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BL
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| Record Number
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889695
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| Main Entry
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McLennan, Andrew
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| Title & Author
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Advanced fixed point theory for economics /\ Andrew McLennan.
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| Publication Statement
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Singapore :: Springer,, [2018]
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, ©2018
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| Page. NO
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1 online resource
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| ISBN
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9789811307102
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: 9811307105
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9789811307096
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9811307091
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; Symbols; Symbols for Extensive Form Game Theory; Part I Overview; 1 Introduction and Summary; 1.1 The Key Concept; 1.2 Historical Background; 1.3 Chapter Contents; 1.3.1 Chapter 2: Planes, Polyhedra, and Polytopes; 1.3.2 Chapter 3: Computing Fixed Points; 1.3.3 Chapter 4: Topologies on Sets; 1.3.4 Chapter 5: Topologies on Functions and Correspondences; 1.3.5 Chapter 6: Metric Space Theory; 1.3.6 Chapter 7: Essential Sets of Fixed Points; 1.3.7 Chapter 8: Retracts; 1.3.8 Chapter 9: Approximation; 1.3.9 Chapter 10: Manifolds; 1.3.10 Chapter 11: Sard's Theorem
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1.3.11 Chapter 12: Degree Theory1.3.12 Chapter 13: The Fixed Point Index; 1.3.13 Chapter 14: Topological Consequences; 1.3.14 Chapter 15: Dynamical Systems; 1.3.15 Chapter 16: Extensive Form Games; 1.3.16 Chapter 17: Monotone Equilibria; Part II Combinatoric Geometry; 2 Planes, Polyhedra, and Polytopes; 2.1 Affine Subspaces; 2.2 Convex Sets and Cones; 2.3 Polyhedra; 2.4 Polytopes and Polyhedral Cones; 2.5 Polyhedral Complexes; 2.6 Simplicial Approximation; 2.7 Graphs; 3 Computing Fixed Points; 3.1 The Axiom of Choice, Subsequences, and Computation; 3.2 Sperner's Lemma; 3.3 The Scarf Algorithm
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3.4 Primitive Sets3.5 The Lemke-Howson Algorithm; 3.6 Implementation and Degeneracy Resolution; 3.7 Using Games to Find Fixed Points; 3.8 Homotopy; 3.9 Remarks on Computation; Part III Topological Methods; 4 Topologies on Spaces of Sets; 4.1 Topological Terminology; 4.2 Spaces of Closed and Compact Sets; 4.3 Vietoris' Theorem; 4.4 Hausdorff Distance; 4.5 Basic Operations on Subsets; 4.5.1 Continuity of Union; 4.5.2 Continuity of Intersection; 4.5.3 Singletons; 4.5.4 Continuity of the Cartesian Product; 4.5.5 The Action of a Function; 4.5.6 The Union of the Elements
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5 Topologies on Functions and Correspondences5.1 Upper and Lower Hemicontinuity; 5.2 The Strong Upper Topology; 5.3 The Weak Upper Topology; 5.4 The Homotopy Principle; 5.5 Continuous Functions; 6 Metric Space Theory; 6.1 Paracompactness; 6.2 Partitions of Unity; 6.3 Topological Vector Spaces; 6.4 Banach and Hilbert Spaces; 6.5 Embedding Theorems; 6.6 Dugundji's Theorem; 7 Essential Sets of Fixed Points; 7.1 The Fan-Glicksberg Theorem; 7.2 Convex Valued Correspondences; 7.3 Convex Combinations of Correspondences; 7.4 Kinoshita's Theorem; 7.5 Minimal Q-Robust Sets; 8 Retracts
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8.1 Kinoshita's Example8.2 Retracts; 8.3 Euclidean Neighborhood Retracts; 8.4 Absolute Neighborhood Retracts; 8.5 Absolute Retracts; 8.6 Domination; 9 Approximation of Correspondences by Functions; 9.1 The Approximation Result; 9.2 Technical Lemmas; 9.3 Proofs of the Propositions; Part IV Smooth Methods; 10 Differentiable Manifolds; 10.1 Review of Multivariate Calculus; 10.2 Smooth Partitions of Unity; 10.3 Manifolds; 10.4 Smooth Maps; 10.5 Tangent Vectors and Derivatives; 10.6 Submanifolds; 10.7 Tubular Neighborhoods; 10.8 Manifolds with Boundary; 10.9 Classification of Compact 1-Manifolds
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| Abstract
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This book develops the central aspect of fixed point theory - the topological fixed point index - to maximal generality, emphasizing correspondences and other aspects of the theory that are of special interest to economics. Numerous topological consequences are presented, along with important implications for dynamical systems. The book assumes the reader has no mathematical knowledge beyond that which is familiar to all theoretical economists. In addition to making the material available to a broad audience, avoiding algebraic topology results in more geometric and intuitive proofs. Graduate students and researchers in economics, and related fields in mathematics and computer science, will benefit from this book, both as a useful reference and as a well-written rigorous exposition of foundational mathematics. Numerous problems sketch key results from a wide variety of topics in theoretical economics, making the book an outstanding text for advanced graduate courses in economics and related disciplines.
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| Subject
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Economics.
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Fixed point theory.
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Economics.
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Fixed point theory.
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MATHEMATICS-- Calculus.
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| Subject
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MATHEMATICS-- Mathematical Analysis.
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| Dewey Classification
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515/.7248
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| LC Classification
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QA329.9
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