Nonstandard Analysis for the Working Mathematician (Mathematics and Its Applications)

Editore: Springer, 2000
ISBN 10: 079236340X / ISBN 13: 9780792363408
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Riassunto: This book is addressed to mathematicians working in analysis and its applications. The aim is to provide an understandable introduction to the basic theory of non-standard analysis and to illuminate some of its most striking applications. Problems are posed in all chapters.
The opening chapter of the book presents a simplified form of the general theory that is suitable for the results of calculus and basic real analysis. The presentation is intended to facilitate the acquisition of basic skills in the subject, so that a reader who begins with no background in mathematical logic should find it relatively easy to continue. The book then proceeds with the full theory.
Following Part I, each chapter takes up a different field for applications, beginning with a gentle introduction that even non-experts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use non-standard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insights into the field under consideration. The particular applications discussed here are in functional analysis including operator theory, probability theory including stochastic processes, and economics including game theory and financial mathematics. In working through this book the reader should gain many new and helpful insights into the enterprise of mathematics.
Audience: This work will be of interest to specialists whose work involves real functions, probability theory, stochastic processes, logic and foundations. Much of the book, in particular the introductory Part I, can be used in a graduate course.

Sinossi: This book is addressed to mathematicians working in analysis and its applications. The aim is to provide an understandable introduction to the basic theory of nonstan­ dard analysis in Part I, and then to illuminate some of its most striking applications. Much of the book, in particular Part I, can be used in a graduate course; problems are posed in all chapters. After Part I, each chapter takes up a different field for the application of nonstandard analysis, beginning with a gentle introduction that even non-experts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use nonstandard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insight into the field under consideration. The applications discussed here are in functional analysis including operator theory, probability theory including stochastic processes, and economics including game theory and financial mathematics. In all of these areas, the intuitive notion of an infinitely small or infinitely large quantity plays an essential and helpful role in the creative process. For example, Brownian motion is often thought of as a random walk with infinitesimal increments; the spectrum of a selfadjoint operator is viewed as the set of "almost eigenvalues"; an ideal economy consists of an infinite number of agents each having an infinitesimal influence on the economy. Already at the level of calculus, one often views the integral as an infinitely large sum of infinitesimal quantities.

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Titolo: Nonstandard Analysis for the Working ...
Casa editrice: Springer
Data di pubblicazione: 2000
Condizione libro: very good

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Loeb, Peter A.; Wolff, Manfred P.H.
Editore: Springer, U.S.A. (2000)
ISBN 10: 0792363418 ISBN 13: 9780792363415
Usato Soft cover Quantità: 1
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Descrizione libro Springer, U.S.A., 2000. Soft cover. Condizione libro: Fine. [MaBx-A]. Softcover, bound in wraps. Unmarked. Codice libro della libreria 066987

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