The Collected Papers of STEPHEN SMALE. Edited by. F. Cucker R. Wong. City University of Hong Kong

Transcription

1 The Collected Papers of STEPHEN SMALE Volume 1 Edited by F. Cucker R. Wong City University of Hong Kong SINGAPORE UNIVERSITY PRESS V W NATIONAL UNIVERSITY OF SINGAPORE \> World Scientific Singapore»New Jersey London Hong Kong

2 Contents VOLUME I Research Themes Luncheon Talk and Nomination for Stephen Smale (R. Bott) Some Recollections of the Early Work of Steve Smale (M. M. Peixoto) Luncheon Talk (R. Thom) Banquet Address at the Smalefest (E. C. Zeeman) Some Retrospective Remarks Part I. Topology The Work of Stephen Smale in Differential Topology (M. Hirsch) A Note on Open Maps A Vietoris Mapping Theorem for Homotopy Regulär Curves on Riemannian Manifolds On the Immersion of Manifolds in Euclidean Space (with R. K. Lashof) Self-Intersections of Immersed Manifolds (with R. K. Lashof) A Classification of Immersions of the Two-Sphere The Classification of Immersions of Spheres in Euclidean Spaces Diffeomorphisms of the 2-Sphere On Involutions of the 3-Sphere (with M. Hirsch) The Generalized Poincare Conjecture in Higher Dimensions On Gradient Dynamical Systems Generalized Poincare's Conjecture in Dimensions Greater Than Four Differentiable and Combinatorial Structures on Manifolds On the Structure of 5-Manifolds On the Structure of Manifolds

3 A Survey of Some Recent Developments in Differential Topology 217 The Story of the Higher Dimensional Poincare Conjecture (What actually happened on the beaches of Rio) 232 Part II. Economics Stephen Smale and the Economic Theory of General Equilibrium (G. Debreu) 243 Global Analysis and Economics, I: Pareto optimum and a generalization of Morse theory 259 Global Analysis and Economics, IIA: Extension of a theorem of Debreu 271 Global Analysis and Economics, III: Pareto optima and price equilibria 285 Global Analysis and Economics, IV: Finiteness and stability of equilibria with general consumption sets and production 296 Global Analysis and Economics, V: Pareto theory with constraints 305 Dynamics in general equilibrium theory 314 Global Analysis and Economics, VI: Geometrie analysis of Pareto Optima and price equilibria under classical hypotheses 321 A Convergent Process of Price Adjustment and Global Newton Methods 335 Exchange Processes with Price Adjustment 349 Some Dynamical Questions in Mathematical Economics 365 An Approach to the Analysis of Dynamic Processes in Economic Systems 368 On Comparative Statics and Bifurcation in Economic Equilibrium Theory 373 The Prisoner's Dilemma and Dynamical Systems Associated to Non-Cooperative Games 380 Global Analysis and Economics 398 Gerard Debreu Wins the Nobel Prize 438 Global Analysis in Economic Theory 440

4 Part III. Miscellaneous Scientists and the Arms Race 445 On the Steps of Moscow University 454 Some Autobiographical Notes 461 Mathematical Problems for the Next Century 480 VOLUME II Part IV. Calculus of Variations (Global Analysis) and PDE's Smale and Nonlinear Analysis: A personal perspective (A. J. Tromba) 491 A Generalized Morse Theory (with R. Palais) 503 Morse Theory and a Non-Linear Generalization of the Dirichlet Problem 511 On the Calculus of Variations 526 An Infinite Dimensional Version of Sard's Theorem 529 On the Morse Index Theorem 535 A correction to "On the Morse Index Theorem" 542 What is Global Analysis? 544 Book Review on "Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold" by Marston Morse 550 Smooth Solutions of the Heat and Wave Equations 561 Part V. Dynamics On the Contribution of Smale to Dynamical Systems (J. Palis) 575 Discussion (S. Newhouse, R. F. Williams and others) 589 Morse Inequalities for a Dynamical System 596 On Dynamical Systems 603 Dynamical Systems and the Topological Conjugacy Problem for Diffeomorphisms 607 Stable Manifolds for Differential Equations and Diffeomorphisms 614

5 A Structurally Stable Differentiable Homeomorphism with an Infinite Number of Periodic Points 634 Diffeomorphisms with Many Periodic Points 636 Structurally Stable Systems Are Not Dense 654 Dynamical Systems on n-dimensional Manifolds 660 Differentiable Dynamical Systems 664 Nongenericity of O-Stability (with R. Abraham) 735 Structural Stability Theorems (with J. Palis) 739 Notes on Differential Dynamical Systems 748 The ß-Stability Theorem 759 Stability and Genericity in Dynamical Systems 768 Beyond Hyperbolicity (with M. Shub) 776 Stability and Isotopy in Discrete Dynamical Systems 781 Differential Equations 785 Dynamical Systems and Turbulence 791 Review of "Catastrophe Theory: Selected Papers, " by E. C. Zeeman 814 On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff 823 On How I Got Started in Dynamical Systems 831 Dynamics Retrospective: Great problems, attempts that failed 836 What is Chaos? 843 Finding a Horseshoe on the Beaches of Rio 859 The Work of Curtis T. McMullen 865 Part VI. Mechanics Steve Smale and Geometrie Mechanics (J. E. Marsden) 871 Topology and Mechanics, I. 889

6 Topology and Mechanics, IL 916 Problems on the Nature of Relative Equilibria in Celestial Mechanics 936 Personal Perspectives on Mathematics and Mechanics 941 Part VII. Biology, Electric Circuits, Mathematical Programming On the Mathematical Foundations of Electrical Circuit Theory 951 A Mathematical Model of Two Cells via Turing's Equation 969 Optimizing Several Functions 979 Sufficient Conditions for an Optimum 986 The Qualitative Analysis of a Difference Equation of Population Growth (with R. F. Williams) 993 On the Differential Equations of Species in Competition 997 The Problem of the Average Speed of the Simplex Method 1000 On the Average Number of Steps of the Simplex Method of Linear Programming 1010 VOLUME III Part VIII. Theory of Computation On the Work of Steve Smale on the Theory of Computation (M. Shub) 1035 The Work of Steve Smale on the Theory of Computation: (L. B lum and F. Cucker) 1056 On Algorithms for Solving/(x) = 0 (with M. Hirsch) 1076 The Fundamental Theorem of Algebra and Complexity Theory 1108 Computational Complexity: On the geometry of polynomials and a theory of cost, Part I (with M. Shub) 1144 On the Efficiency of Algorithms of Analysis 1180 Computational Complexity: On the geometry of polynomials and a theory of cost, Part II (with M. Shub) 1215 On the Existence of Generally Convergent Algorithms (with M. Shub) 1232

7 Newton's Method Estimates from Data at One Point 1242 On the Topology of Algorithms, I Algorithms for Solving Equations 1263 The Newtonian Contribution to Our Understanding of the Computer 1287 On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, recursive functions and universal machines (with L. Blum and M. Shub) 1293 Some Remarks on the Foundations of Numerical Analysis 1339 Theory of Computation 1349 Complexity of Bezout's Theorem I: Geometrie aspects (with M. Shub) 1359 Complexity of Bezout's Theorem II: Volumes and probabilities (with M. Shub) 1402 Complexity of Bezout's Theorem III: Condition number and packing (with M. Shub) 1421 Complexity of Bezout's Theorem IV: Probability of success; Extensions (with M. Shub) 1432 Complexity of Bezout's Theorem V: Polynomial time (with M. Shub) 1453 The Gödel Incompleteness Theorem and Decidability over a Ring (with L. Blum) 1477 Separation of Complexity Classes in Koiran's Weak Model (with F. Cucker and M. Shub) 1496 On the Intractability of Hilbert's Nullstellensatz and an Algebraic Version of "NP*P1" (with M. Shub) 1508 Complexity and Real Computation: A Manifesto (with L. Blum, F. Cucker and M. Shub) 1516 Algebraic Settings for the Problem "/VW?" (with L. Blum, F. Cucker and M. Shub) 1540 Complexity Theory and Numerical Analysis 1560 Some Lower Bounds for the Complexity of Continuation Methods (with J.-P. Dedieu) 1589

8 A Polynomial Time Algorithm for Diophantine Equations in One Variable (with F. Cucker and P. Koiran) 1601 Complexity Estimates Depending on Condition and Round-off Error (with F. Cucker) 1610