Synopses & Reviews
This book presents a careful and detailed introduction to the methodology of nonstandard analysis and the foundations of its use in analysis, topology, probability theory and stochastic analysis. Further articles expound recent, more advanced applications in functional analysis, stochastic differential equations, mathematical physics and mathematical finance theory. All authors are world leaders in the subject. Audience: All mathematicians at postgraduate level and beyond who wish to learn the basics of nonstandard analysis and its role in current mathematical research.
Synopsis
1 More than thirty years after its discovery by Abraham Robinson, the ideas and techniques of Nonstandard Analysis (NSA) are being applied across the whole mathematical spectrum, as well as constituting an im portant field of research in their own right. The current methods of NSA now greatly extend Robinson's original work with infinitesimals. However, while the range of applications is broad, certain fundamental themes re cur. The nonstandard framework allows many informal ideas (that could loosely be described as idealisation) to be made precise and tractable. For example, the real line can (in this framework) be treated simultaneously as both a continuum and a discrete set of points; and a similar dual ap proach can be used to link the notions infinite and finite, rough and smooth. This has provided some powerful tools for the research mathematician - for example Loeb measure spaces in stochastic analysis and its applications, and nonstandard hulls in Banach spaces. The achievements of NSA can be summarised under the headings (i) explanation - giving fresh insight or new approaches to established theories; (ii) discovery - leading to new results in many fields; (iii) invention - providing new, rich structures that are useful in modelling and representation, as well as being of interest in their own right. The aim of the present volume is to make the power and range of appli cability of NSA more widely known and available to research mathemati cians."
Table of Contents
Preface. Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions; C.W. Henson. Nonstandard Real Analysis; N.J. Cutland. Nonstandard Analysis and Topology; P.A. Loeb. Loeb Measure and Probability; D.A. Ross. An Introduction to Nonstandard Functional Analysis; M.P.H. Wolff. Applications of Nonstandard Analysis in Ordinary Differential Equations; E. Benoit. Better Nonstandard Universes with Applications; R. Jin. Internal Martingales and Stochastic Integration; T. Lindstrøm. Stochastic Differential Equations with Extra Properties; H.J. Keisler. Hyperfinite Mathematical Finance; P.E. Kopp. Applications of NSA To Mathematical Physics; L. Arkeryd. A Nonstandard Approach to Hydromechanics: Navier-Stokes Equations; M. Capinski. Index.