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

ISBN-13: 9781461394495 / Angielski / Miękka / 2011 / 300 str.

Joel Franklin

Opis

In 1924 the firm of Julius Springer published the first volume of Methods of Mathematical Physics by Richard Courant and David Hilbert. In the preface, Courant says this: Since the sevc nteenth century, physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinement and emphasized the postulational side of mathematics, anel at times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes ofmathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific deve10pment may split into sm aller and smaUer rivulets and dry out. It seems therefore important to direct our efforts toward reuniting divergent trends by ciarifying the common features and interconnections of many distinct and diverse scientific facts. Only thus can the student attain some mastery of the material and the basis be prepared for further organic development of research. The present work is designed to serve this purpose for the field of mathe- matical physics . . . . Completeness is not attempted, but it is hoped that access to a rich and important field will be facilitated by the book. When I was a student, the book of Courant and Hilbert was my bible.

Spis treści

I Linear Programming.- 1 Introduction to Linear Programming.- 2 Linear Programs and Their Duals.- 3 How the Dual Indicates Optimality.- 4 Basic Solutions.- 5 The Idea of the Simplex Method.- 6 Separating Planes for Convex Sets.- 7 Finite Cones and the Farkas Alternative.- 8 The Duality Principle.- 9 Perturbations and Parametric Programming.- 10 The Simplex Tableau Algorithm.- 11 The Revised Simplex Algorithm.- 12 A Simplex Algorithm for Degenerate Problems.- 13 Multiobjective Linear Programming.- 14 Zero-Sum, Two-Person Games.- 15 Integer Programming: Gomory’s Method.- 16 Network Flows.- 17 Assignment and Shortest-Route Problems.- 18 The Transportation Problem.- II Nonlinear Programming.- 1 Wolfe’s Method for Quadratic Programming.- 2 Kuhn-Tucker Theory.- 3 Geometric Programming.- III Fixed-Point Theorems Contents.- 1 Introduction to Fixed Points; Contraction Mappings.- 2 Garsia’s Proof of the Brouwer Fixed-Point Theorem.- 3 Milnor’s Proof of the Brouwer Fixed-Point Theorem.- 4 Barycentric Coordinates, Sperner’s Lemma, and an Elementary Proof of the Brouwer Fixed-Point Theorem.- 5 The Schauder Fixed-Point Theorem.- 6 Kakutani’s Fixed-Point Theorem and Nash’s Theorem for n-Person Games.

O autorze

Joel Franklin is an Assistant Professor in the Physics department of Reed College. His work spans a variety of fields, including stochastic Hamiltonian systems (both numerical and mathematical), theoretical extensions of general relativity, and their observational implications.