( k \) -forms form a module over the algebra of scalar functions on \( \Omega \) in a natural way.
Let \( {\tau }^{k} \... |
1329_[肖梁] Abstract Algebra (2022F) | Definition 8.1.1 |
Definition 8.1.1. For \( x, y \in G \), define \( \left\lbrack {x, y}\right\rbrack \mathrel{\text{:=}} {x}^{-1}{y}^{-1}{xy} \), the commutator of \( x \) and \( y \) . Let \( {G}^{\text{der }} = {G}^{\prime } \mathrel{\text{:=}} \langle \left\lbrack {x, y}\right\rbrack \mid x, y \in G\rangle \) be the subgroup of \( ... |
1172_(GTM8)Axiomatic Set Theory | Definition 13.1 |
Definition 13.1. Let \( \mathbf{B} = \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) be a complete Boolean algebra. Then \( {V}_{\alpha }^{\left( \mathbf{B}\right) } \) is defined by recursion with respect to \( \alpha \) as follows:
\[
{V}_{0}^{\left( \mathbf{B}\right) } \triangleq 0
\]
\[
{V}_{\alpha }^... |
109_The rising sea Foundations of Algebraic Geometry | Definition 7.2 |
Definition 7.2. A family \( {\left( {X}_{\alpha }\right) }_{\alpha \in \Phi } \) of subgroups of \( {\operatorname{Aut}}_{0}\Delta \) is called a system of pre-root groups if it satisfies the following three conditions:
(1) \( {X}_{\alpha } \) fixes \( \alpha \) pointwise for each \( \alpha \in \Phi \) .
(2) For ea... |
1063_(GTM222)Lie Groups, Lie Algebras, and Representations | Definition 3.40 |
Definition 3.40. If \( G \) is a matrix Lie group with Lie algebra \( \mathfrak{g} \), then the exponential map for \( G \) is the map
\[
\exp : \mathfrak{g} \rightarrow G\text{.}
\]
That is to say, the exponential map for \( G \) is the matrix exponential restricted to the Lie algebra \( \mathfrak{g} \) of \( G \)... |
111_Three Dimensional Navier-Stokes Equations-James_C._Robinson,_Jos_L._Rodrigo,_Witold_Sadows(z-lib.org | Definition 6.42 |
Definition 6.42 ([813]). A finite family \( {\left( {S}_{i}\right) }_{i \in I}\left( {I \mathrel{\text{:=}} {\mathbb{N}}_{k}}\right) \) of closed subsets of a normed space \( X \) is said to be allied at \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \) if whenever \( {x}_{n, i}^{ * } \in \) \(... |
1097_(GTM253)Elementary Functional Analysis | Definition 1.12 |
Definition 1.12. Let \( X \) be a normed linear space. If \( X \) is complete in the metric \( d \) defined from the norm by \( d\left( {x, y}\right) = \parallel x - y\parallel \), we call \( X \) a Banach space.
All of the above examples of normed linear spaces are Banach spaces. We will not stop to prove this now,... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 23.52 |
Definition 23.52 If \( f \) is a function on \( N \) for which \( {X}_{f} \) preserves \( \bar{P} \), let \( Q\left( f\right) \) be the operator on the half-form Hilbert space of \( P \) given by
\[
Q\left( f\right) s = \left( {{Q}_{\text{pre }}\left( f\right) \mu }\right) \otimes \nu - i\hslash \mu \otimes {\mathca... |
18_Algebra Chapter 0 | Definition 2.1 |
Definition 2.1. Let \( A \) be a set. The symmetric group, or group of permutations of \( A \), denoted \( {S}_{A} \), is the group \( {\operatorname{Aut}}_{\text{Set }}\left( A\right) \) . The group of permutations of the set \( \{ \mathbf{1},\ldots ,\mathbf{n}\} \) is denoted by \( {S}_{n} \) .
The terminology is ... |
1167_(GTM73)Algebra | Definition 3.10 |
Definition 3.10. Let \( \mathrm{X} \) be a nonempty subset of a commutative ring \( \mathrm{R} \) . An element \( \mathrm{d}\varepsilon \mathrm{R} \) is a greatest common divisor of \( \mathrm{X} \) provided:
(i) \( \mathrm{d} \mid \mathrm{a} \) for all \( \mathrm{a}\varepsilon \mathrm{X} \) ;
(ii) \( \mathrm{c} \m... |
1172_(GTM8)Axiomatic Set Theory | Definition 13.8 |
Definition 13.8. \( {\mathbf{V}}_{\alpha }^{\left( \mathbf{B}\right) } = \left\langle {{V}_{\alpha }^{\left( \mathbf{B}\right) }, \equiv ,\bar{ \in }}\right\rangle \) is defined by
\[
\llbracket u = v{\rrbracket }_{\alpha } \triangleq \llbracket u = v\rrbracket
\]
\[
\llbracket u \in v{\rrbracket }_{\alpha } \trian... |
1139_(GTM44)Elementary Algebraic Geometry | Definition 3.1 |
Definition 3.1. Let \( R \) be a ring and suppose the associated p.o. set \( \left( {\mathcal{I}, \subset }\right) \) satisfies the a.c.c.-that is, each strictly ascending chain of ideals of \( R \) , \( {\mathfrak{a}}_{1} \subsetneqq {\mathfrak{a}}_{2} \subsetneqq \ldots \), terminates after finitely many steps. The... |
1167_(GTM73)Algebra | Definition 1.3 |
Definition 1.3. A nonzero element a in a ring \( \mathrm{R} \) is said to be a left [resp. right] zero divisor if there exists a nonzero \( \mathrm{b}\varepsilon \mathrm{R} \) such that \( \mathrm{{ab}} = 0 \) [resp. \( \mathrm{{ba}} = 0 \) ]. \( A \) zero divisor is an element of \( \mathrm{R} \) which is both a lef... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 4.2 |
Definition 4.2 Suppose \( X \) is a labeled cell complex, by which we mean that its \( r \) vertices have labels that are vectors \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{r} \) in \( {\mathbb{N}}^{n} \) . The label on an arbitrary face \( F \) of \( X \) is the exponent \( {\mathbf{a}}_{F} \) on the least common mul... |
1094_(GTM250)Modern Fourier Analysis | Definition 2.3.5 |
Definition 2.3.5. Let \( 0 < p, q < \infty \) . A sequence of complex numbers \( r = {\left\{ {r}_{Q}\right\} }_{Q \in \mathcal{D}} \) is called an \( \infty \) -atom for \( {\dot{f}}_{p}^{\alpha, q} \) if there exists a dyadic cube \( {Q}_{0} \) such that
(a) \( {r}_{Q} = 0 \) if \( Q \nsubseteq {Q}_{0} \) ;
(b) \... |
108_The Joys of Haar Measure | Definition 3.3.14 |
Definition 3.3.14. Let \( K \) be a number field. An ideal \( I \) of \( {\mathbb{Z}}_{K} \) is a sub- \( {\mathbb{Z}}_{K} \) -module of \( {\mathbb{Z}}_{K} \) ; in other words, it is an additive subgroup of \( {\mathbb{Z}}_{K} \) such that \( {\alpha x} \in I \) for all \( \alpha \in {\mathbb{Z}}_{K} \) and \( x \in... |
1116_(GTM270)Fundamentals of Algebraic Topology | Definition 4.2.13 |
Definition 4.2.13. Let \( X \) be a CW-complex. The cellular chain complex of \( X \) is defined by
\[
{C}_{n}^{\text{cell }}\left( X\right) = {H}_{n}\left( {{X}^{n},{X}^{n - 1}}\right)
\]
with \( {\partial }_{n} : {C}_{n}^{\text{cell }}\left( X\right) \rightarrow {C}_{n - 1}^{\text{cell }}\left( X\right) \) the co... |
109_The rising sea Foundations of Algebraic Geometry | Definition 8.10 |
Definition 8.10. A set of twin roots \( \Psi \subseteq \Phi \) is said to be convex if it has the form \( \Psi = \Psi \left( \mathcal{M}\right) \) for some pair \( \mathcal{M} \) with \( {\mathcal{M}}_{ + } \) and \( {\mathcal{M}}_{ - } \) both nonempty.
Remarks 8.11. (a) Recall that a twin root \( \alpha = \left( {... |
1167_(GTM73)Algebra | Definition 7.4 |
Definition 7.4. Let \( \mathrm{S} \) be a nonempty set of automorphisms of a field \( \mathrm{F} \) . \( \mathrm{S} \) is linearly independent provided that for any \( {\mathrm{a}}_{1},\ldots ,{\mathrm{a}}_{\mathrm{n}}\varepsilon \mathrm{F} \) and \( {\sigma }_{1},\ldots ,{\sigma }_{\mathrm{n}}\varepsilon \mathrm{S}\... |
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