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Nonstandard Analysis

Theory and Applications

  • Book
  • © 1997

Overview

Editors:
  1. Leif O. Arkeryd

    1. University of Gothenburg, Sweden

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  2. Nigel J. Cutland

    1. University of Hull, England

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  3. C. Ward Henson

    1. University of Illinois, Urbana-Champaign, USA

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Part of the book series: Nato Science Series C: (ASIC, volume 493)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Foundations of Nonstandard Analysis

    • C. Ward Henson
    Pages 1-49

  3. Nonstandard Real Analysis

    • Nigel J. Cutland
    Pages 51-76

  4. Nonstandard Analysis and Topology

    • Peter A. Loeb
    Pages 77-89

  5. Loeb Measure and Probability

    • David A. Ross
    Pages 91-120

  6. An Introduction to Nonstandard Functional Analysis

    • Manfred P. H. Wolff
    Pages 121-151

  7. Applications of Nonstandard Analysis in Ordinary Differential Equations

    • E. Benoit
    Pages 153-182

  8. Better Nonstandard Universes with Applications

    • R. Jin
    Pages 183-208

  9. Internal Martingales and Stochastic Integration

    • Tom Lindstrøm
    Pages 209-258

  10. Stochastic Differential Equations with Extra Properties

    • H. Jerome Keisler
    Pages 259-277

  11. Hyperfinite Mathematical Finance

    • P. Ekkehard Kopp
    Pages 279-307

  12. Applications of NSA to Mathematical Physics

    • Leif Arkeryd
    Pages 309-339

  13. A Nonstandard Approach to Hydromechanics

    • M. CapiÅ„ski
    Pages 341-355

  14. Back Matter

    Pages 357-366
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Keywords

About this book

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.

Editors and Affiliations

  • University of Gothenburg, Sweden

    Leif O. Arkeryd

  • University of Hull, England

    Nigel J. Cutland

  • University of Illinois, Urbana-Champaign, USA

    C. Ward Henson

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