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In 1961 Robinson introduced an entirely new version of the theory of infinitesimals, which he called 'Nonstandard analysis'. 'Nonstandard' here refers to the nature of new fields of numbers as defined by nonstandard models of the first-order theory of the reals. This system of numbers was closely related to the ring of Schmieden and Laugwitz, developed independently a few years earlier. During the last thirty years the use of nonstandard models in mathematics has taken its rightful place among the various methods employed by mathematicians. The contributions in this volume have been selected to present a panoramic view of the various directions in which nonstandard analysis is advancing, thus serving as a source of inspiration for future research. Papers have been grouped in sections dealing with analysis, topology and topological groups; probability theory; and mathematical physics. This volume can be used as a complementary text to courses in nonstandard analysis, and will be of interest to graduate students and researchers in both pure and applied mathematics and physics.

Nonstandard mathematical analysis --- Congresses. --- Congresses --- Nonstandard mathematical analysis - Congresses

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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.

Nonstandard mathematical analysis --- Congresses. --- Mathematical analysis [Nonstandard ] --- Congresses --- Mathematical analysis. --- Analysis (Mathematics). --- Probabilities. --- Functional analysis. --- Applied mathematics. --- Engineering mathematics. --- Fluids. --- Analysis. --- Probability Theory and Stochastic Processes. --- Functional Analysis. --- Applications of Mathematics. --- Fluid- and Aerodynamics. --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Engineering --- Engineering analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- 517.1 Mathematical analysis --- Analysis, Nonstandard mathematical --- Mathematical analysis, Nonstandard --- Non-standard analysis --- Nonstandard analysis --- Model theory --- Nonstandard mathematical analysis - Congresses