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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
ISBN：
978-7-04-036741-6
版面字数：
210.000千字
开本：
16开
全书页数：
185页
装帧形式：
精装
重点项目：
暂无
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
  • 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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