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《线性微分方程的伽罗瓦理论》(Galois Theory of Linear Differential Equations)清晰版[PDF]

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  • 摘要:
    出版社Springer
    发行时间2003年
    语言英文
  • 时间: 2010/05/31 21:51:55 发布 | 2010/06/01 01:10:44 更新
  • 分类: 图书  教育科技 

Jean_Pierre

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中文名线性微分方程的伽罗瓦理论
原名Galois Theory of Linear Differential Equations
资源格式PDF
版本清晰版
出版社Springer
书号3540442286
发行时间2003年
地区美国
语言英文
简介

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内容简介

本书专门论述线性微分方程的伽罗瓦理论,涉及诸多方面:代数理论(尤其足微分伽罗瓦理论)、形式理论、分类、有限项可解性判定算法、单值性、希尔伯特21问题、渐近性和可求和性、反问题以及具正特征值的线性微分方程。附录是本书所用到的代数几何、线性代数群、层及Tannakian范畴中的一些概念。.

本书将成为该领域所有数学家和研究生的标准参考书。

内容截图

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目录

Algebraic Theory .
1 Picard-Vessiot Theory
1.1 Differential Rings and Fields
1.2 Linear Differential Equations
1.3 Picard-Vessiot Extensions
1.4 The Differential Galois Group
1.5 Liouvillian Extensions
2 Differential Operators and Differential Modules
2.1 The Ring g) = k[a] of Differential Operators
2.2 Constructions with Differential Modules
2.3 Constructions with Differential Operators
2.4 Differential Modules and Representations
3 Formal Local Theory
3.1 Formal Classification of Differential Equations
3.2 The Universal Picard-Vessiot Ring of K
3.3 Newton Polygons
4 Algorithmic Considerations
4.1 Rational and Exponential Solutions
4.2 Factoring Linear Operators
4.3 Liouvillian Solutions
4.3.1 Group Theory
4.3.2 Liouvillian Solutions for a Differential Module
4.3.3 Liouvillian Solutions for a Differential Operator
4.3.4 Second Order Equations
4.3.5 Third Order Equations
4.4 Finite Differential Galois groups
4.4.1 Generalities on Scalar Fuchsian Equations
4.4.2 Restrictions on the Exponents
4.4.3 Representations of Finite Groups
4.4.4 A Calculation of the Accessory Parameter
4.4.5 Examples
Analytic Theory
5 Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group
5.1 Monodromy of a Differential Equation
5.2 A Solution of the Inverse Problem
5.3 The Riemann-Hilbert Problem
6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem
6.1 Differentials and Connections
6.2 Vector Bundles and Connections
6.3 Fuchsian Equations
6.4 The Riemann-Hilbert Problem, Weak Form
6.5 Irreducible Connections
6.6 Counting Fuchsian Equations
7 Exact Asymptotics
7.1 Introduction and Notation
7.2 The Main Asymptotic Existence Theorem
7.3 The Inhomogeneous Equation of Order One
7.4 The Sheaves A,A0,A1/k,A01/k
7.5 The Equation (8 - q)f = g Revisited
7.6 The Laplace and Borel Transforms
7.7 The k-Summation Theorem ..
7.8 The Multisummation Theorem
8 Stokes Phenomenon and Differential Galois Groups
8.1 Introduction
8.2 The Additive Stokes Phenomenon
8.3 Construction of the Stokes Matrices
9 Stokes Matrices and Meromorphic Classification
9.1 Introduction
9.2 The Category Gr2
9.3 The Cohomology Set H1(S1, STS)
9.4 Explicit l-cocycles for H](Sl, STS)
9.5 H1(S1, STS) as an Algebraic Variety
10 Universal Picard-Vessiot Rings and Galois Groups
10.1 Introduction
10.2 Regular Singular Differential Equations
10.3 Formal Differential Equations
10.4 Meromorphic Differential Equations
11 Inverse Problems
11.1 Introduction
11.2 The Inverse Problem for C((z))
11.3 Some Topics on Linear Algebraic Groups
11.4 The Local Theorem
11.5 The Global Theorem
11.6 More on Abhyankar's Conjecture
11.7 The Constructive Inverse Problem
12 Moduli for Singular Differential Equations
12.1 Introduction
12.2 The Moduli Functor
12.3 An Example
12.4 Unramified Irregular Singularities
12.5 The Ramified Case
12.6 The Meromorphic Classification
13 Positive Characteristic
13.1 Classification of Differential Modules
13.2 Algorithmic Aspects
13.3 Iterative Differential Modules
Appendices
A Algebraic Geometry
A.1 Affine Varieties
A.2 Linear Algebraic Groups
B Tannakian Categories
B.1 Galois Categories
B.2 Affine Group Schemes
B.3 Tannakian Categories
C Sheaves and Cohomology
C.1 Sheaves: Definition and Examples
C.2 Cohomology of Sheaves
D Partial Differential Equations
D.1 The Ring of Partial Differential Operators
D.2 Picard-Vessiot Theory and Some Remarks
Bibliography
List of Notation
Index

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