Joel N. Franklin

Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems

Classics in Applied Mathematics 37

Many advances have taken place in the field of combinatorial algorithms since Methods of Mathematical Economics first appeared two decades ago. Despite these advances and the development of new computing methods, several basic theories and methods remain important today for understanding mathematical programming and fixed-point theorems. In this easy-to-read classic, readers learn Wolfe's method, which remains useful for quadratic programming, and the Kuhn-Tucker theory, which underlies quadratic programming and most other nonlinear programming methods. In addition, the author presents multiobjective linear programming, which is being applied in environmental engineering and the social sciences.

The book presents many useful applications to other branches of mathematics and to economics, and it contains many exercises and examples. The advanced mathematical results are proved clearly and completely. By providing the necessary proofs and presenting the material in a conversational style, Franklin made Methods of Mathematical Economics extremely popular among students. The addition of a list of errata, new to this edition, should add to the book's popularity as well as its usefulness both in the classroom and for individual study.

The book has three chapters: "Linear Programming," "Nonlinear Programming," and "Fixed-Point Theorems." The first and third chapters include the economic equilibrium theorems of von Neumann and of J. F. Nash, while the second chapter includes Kuhn-Tucker theory and Wolfe's simplex algorithm for quadratic programming. The book concludes with easy, elementary proofs of the famous theorems of Brouwer, of Kakutani, and of Schauder. These fundamental results are usually proved only in advanced texts in topology, economic theory, and nonlinear analysis.

Audience

This book is intended for undergraduate and graduate students of mathematics and economics; it requires no background in these areas except an understanding of elementary calculus and linear algebra.

Contents

Preface to the Classics Edition; Preface; Errata; Chapter 1: Linear Programming. Introduction to Linear Programming; Linear Programs and Their Duals; How the Dual Indicates Optimality; Basic Solutions; The Idea of the Simplex Methods; Separating Planes for Convex Sets; Finite Cones and the Farkas Alternative; The Duality Principle; Perturbations and Parametric Programming; The Simplex Tableau Algorithm; The Revised Simplex Algorithm; A Simplex Algorithm for Degenerate Problems; Multiobjective Linear Programming; Zero-Sum, Two-Person Games; Integer Programming: Gomory's Method; Network Flows; Assignment and Shortest-Route Problems; The Transportation Problem; Chapter 2: Nonlinear Programming. Wolfe's Method for Quadratic Programming; Kuhn-Tucker Theory; Geometric Programming; Chapter 3: Fixed-Point Theorems. Introduction to Fixed Points; Contraction Mappings; Garsia's Proof of the Brouwer Fixed-Point Theorem; Milnor's Proof of the Brouwer Fixed-Point Theorem; Barycentric Coordinates, Sperner's Lemma, and an Elementary Proof of the Brouwer Fixed-Point Theorem; The Schauder Fixed-Point Theorem; Kakutani's Fixed-Point Theorem and Nash's Theorem for n-Person Games; Index.

2002 / xviii + 297 pages / Softcover / ISBN 0-89871-509-1

Philip Hartman

Ordinary Differential Equations: Second Edition

Classics in Applied Mathematics 38

Ordinary Differential Equations covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition to these results, the text illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Unlike many texts, which supply only the standard simplified theorems, Ordinary Differential Equations presents the basic theory of ODEs in a general way, making it a valuable reference.

This SIAM reissue of the 1982 second edition covers invariant manifolds, perturbations, and dichotomies, making the text relevant to current studies of geometrical theory of differential equations and dynamical systems. In particular, Ordinary Differential Equations includes the proof of the Hartman-Grobman theorem on the equivalence of a nonlinear to a linear flow in the neighborhood of a hyperbolic stationary point, as well as theorems on smooth equivalences, the smoothness of invariant manifolds, and the reduction of problems on ODEs to those on "maps" (Poincar・.

Ordinary Differential Equations is based on the author's lecture notes from courses on ODEs taught to advanced undergraduate and graduate students in mathematics, physics, and engineering. The book, which remains as useful today as when it was first published, includes an excellent selection of exercises varying in difficulty from routine examples to more challenging problems. These exercises show extensions of the techniques in question and serve to introduce the reader to the literature in this area. New to this SIAM Classics edition is an Errata section listing corrections to minor errors in the 1982 edition. An extensive bibliography (up to 1980) is provided.

Audience

Readers should have knowledge of matrix theory and the ability to deal with functions of real variables.

Contents

Foreword to the Classics Edition; Preface to the First Edition; Preface to the Second Edition; Errata; I: Preliminaries; II: Existence; III: Differential Inqualities and Uniqueness; IV: Linear Differential Equations; V: Dependence on Initial Conditions and Parameters; VI: Total and Partial Differential Equations; VII: The Poincar・Bendixson Theory; VIII: Plane Stationary Points; IX: Invariant Manifolds and Linearizations; X: Perturbed Linear Systems; XI: Linear Second Order Equations; XII: Use of Implicity Function and Fixed Point Theorems; XIII: Dichotomies for Solutions of Linear Equations; XIV: Miscellany on Monotomy; Hints for Exercises; References; Index

2002 / xx + 612 pages / Softcover / ISBN 0-89871-510-5

Michael D. Intriligator

Mathematical Optimization and Economic Theory

Classics in Applied Mathematics 39

Mathematical Optimization and Economic Theory provides a self-contained introduction to and survey of mathematical programming and control techniques and their applications to static and dynamic problems in economics, respectively. It is distinctive in showing the unity of the various approaches to solving problems of constrained optimization that all stem back directly or indirectly to the method of Lagrange multipliers. In the 30 years since its initial publication, there have been many more applications of these mathematical techniques in economics, as well as some advances in the mathematics of programming and control. Nevertheless, the basic techniques remain the same today as when the book was originally published. Thus, it continues to be useful not only to its original audience of advanced undergraduate and graduate students in economics, but also to mathematicians and other researchers who are interested in learning about the applications of the mathematics of optimization to economics.

The book is distinctive in that it covers in some depth both static programming problems and dynamic control problems of optimization and the techniques of their solution. It also clearly presents many applications of these techniques to economics, and it shows why optimization is important for economics. Many cchallenging problems for both students and researchers are included.

Audience

This book should be of interest to various readers in the fields of economics and mathematics as both a textbook and a general reference book. Mathematicians and other researchers who are interested in learning about the applications of mathematical optimization in economics, as well as students at the advanced undergraduate and beginning graduate level, will benefit from the information that this book offers. The mathematical level is kept as elementary as possible. A basic knowledge of analysis and matrix algebra is recommended. Two appendices summarize the necessary mathematics.

Contents

Preface to the Classics Edition; Preface; Part One: Introduction. Economizing and the Economy; Part Two: Static Optimization. The Mathematical Programming Problem; Classical Programming; Nonlinear Programming; Linear Programming; Game Theory; Part Three: Applications of Static Optimization. Theory of the Household; Theory of the Firm; General Equilibrium; Welfare Economics; Part Four: Dynamic Optimization. The Control Problem; Calculus of Variations; Dynamic Programming; Maximum Principle; Differential Games; Part Five: Applications of Dynamic Optimization. Optimal Economic Growth; Appendix A: Analysis; Appendix B: Matrices; Index.

2002 / xx + 508 pages / Softcover / ISBN 0-89871-511-3

Philippe G. Ciarlet

The Finite Element Method for Elliptic Problems

Classics in Applied Mathematics 40

The Finite Element Method for Elliptic Problems is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty.

Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters.

Audience

This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.

Contents
Preface to the Classics Edition; Preface; General Plan and Interdependence Table; Chapter 1: Elliptic Boundary Value Problems; Chapter 2: Introduction to the Finite Element Method; Chapter 3: Conforming Finite Element Methods for Second Order Problems; Chapter 4: Other Finite Element Methods for Second-Order Problems; Chapter 5: Application of the Finite Element Method to Some Nonlinear Problems; Chapter 6: Finite Element Methods for the Plate Problem; Chapter 7: A Mixed Finite Element Method; Chapter 8: Finite Element Methods for Shells; Epilogue: Some "real-life" finite element model examples; Bibliography; Glossary of Symbols; Index.

April 2002 / xxviii + 530 pages / Softcover / ISBN 0-89871-514-8