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........ published in NEWSLETTER # 63

NONSTANDARD ANALYSIS: THEORY AND APPLICATIONS
by Professor N.J. Cutland, University of Hull, Hull (U.K.)

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 important 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 recur. 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 approach 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 (NATO ASI SERIES C493) is to make the power and range of applicability of NSA more widely known and available to research mathematicians. The lecture notes provided to students at the NATO ASI Nonstandard Analysis and its Applications, Edinburgh 1996 have been reworked to provide a multi-authored text book that offers a careful introduction to NSA and a view of the current "state of the art" - both the foundations of the subject and its role in current mathematical research.

The first four articles (Henson, Cutland, Loeb, Ross) cover the fundamentals of NSA, and are designed to equip the reader for the study of more advanced theory and applications. They include basic material on nonstandard models, and the fundamental applications to real analysis, topology and measure theory - including the Loeb measure construction.

Other articles cover more advanced applications, and provide an up-to-date picture of current activity in the applications of nonstandard techniques. These include: an introduction to nonstandard functional analysis (Wolff), and applications to ODEs (Benoit) - the latter using the IST approach to nonstandard analysis. The paper by Jin surveys the additional properties one can have in a nonstandard universe, and the way such extra properties can be usefully employed. Lindstrom introduces one of the most fruitful general areas of application - stochastic analysis and martingales - and this theme is continued with Keisler's paper giving new applications and techniques for the solution of stochastic DEs. Kopp describes the way in which nonstandard techniques help to understand modern (stochastic) finance theory. The final paper give a sample of recent applications in mathematical physics - Arkeryd in the field of kinetic and the Schrodinger equation, Capinski in the area of hydromechanics (including stochastic settings).
Reference books: C493

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