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As in ordinary language, metaphors may be used in mathematics to explain agiven phenomenon by associating it with another which is (or is considered 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 construction 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 construction- of the mathematical corpusobeys its own logic, like that of literature, 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 consensus - are used todevelop aesthetic criteria).

Introduction

Part I. Nonlinear Analysis: Theory

1. Minimisation Problems: General Theorems

1.1 Introduction

1.2 Definitions

1.3 Epigraph

1.4 Lower Sections

1.5 Lower Semi-continuous Functions

1.6 Lower Semi-compact Functions

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

1.8 Application to Fixed-point Theorems

2. Convex Functions and Proximation， Projection and

Separation Theorems

2.1 Introduction

2.2 Definitions

2.3 Examples of Convex Functions

2.4 Continuous Convex Functions

2.5 The Proximation Theorem

2.6 Separation Theorems

2.6 Separation Theorems

3. Conjugate Functions and Convex Minimisation Problems

3.1 Introduction

3.2 Characterisation of Convex Lower Semi-continuous Functions

3.3 Fenchel's Theorem

3.4 Properties of Conjugate Functions

3.5 Support Functions

4. Subdifferentials of Convex Functions

4.1 Introduction

4.2 Definitions

4.3 Subdifferentiability of Convex Continuous Functions

4.4 Subdifferentiability of Convex Lower Semi-continuous Functions

4.5 Subdifferential Calculus

4.6 Tangent and Normal Cones

5. Marginal Properties of Solutions of Convex Minimisation

Problems

5.1 Introduction

5.2 Fermat's Rule

5.3 Minimisation Problems with Constraints

5.4 Principle of Price Decentralisation

5.5 Regularisation and Penalisation

6. Generalised Gradients of Locally Lipschitz Functions

6.1 Introduction

6.2 Definitions

6.3 Elementary Properties

6.4 Generalised Gradients

6.5 Normal and Tangent Cones to a Subset

6.6 Fermat's Rule for Minimisation Problems with Constraints

7. Two-person Games. Fundamental Concepts and Examples

7.1 Introduction

7.2 Decision Rules and Consistent 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 Conservative 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 Inclusions

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. Solutions 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 another which is (or is considered 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 construction 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 construction- of the mathematical corpusobeys its own logic, like that of literature, 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 consensus - are used todevelop aesthetic criteria).

　　本书为英文版。