最優(yōu)和平衡

內容概要

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).
　　本書為英文版。

書籍目錄

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 Theorems3. Conjugate Functions and Convex Minimisation Problems3.1 Introduction3.2 Characterisation of Convex Lower Semi-continuous Functions3.3 Fenchel's Theorem3.4 Properties of Conjugate Functions3.5 Support Functions4. Subdifferentials of Convex Functions4.1 Introduction 4.2 Definitions4.3 Subdifferentiability of Convex Continuous Functions4.4 Subdifferentiability of Convex Lower Semi-continuous Functions4.5 Subdifferential Calculus4.6 Tangent and Normal Cones5. Marginal Properties of Solutions of Convex MinimisationProblems5.1 Introduction5.2 Fermat's Rule5.3 Minimisation Problems with Constraints5.4 Principle of Price Decentralisation5.5 Regularisation and Penalisation6. Generalised Gradients of Locally Lipschitz Functions6.1 Introduction6.2 Definitions6.3 Elementary Properties6.4 Generalised Gradients6.5 Normal and Tangent Cones to a Subset6.6 Fermat's Rule for Minimisation Problems with Constraints7. Two-person Games. Fundamental Concepts and Examples7.1 Introduction7.2 Decision Rules and Consistent Pairs of Strategies7.3 Brouwer's Fixed-point Theorem (1910)7.4 The Need to Convexify: Mixed Strategies7.5 Games in Normal (Strategic) Form7.6 Pareto Optima7.7 Conservative Strategies7.8 Some Finite Games7.9 Cournot's Duopoly8. Two-person Zero-sum Games: Theorems of Von Neumannand Ky Fan9. Solution of Nonlinear Equatioris and Inclusions10. Introduction to the Theory of Economic Equilibrium11. The Von Neumann Growth Model12. n-person Games13. Cooperative Games and Fuzzy GamesPart Ⅱ. Nonlinear Analysis: Examples15. Statements of Problems16. Solutions to ProblemsAppendix17. Compendium of ResultsReferencesSubject Index

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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).

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