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values | def_number stringlengths 12 19 | text stringlengths 5.47k 10k |
|---|---|---|
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... |
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