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As in ordinary language, metaphors may be used in mathematics to explain agiven phenomenon by associating it with anothe***hich is (or is c***idered tobe) more familiar. It is this sense of familiarity, whether individual or collective,innate or acquired by education, which enables one to convince oneself that onehas understood the phenomenon in question. Contrary to popular opinion, mathematics is not simply a richer or moreprecise language. Mathematical reasoning is a separate faculty possessed by allhuman brains, just like the ability to compose or listen to music, to paint orlook at paintings, to believe in and follow cultural or moral codes, etc. But it is impossible (and dangerous) to compare these various facultieswithin a hierarchical framework; in particular, one cannot speak of thesuperiority of the language of mathematics. Naturally, the c***truction of mathematical metaphors requires theautonomous development of the discipline to provide theories which may besubstituted for or associated with the phenomena to be explained. This is thedomain of pure mathematics. The c***truction- of the mathematical corpusobeys its own logic, like that of li***ture, music or art. In all these domains,an aesthetic satisfaction is at once the objective of the creative activity and asignal which enables one to recognise successful works. (Likewise, in all thesedomains, fashionable phenomena - reflecting social c***ensus - are used todevelop aesthetic criteria).

Introduction

Part I. Nonlinear Analysis: Theory

1. Minimisation Problems: General Theorems

1.1 Introduction

1.2 Definiti***

1.3 Epigraph

1.4 Lower Secti***

1.5 Lower Semi-continuous Functi***

1.6 Lower Semi-compact Functi***

1.7 Approximate Minimisation of Lower Semi-continuous Functi*** on a Complete Space

1.8 Application to Fixed-point Theorems

2. Convex Functi*** and Proximation， Projection and

Separation Theorems

2.1 Introduction

2.2 Definiti***

2.3 Examples of Convex Functi***

2.4 Continuous Convex Functi***

2.5 The Proximation Theorem

2.6 Separation Theorems

2.6 Separation Theorems

3. Conjugate Functi*** and Convex Minimisation Problems

3.1 Introduction

3.2 Characterisation of Convex Lower Semi-continuous Functi***

3.3 Fenchel's Theorem

3.4 Properties of Conjugate Functi***

3.5 Support Functi***

4. Subdifferentials of Convex Functi***

4.1 Introduction

4.2 Definiti***

4.3 Subdifferentiability of Convex Continuous Functi***

4.4 Subdifferentiability of Convex Lower Semi-continuous Functi***

4.5 Subdifferential Calculus

4.6 Tangent and Normal Cones

5. Marginal Properties of Soluti*** of Convex Minimisation

Problems

5.1 Introduction

5.2 Fermat's Rule

5.3 Minimisation Problems with C***traints

5.4 Principle of Price Decentralisation

5.5 Regularisation and Penalisation

6. Generalised Gradients of Locally Lipschitz Functi***

6.1 Introduction

6.2 Definiti***

6.3 Elementary Properties

*** Generalised Gradients

6.5 Normal and Tangent Cones to a Subset

6.6 Fermat's Rule for Minimisation Problems with C***traints

7. Two-person Games. Fundamental Concepts and Examples

7.1 Introduction

7.2 Decision Rules and C***istent Pairs of Strategies

7.3 Brouwer's Fixed-point Theorem (1910)

7.4 The Need to Convexify: Mixed Strategies

7.5 Games in Normal (Strategic) Form

7.6 Pareto Optima

7.7 C***ervative Strategies

7.8 Some Finite Games

7.9 Cournot's Duopoly

8. Two-person Zero-sum Games: Theorems of Von Neumannand Ky Fan

9. Solution of Nonlinear Equatioris and Inclusi***

10. Introduction to the Theory of Economic Equilibrium

11. The Von Neumann Growth Model

12. n-person Games

13. Cooperative Games and Fuzzy Games

Part Ⅱ. Nonlinear Analysis: Examples

15. Statements of Problems

16. Soluti*** to Problems

Appendix

17. Compendium of Results

References

Subject Index

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书籍介绍

As in ordinary language, metaphors may be used in mathematics to explain agiven phenomenon by associating it with anothe***hich is (or is c***idered tobe) more familiar. It is this sense of familiarity, whether individual or collective,innate or acquired by education, which enables one to convince oneself that onehas understood the phenomenon in question. Contrary to popular opinion, mathematics is not simply a richer or moreprecise language. Mathematical reasoning is a separate faculty possessed by allhuman brains, just like the ability to compose or listen to music, to paint orlook at paintings, to believe in and follow cultural or moral codes, etc. But it is impossible (and dangerous) to compare these various facultieswithin a hierarchical framework; in particular, one cannot speak of thesuperiority of the language of mathematics. Naturally, the c***truction of mathematical metaphors requires theautonomous development of the discipline to provide theories which may besubstituted for or associated with the phenomena to be explained. This is thedomain of pure mathematics. The c***truction- of the mathematical corpusobeys its own logic, like that of li***ture, music or art. In all these domains,an aesthetic satisfaction is at once the objective of the creative activity and asignal which enables one to recognise successful works. (Likewise, in all thesedomains, fashionable phenomena - reflecting social c***ensus - are used todevelop aesthetic criteria).

　　本书为英文版。