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  • 非线性泛函分析及其应用,第3卷,变分法及最优化[平装]
  • 共1个商家     63.10元~63.10
  • 作者:宰德勒(作者)
  • 出版社:世界图书出版公司;第1版(2009年8月1日)
  • 出版时间:
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  • ISBN:9787510005220

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    《非线性泛函分析及其应用,第3卷,变分法及最优化》的写作起点很低,具备本科数学水平就可以读。

    作者简介

    作者:(德国)宰德勒

    目录

    Introduction to the Subject
    General Basic Ideas
    CHAPTER 37
    Introductory Typical Examples
    37.1. Real Functions in R
    37.2. Convex Functions in R
    37.3. Real Functions in RN, Lagrange Multipliers, Saddle Points, and
    Critical Points
    37.4. One-Dimensional Classical Variational Problems and Ordinary
    Differential Equations, Legendre Transformations, the
    Hamilton-Jaeobi Differential Equation, and the Classical
    Maximum Principle
    37.5. Multidimensional Classical Variational Problems and Elliptic
    Partial Differential Equations
    37.6. Eigenvalue Problems for Elliptic Differential Equations and
    Lagrange Multipliers
    37.7. Differential Inequalities and Variational Inequalities
    37.8. Game Theory and Saddle Points, Nash Equilibrium Points and
    Pareto Optimization
    37.9. Duality between the Methods of Ritz and Trefftz, Two-Sided
    Error Estimates
    37.10. Linear OptimiTation in R N, Lag, range Multipliers, and Duality
    37.11. Convex Optimization and Kuhn-Tucker Theory
    37.12. Approximation Theory, the Least-Squares Method, Deterministic
    and Stochastic Compensation Analysis
    37.13. Approximation Theory and Control Problems
    37.14. Pseudoinverses, Ill-Posed Problems and Tihonov Regularization
    37.15. Parameter Identification
    37.16. Chebyshev Approximation and Rational Approximation
    37.17. Linear Optimization in Infinite-Dimehsional Spaces, Chebyshev
    Approximation, and Approximate Solutions for Partial
    Differential Equations
    37.18. Splines and Finite Elements
    37.19. Optimal Quadrature Formulas
    37.20. Control Problems, Dynamic Optimization, and the Bellman
    Optimization Principle
    37.21. Control Problems, the Pontrjagin Maximum Principle, and the
    Bang-Bang Principle
    37.22. The Synthesis Problem for Optimal Control
    37.23. Elementary Provable Special Case of the Pontrjagin Maximum
    Principle
    37.24. Control with the Aid of Partial Differential Equations
    37.25. Extremal Problems with Stochastic Influences
    37.26. The Courant Maximum-Minimum Principle. Eigenvalues,
    Critical Points, and the Basic Ideas of the Ljusternik-Schnirelman
    Theory
    37.27. Critical Points and the Basic Ideas of the Morse Theory
    37.28. Singularities and Catastrophe Theory
    37.29. Basic Ideas for the Construction of Approximate Methods for
    Extremal Problems
    TWO FUNDAMENTAL EXISTENCE AND UNIQUENESS
    PRINCIPLES

    CHAPIER 38
    Compactness and Extremal Principles
    38.1. Weak Convergence and Weak* Convergence
    38.2. Sequential Lower Semicontinuous and Lower Semicontinuous
    Functionals
    38.3. Main Theorem for Extremal Problems
    38.4. Strict Convexity and Uniqueness
    38.5. Variants of the Main Theorem
    38.6. Application to Quadratic Variational Problems
    38.7. Application to Linear Optimization and the Role of Extreme
    Points
    38.8. Quasisolutions of Minimum Problems
    38.9. Application to a Fixed-Point Theorem
    38.10. The Palais-Smale Condition and a General Minimum Principle
    38.11. The Abstract Entropy Principle

    CHAPTER 39
    Convexity and Extremal Principles
    39.1. The Fundamental Principle of Geometric Functional Analysis
    39.2. Duality and the Role of Extreme Points in Linear Approximation
    Theory
    39.3. Interpolation Property of Subspaces and Uniqueness
    39.4. Ascent Method and the Abstract Alternation Theorem
    39.5. Application to Chebyshev Approximation
    EXTREMAL PROBLEMS WITHOUT SIDE CONDITIONS

    CHAPTER 40
    Free Local Extrema of Differentiable Functionals and the Calculus
    of Variations
    40.1. n th Variations, G-Derivative, and F-Derivative
    40.2. Necessary and Sufficient Conditions for Free Local Extrema
    40.3. Sufficient Conditions by Means of Comparison Functionals and
    Abstract Field Theory
    40.4. Application to Real Functions in RN
    40.5. Application to Classical Multidimensional Variational Problems
    in Spaces of Continuously Differentiable Functions
    40.6. Accessory Quadratic Variational Problems and Sufficient
    Eigenvalue Criteria for Local Extrema
    40.7. Application to Necessary and Sufficient Conditions for Local
    Extrema for Classical One-Dimensional Variational Problems

    CHAPTER 41
    Potential Operators
    41.1. Minimal Sequences
    41.2. Solution of Operator Equations by Solving Extremal Problems
    41.3. Criteria for Potential Operators
    41.4. Criteria for the Weak Sequential Lower Semicontinuity of
    Functionals
    41.5. Application to Abstract Hammerstein Equations with Symmetric
    Kernel Operators
    41.6. Application to Hammerstein Integral Equations

    CHAPTER 42
    Free Minima for Convex Functionals, Ritz Method and the
    Gradient Method
    42.1. Convex Functionals and Convex Sets
    42.2. Real Convex Functions
    42.3. Convexity of F, Monotonicity of F', and the Definiteness of the
    Second Variation
    42.4. Monotone Potential Operators
    42.5. Free Convex Minimum Problems and the Ritz Method
    42.6. Free Convex Minimum Problems and the Gradient Method
    42.7. Application to Variational Problems and Quasilinear Elliptic
    Differential Equations in Sobolev Spaces
    EXTREMAL PROBLEMS WITH SMOOTH SIDE CONDITIONS

    CHAPTER 43
    Lagrange Multipliers and Eigenvalue Problems
    43.1. The Abstract Basic Idea of Lagrange Multipliers
    43.2. Local Extrema with Side Conditions
    43.3. Existence of an Eigenvector Via a Minimum Problem
    43.4. Existence of a Bifurcation Point Via a Maximum Problem
    43.5. The Galerkin Method for Eigenvalue Problems
    43.6. The Generalized Implicit Function Theorem and Manifolds in
    B-Spaces
    43.7. Proof of Theorem 43.C
    43.8. Lagrange Multipliers
    43.9. Critical Points and Lagrange Multipliers
    43.10. Application to Real Functions in RN
    43.11. Application to Information Theory
    43.12. Application to Statistical Physics. Temperature as a Lagrange
    Multiplier
    43.13. Application to Variational Problems with Integral Side Conditions
    43.14. Application to Variational Problems with Differential Equations
    as Side Conditions

    CHAPTER 44
    Ljustemik-Schnirelman Theory and the Existence of
    Several Eigenvectors
    44.1. The Courant Maximum-Minimum Principle
    44.2. The Weak and the Strong Ljustemik Maximum-Minimum
    Principle for the Construction of Critical Points
    44.3. The Genus of Symmetric Sets
    44.4. The Palais-Smale Condition
    44.5. The Main Theorem for Eigenvalue Problems in Infinite-
    Dimensional B-spaces
    44.6. A Typical Example
    44.7. Proof of the Main Theorem
    ……
    CHAPTER 45
    CHAPTER 46
    CHAPTER 47
    CHAPTER 48
    CHAPTER 49
    CHAPTER 50
    CHAPTER 51
    CHAPTER 52
    CHAPTER 53
    CHAPTER 54
    CHAPTER 55
    CHAPTER 56
    CHAPTER 57
    Index

    序言

    自1932年,波兰数学家Banach发表第一部泛函分析专著“Theorie des operations lineaires”以来,这一学科取得了巨大的发展,它在其他领域的应用也是相当成功。如今,数学的很多领域没有了泛函分析恐怕寸步难行,不仅仅在数学方面,在理论物理方面的作用也具有同样的意义,M.Reed和B.Simon的“Methods of Modern MathematicalPhysjcs”在前言中指出:“自1926年以来,物理学的前沿已与日俱增集中于量子力学,以及奠定于量子理论的分支:原子物理、核物理固体物理、基本粒子物理等,而这些分支的中心数学框架就是泛函分析。”所以,讲述泛函分析的文献已浩如烟海。而每个时代,都有这个领域的代表性作品。例如上世纪50年代,F.Riesz和Sz.-Nagy的《泛函分析讲义》(中译版,科学出版社,1985),就是那个时代的一部具有代表性的著作;而60年代,N.Dunford和J.Schwartz的三大卷“Linear Operators”则是泛函分析学发展到那个时代的主要成果和应用的一个较全面的总结。泛函分析一经产生,它的发展就受到量子力学的强有力的推动,上世纪70年代,M.Reed和B.Simon的4卷“Methods 0f M0dern Mathematical Physics”是泛函分析对于量子力学应用的一个很好的总结。

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