The collected papers of Stephen Smale Volume 2 / edited by F. Cucker, R. Wong.

Format:

Book

Author/Creator:

Smale, Stephen, 1930-

Contributor:

Cucker, Felipe, 1958-

Wong, Roderick, 1944-

Standardized Title:

Works. 2000

Language:

English

Subjects (All):

Mathematics.

Computer science.

Economics.

Physical Description:

1 online resource (557 p.)

Place of Publication:

Singapore : Singapore University Press ; River Edge, NJ : World Scientific, c2000.

Language Note:

English

Summary:

Stephen Smale is one of the great mathematicians of the 20th century. His work encompasses a wide variety of subjects: differential topology, dynamical systems, calculus of variations, theory of computation, mechanics and mathematical economy. In all these subjects he has left the imprint of a collection of fundamental results. He has obtained several distinctions, including the Fields Medal, the Veblen Prize, the Chauvenet Prize, the von Neumann Award and the National Medal of Science.This invaluable book contains the collected papers of Stephen Smale. These are divided into eight groups: top

Contents:

Contents; Part IV. Calculus of Variations (Global Analysis) and PDE's; Smale and Nonlinear Analysis: A Personal Perspective; References; A GENERALIZED MORSE THEORY; 1. Abstract theory.; 2. Manifolds of maps.; 3. Variational problems.; 4. A generalization of the Morse theory of geodesics.; 5. A nonlinear Dirichlet problem.; REFERENCES; Morse Theory and a Non-Linear Generalization of the Dirichlet Problem; REFERENCES; ON THE CALCULUS OF VARIATIONS; REFERENCES; AN INFINITE DIMENSIONAL VERSION OF SARD'S THEOREM.; Section 1.; Section 2.; Section 3.; REFERENCES.; On the Morse Index Theorem

Main Theorem.REFERENCES; Corrigendum On the Morse Index Theorem; WHAT IS GLOBAL ANALYSIS?; References; BOOK REVIEWS; REFERENCES; Smooth solutions of the heat and wave equations; Section 1; Main theorem; Section 2; Fundamental theorem of elliptic theory; Eigenfunction theorem; Section 3; Section 4; REFERENCES; Part V. Dynamics; On the Contribution of Smale to Dynamical Systems; References; Discussion; MORSE INEQUALITIES FOR A DYNAMICAL SYSTEM; 1. Introduction.; 2. Construction of the stable and unstable manifolds.; 3. Implications of (1)-(5).; 4. On Morse theory.; 5. Proof of the main theorem.

REFERENCESON DYNAMICAL SYSTEMS; 1. General considerations; 2. Morse relations for X in C; 3. On structural stability; BIBLIOGRAPHY; DYNAMICAL SYSTEMS AND THE TOPOLOGICAL CONJUGACY PROBLEM FOR DIFFEOMORPHISMS; REFERENCES; STABLE MANIFOLDS FOR DIFFERENTIAL EQUATIONS AND DIFFEOMORPHISMS; 1. Preliminaries.; 2. Cross-sections.; 2.1. LEMMA.; 2.2. THEOREM.; 2.3. THEOREM.; 2.4. THEOREM.; 2.5. LEMMA.; 2.6. LEMMA.; 2.7. LEMMA.; 3. Local Diffeomorphisms.; 3.1. THEOREM.; 3.2. THEOREM.; 3.3. THEOREM.; 3.4. LEMMA.; 4. Stable manifolds of a periodic orbit.; 4.1. THEOREM.; 5. Elementary periodic points.

5.1 THEOREM.5.2 THEOREM.; 5.3 LEMMA.; REMARK.; 6. Normal intersection.; 6.1. THEOREM.; 6.2. THEOREM.; 6.4. LEMMA.; 7. Elementary singularities of a vector field.; 7.2 THEOREM.; 7.3 LEMMA.; 7.4 LEMMA.; 8. Elementary closed orbits.; 8.1 THEOREM.; 8.2 THEOREM.; 9. Stable manifolds for a differential equation.; 9.1. THEOREM; 9.2. THEOREM; 9.3. THEOREM; 9.4. THEOREM; REFERENCES; A STRUCTURALLY STABLE DIFFERENTIABLE HOMEOMORPHISM WITH AN INFINITE NUMBER OF PERIODIC POINTS; Reference; Diffeomorphisms with Many Periodic Points; Introduction; 1; 2; 3; 4; 5; 6; 7; 8; 9; REFERENCES

STRUCTURALLY STABLE SYSTEMS ARE NOT DENSE.REFERENCES.; Dynamical Systems on n-Dimensional Manifolds; REFERENCES; DIFFERENTIABLE DYNAMICAL SYSTEMS; PART I. DIFFEOMORPHISMS; I.1. Introduction to conjugacy problems for diffeomorphisms.; I.2. The simplest examples.; I.3. Anosov diffeomorphisms.; I.4. The zeta function of a diffeomorphism.; I.5. Shift automorphisms and homoclinic points.; I.6. Unification.; I.7. Our goal in this section is to give at least a full sketch of proofs of the spectral decomposition theorem (6.2) and its corollary.

I.8. The goal of this section is to sketch the proofs of Theorems (6.5) and (6.6).

Notes:

Description based upon print version of record.

Includes bibliographical references.

ISBN:

9789812792822

9812792821