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变换群和李代数（英文版) (Transformation Groups and Lie Algebras)

样章
作者：
Nail Ibragimov
定价：
59.00 元
版面字数：
210千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2013年
ISBN：
978-7-04-036741-6
物料号：
36741-00
出版时间：
2013-03-11
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数学
暂无
  • 目录
    • Front Matter
      • Part I Local Transformation Groups
        • 1 Preliminaries
          • 1.1 Changes of frames of reference and point transformations
            • 1.1.1 Translations
              • 1.1.2 Rotations
                • 1.1.3 Galilean transformation
                • 1.2 Introduction of transformation groups
                  • 1.2.1 Definitions and examples
                    • 1.2.2 Different types of groups
                    • 1.3 Some useful groups
                      • 1.3.1 Finite continuous groups on the straight line
                        • 1.3.2 Groups on the plane
                          • 1.3.3 Groups in IRn
                          • Exercises to Chapter 1
                          • 2 One-parameter groups and their invariants
                            • 2.1 Local groups of transformations
                              • 2.1.1 Notation and definition
                                • 2.1.2 Groups written in a canonical parameter
                                  • 2.1.3 Infinitesimal transformations and generators
                                    • 2.1.4 Lie equations
                                      • 2.1.5 Exponential map
                                        • 2.1.6 Determination of a canonical parameter
                                        • 2.2 Invariants .
                                          • 2.2.1 Definition and infinitesimal test
                                            • 2.2.2 Canonical variables
                                              • 2.2.3 Construction of groups using canonical variables
                                                • 2.2.4 Frequently used groups in the plane
                                                • 2.3 Invariant equations
                                                  • 2.3.1 Definition and infinitesimal test
                                                    • 2.3.2 Invariant representation of invariant manifolds
                                                      • 2.3.3 Proof of Theorem 2.9
                                                        • 2.3.4 Examples on Theorem 2.9
                                                        • Exercises to Chapter 2
                                                        • 3 Groups admitted by differential equations
                                                          • 3.1 Preliminaries
                                                            • 3.1.1 Differential variables and functions
                                                              • 3.1.2 Point transformations
                                                                • 3.1.3 Frame of differential equations
                                                                • 3.2 Prolongation of group transformations
                                                                  • 3.2.1 One-dimensional case
                                                                    • 3.2.2 Prolongation with several differential variables
                                                                      • 3.2.3 General case
                                                                      • 3.3 Prolongation of group generators
                                                                        • 3.3.1 One-dimensional case
                                                                          • 3.3.2 Several differential variables
                                                                            • 3.3.3 General case
                                                                            • 3.4 First definition of symmetry groups
                                                                              • 3.4.1 Definition
                                                                                • 3.4.2 Examples
                                                                                • 3.5 Second definition of symmetry groups
                                                                                  • 3.5.1 Definition and determining equations
                                                                                    • 3.5.2 Determining equation for second-order ODEs
                                                                                      • 3.5.3 Examples on solution of determining equations
                                                                                      • Exercises to Chapter 3
                                                                                      • 4 Lie algebras of operators
                                                                                        • 4.1 Basic definitions
                                                                                          • 4.1.1 Commutator
                                                                                            • 4.1.2 Properties of the commutator
                                                                                              • 4.1.3 Properties of determining equations
                                                                                                • 4.1.4 Lie algebras
                                                                                                • 4.2 Basic properties
                                                                                                  • 4.2.1 Notation
                                                                                                    • 4.2.2 Subalgebra and ideal
                                                                                                      • 4.2.3 Derived algebras
                                                                                                        • 4.2.4 Solvable Lie algebras
                                                                                                        • 4.3 Isomorphism and similarity
                                                                                                          • 4.3.1 Isomorphic Lie algebras
                                                                                                            • 4.3.2 Similar Lie algebras
                                                                                                            • 4.4 Low-dimensional Lie algebras
                                                                                                              • 4.4.1 One-dimensional algebras
                                                                                                                • 4.4.2 Two-dimensional algebras in the plane
                                                                                                                  • 4.4.3 Three-dimensional algebras in the plane
                                                                                                                    • 4.4.4 Three-dimensional algebras in IR3
                                                                                                                    • 4.5 Lie algebras and multi-parameter groups
                                                                                                                      • 4.5.1 Definition of multi-parameter groups
                                                                                                                        • 4.5.2 Construction of multi-parameter groups
                                                                                                                        • Exercises to Chapter 4
                                                                                                                        • 5 Galois groups via symmetries
                                                                                                                          • 5.1 Preliminaries
                                                                                                                            • 5.2 Symmetries of algebraic equations
                                                                                                                              • 5.2.1 Determining equation
                                                                                                                                • 5.2.2 First example
                                                                                                                                  • 5.2.3 Second example
                                                                                                                                    • 5.2.4 Third example
                                                                                                                                    • 5.3 Construction of Galois groups
                                                                                                                                      • 5.3.1 First example
                                                                                                                                        • 5.3.2 Second example
                                                                                                                                          • 5.3.3 Third example
                                                                                                                                            • 5.3.4 Concluding remarks
                                                                                                                                          • Assignment to Part I
                                                                                                                                            • Part II Approximate Transformation Groups
                                                                                                                                              • 6 Preliminaries
                                                                                                                                                • 6.1 Motivation
                                                                                                                                                  • 6.2 A sketch on Lie transformation groups
                                                                                                                                                    • 6.2.1 One-parameter transformation groups
                                                                                                                                                      • 6.2.2 Canonical parameter
                                                                                                                                                        • 6.2.3 Group generator and Lie equations
                                                                                                                                                          • 6.2.4 Exponential map
                                                                                                                                                          • 6.3 Approximate Cauchy problem
                                                                                                                                                            • 6.3.1 Notation
                                                                                                                                                              • 6.3.2 Definition of the approximate Cauchy problem
                                                                                                                                                            • 7 Approximate transformations
                                                                                                                                                              • 7.1 Approximate transformations defined
                                                                                                                                                                • 7.2 Approximate one-parameter groups
                                                                                                                                                                  • 7.2.1 Introductory remark
                                                                                                                                                                    • 7.2.2 Definition of one-parameter approximate transformation groups
                                                                                                                                                                      • 7.2.3 Generator of approximate transformation group
                                                                                                                                                                      • 7.3 Infinitesimal description
                                                                                                                                                                        • 7.3.1 Approximate Lie equations
                                                                                                                                                                          • 7.3.2 Approximate exponential map
                                                                                                                                                                          • Exercises to Chapter 7
                                                                                                                                                                          • 8 Approximate symmetries
                                                                                                                                                                            • 8.1 Definition of approximate symmetries
                                                                                                                                                                              • 8.2 Calculation of approximate symmetries
                                                                                                                                                                                • 8.2.1 Determining equations
                                                                                                                                                                                  • 8.2.2 Stable symmetries
                                                                                                                                                                                    • 8.2.3 Algorithm for calculation
                                                                                                                                                                                    • 8.3 Examples .
                                                                                                                                                                                      • 8.3.1 First example
                                                                                                                                                                                        • 8.3.2 Approximate commutator and Lie algebras
                                                                                                                                                                                          • 8.3.3 Second example
                                                                                                                                                                                            • 8.3.4 Third example
                                                                                                                                                                                            • Exercises to Chapter 8
                                                                                                                                                                                            • 9 Applications
                                                                                                                                                                                              • 9.1 Integration of equations with a small parameter using approximate symmetries
                                                                                                                                                                                                • 9.1.1 Equation having no exact point symmetries
                                                                                                                                                                                                  • 9.1.2 Utilization of stable symmetries
                                                                                                                                                                                                  • 9.2 Approximately invariant solutions
                                                                                                                                                                                                    • 9.2.1 Nonlinear wave equation
                                                                                                                                                                                                      • 9.2.2 Approximate travelling waves of KdV equation
                                                                                                                                                                                                      • 9.3 Approximate conservation laws
                                                                                                                                                                                                        • Exercises to Chapter 9
                                                                                                                                                                                                        • Assignment to Part II
                                                                                                                                                                                                          • Bibliography
                                                                                                                                                                                                            • Index

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