不动点在经济学中的应用：Advanced Fixed Point Theory for Economics

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Andrew McLennan写了本关于不动点再经济学中应用的书。Advanced Fixed Point Theory for Economics。比较高级，很早还有本Border85年的类似的书，稍微简单一点。 advanced_fp.pdf (1.68 MB, 需要: 1 个论坛币)

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关键词：Economics Advanced Economic ADVANCE econom 经济学

请问帖子的内容是不是从新浪微播粘贴过来的？

楼主，能不能介绍详细一点，能贴一下该书的目录给大家看看吧

cunxws 发表于 2013-10-10 07:59
请问帖子的内容是不是从新浪微播粘贴过来的？

额啊，是哈

elite7502 发表于 2013-10-10 08:07
楼主，能不能介绍详细一点，能贴一下该书的目录给大家看看吧

Contents
1 Introduction and Summary 2
1.1 The First Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . 3
1.2 “Fixing” Kakutani’s Theorem . . . . . . . . . . . . . . . . . . . . . 5
1.3 Essential Sets of Fixed Points . . . . . . . . . . . . . . . . . . . . . 7
1.4 Index and Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 The Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.3 The Fixed Point Index . . . . . . . . . . . . . . . . . . . . . 15
1.5 Topological Consequences . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
I Topological Methods 22
2 Planes, Polyhedra, and Polytopes 23
2.1 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Convex Sets and Cones . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Polyhedral Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Computing Fixed Points 35
3.1 The Lemke-Howson Algorithm . . . . . . . . . . . . . . . . . . . . . 36
3.2 Implementation and Degeneracy Resolution . . . . . . . . . . . . . 44
3.3 Using Games to Find Fixed Points . . . . . . . . . . . . . . . . . . 49
3.4 Sperner’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 The Scarf Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Remarks on Computation . . . . . . . . . . . . . . . . . . . . . . . 59
4 Topologies on Spaces of Sets 66
4.1 Topological Terminology . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Spaces of Closed and Compact Sets . . . . . . . . . . . . . . . . . . 67
4.3 Vietoris’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Hausdorff Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Basic Operations on Subsets . . . . . . . . . . . . . . . . . . . . . . 71
iiCONTENTS iii
4.5.1 Continuity of Union . . . . . . . . . . . . . . . . . . . . . . 71
4.5.2 Continuity of Intersection . . . . . . . . . . . . . . . . . . . 71
4.5.3 Singletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.4 Continuity of the Cartesian Product . . . . . . . . . . . . . 72
4.5.5 The Action of a Function . . . . . . . . . . . . . . . . . . . . 73
4.5.6 The Union of the Elements . . . . . . . . . . . . . . . . . . . 74
5 Topologies on Functions and Correspondences 76
5.1 Upper and Lower Semicontinuity . . . . . . . . . . . . . . . . . . . 77
5.2 The Strong Upper Topology . . . . . . . . . . . . . . . . . . . . . . 78
5.3 The Weak Upper Topology . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 The Homotopy Principle . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Metric Space Theory 85
6.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Banach and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 EmbeddingTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.6 Dugundji’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Retracts 96
7.1 Kino*****a’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2 Retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Euclidean Neighborhood Retracts . . . . . . . . . . . . . . . . . . . 99
7.4 Absolute Neighborhood Retracts . . . . . . . . . . . . . . . . . . . 100
7.5 Absolute Retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.6 Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8 Essential Sets of Fixed Points 106
8.1 The Fan-Glicksberg Theorem . . . . . . . . . . . . . . . . . . . . . 107
8.2 Convex Valued Correspondences . . . . . . . . . . . . . . . . . . . . 109
8.3 Kino*****a’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9 Approximation of Correspondences 114
9.1 The Approximation Result . . . . . . . . . . . . . . . . . . . . . . . 114
9.2 Extending from the Boundary of a Simplex . . . . . . . . . . . . . . 115
9.3 Extending to All of a Simplicial Complex . . . . . . . . . . . . . . . 117
9.4 Completing the Argument . . . . . . . . . . . . . . . . . . . . . . . 119
II Smooth Methods 123
10 Differentiable Manifolds 124
10.1 Review of Multivariate Calculus . . . . . . . . . . . . . . . . . . . . 125
10.2 Smooth Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . 127CONTENTS 1
10.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.4 Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.5 Tangent Vectors and Derivatives . . . . . . . . . . . . . . . . . . . . 132
10.6 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.7 Tubular Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.8 Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . . . 142
10.9 Classification of Compact 1-Manifolds . . . . . . . . . . . . . . . . . 145
11 Sard’s Theorem 149
11.1 Sets of Measure Zero . . . . . . . . . . . . . . . . . . . . . . . . . . 150
11.2 A Weak Fubini Theorem . . . . . . . . . . . . . . . . . . . . . . . . 152
11.3 Sard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.4 Measure Zero Subsets of Manifolds . . . . . . . . . . . . . . . . . . 156
11.5 Genericity of Transversality . . . . . . . . . . . . . . . . . . . . . . 157
12 Degree Theory 162
12.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
12.2 Induced Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 167
12.3 The Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.4 Composition and Cartesian Product . . . . . . . . . . . . . . . . . . 173
13 The Fixed Point Index 175
13.1 Axioms for an Index on a Single Space . . . . . . . . . . . . . . . . 176
13.2 Multiple Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.3 The Index for Euclidean Spaces . . . . . . . . . . . . . . . . . . . . 179
13.4 Extension by Commutativity . . . . . . . . . . . . . . . . . . . . . . 181
13.5 Extension by Continuity . . . . . . . . . . . . . . . . . . . . . . . . 188
III Applications and Extensions 192
14 Topological Consequences 193
14.1 Euler, Lefschetz, and Eilenberg-Montgomery . . . . . . . . . . . . . 194
14.2 The Hopf Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
14.3 More on Maps Between Spheres . . . . . . . . . . . . . . . . . . . . 199
14.4 Invariance of Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 205
14.5 Essential Sets Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 206
15 Vector Fields and their Equilibria 210
15.1 Euclidean Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 211
15.2 Dynamics on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . 212
15.3 The Vector Field Index . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4 Dynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.5 The Converse Lyapunov Problem . . . . . . . . . . . . . . . . . . . 221
15.6 A Necessary Condition for Stability . . . . . . . . . . . . . . . . . . 225

elite7502 发表于 2013-10-10 08:07
楼主，能不能介绍详细一点，能贴一下该书的目录给大家看看吧

Preface
Over two decades ago now I wrote a rather long survey of the mathematical
theory of fixed points entitled Selected Topics in the Theory of Fixed Points. It
had no content that could not be found elsewhere in the mathematical literature,
but nonetheless some economists found it useful. Almost as long ago, I began
work on the project of turning it into a proper book, and finally that project is
coming to fruition. Various events over the years have reinforced my belief that the
mathematics presented here will continue to influence the development of theoretical
economics, and have intensified my regret about not having completed it sooner.
There is a vast literature on this topic, which has influenced me in many ways,
and which cannot be described in any useful way here. Even so, I should say
something about how the present work stands in relation to three other books on
fixed points. Fixed Point Theorems with Applications to Economics and Game
Theory by Kim Border (1985) is a complement, not a substitute, explaining various
forms of the fixed point principle such as the KKMS theorem and some of the
many theorems of Ky Fan, along with the concrete details of how they are actually
applied in economic theory. Fixed Point Theory by Dugundji and Granas (2003) is,
even more than this book, a comprehensive treatment of the topic. Its fundamental
point of view (applications to nonlinear functional analysis) audience (professional
mathematicians) and technical base (there is extensive use of algebraic topology)
are quite different, but it is still a work with much to offer to economics. Particularly
notable is the extensive and meticulous information concerning the literature and
history of the subject, which is full of affection for the theory and its creators. The
book that was, by far, the most useful to me, is The Lefschetz Fixed Point Theorem
by Robert Brown (1971). Again, his approach and mine have differences rooted in
the nature of our audiences, and the overall objectives, but at their cores the two
books are quite similar, in large part because I borrowed a great deal.
I would like to thank the many people who, over the years, have commented
favorably on Selected Topics. It is a particular pleasure to acknowledge some very
detailed and generous written comments by Klaus Ritzberger. This work would not
have been possible without the support and affection of my families, both present
and past, for which I am forever grateful.

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