Applied Nonstandard Analysis

Author: Martin Davis

Publisher: Courier Corporation

ISBN: 0486152340

Category: Mathematics

Page: 208

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This applications-oriented text assumes no knowledge of mathematical logic in its development of nonstandard analysis techniques and their applications to elementary real analysis and topological and Hilbert space. 1977 edition.

Nonstandard Analysis

Theory and Applications

Author: Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson

Publisher: Springer Science & Business Media

ISBN: 9401155445

Category: Mathematics

Page: 366

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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 Analysis

Author: Martin Väth

Publisher: Springer Science & Business Media

ISBN: 3764377739

Category: Mathematics

Page: 252

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Presents an introduction into Robinson's nonstandard analysis - an application of model theory in analysis. It contains a discussion of the preliminaries to Nonstandard Models; Nonstandard Real Analysis; Enlargements and Saturated Models; Functionals, Generalized Limits and Additive Measures; and Nonstandard Topology and Functional Analysis.

Nonstandard Analysis and Its Applications

Author: Nigel Cutland

Publisher: Cambridge University Press

ISBN: 052135109X

Category: Mathematics

Page: 346

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This textbook is an introduction to non-standard analysis and to its many applications. Non standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology. The book arises from a conference held in July 1986 at the University of Hull which was designed to provide both an introduction to the subject through introductory lectures, and surveys of the state of research. The first part of the book is devoted to the introductory lectures and the second part consists of presentations of applications of NSA to dynamical systems, topology, automata and orderings on words, the non- linear Boltzmann equation and integration on non-standard hulls of vector lattices. One of the book's attractions is that a standard notation is used throughout so the underlying theory is easily applied in a number of different settings. Consequently this book will be ideal for graduate students and research mathematicians coming to the subject for the first time and it will provide an attractive and stimulating account of the subject.

Nonstandard analysis

Author: Alain Robert

Publisher: John Wiley & Sons Inc

ISBN: 9780471917038

Category: Business & Economics

Page: 156

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Infinitesimals have been hotly debated since the invention of calculus 300 years ago. This accessible treatment of nonstandard analysis (NSA) presents an elementary, yet rigorous account of the theory of infinitesimals and derives some known mathematical results in a nonclassical way. Provides recent solutions to mathematical questions which remained unsolved until the advent of NSA. Treatment is self-contained and includes exercises with detailed solutions. ``This wonderful little simple and brilliant, deep and witty, short and far-ranging...''(--J.-M. Ley-Leblond, European Journal of Physics)

Non-standard Analysis

Author: Abraham Robinson

Publisher: Princeton University Press

ISBN: 9780691044903

Category: Mathematics

Page: 293

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Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body.

Nonstandard Methods in the Calculus of Variations

Author: Curtis Tuckey

Publisher: CRC Press

ISBN: 9780582231801

Category: Mathematics

Page: 112

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This monograph is unique in its treatment of the application of methods of nonstandard analysis to the theory of curves in the calculus of variations. It will be of particular value to researchers in the calculus of variations and optimal control theory.

A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia

Author: Jacques Fleuriot

Publisher: Springer Science & Business Media

ISBN: 9781852334666

Category: Mathematics

Page: 140

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Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.

Nonstandard Analysis in Practice

Author: Francine Diener,Marc Diener

Publisher: Springer Science & Business Media

ISBN: 9783540602972

Category: Mathematics

Page: 250

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This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals.

Nonstandard Analysis, Axiomatically

Author: Vladimir Kanovei,Michael Reeken

Publisher: Springer Science & Business Media

ISBN: 366208998X

Category: Mathematics

Page: 410

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In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.

The Strength of Nonstandard Analysis

Author: Imme van den Berg,Vitor Neves

Publisher: Springer Science & Business Media

ISBN: 3211499059

Category: Mathematics

Page: 401

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This book reflects the progress made in the forty years since the appearance of Abraham Robinson’s revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

An Introduction to Nonstandard Real Analysis

Author: Albert E. Hurd,Peter A. Loeb

Publisher: Academic Press

ISBN: 9780080874371

Category: Mathematics

Page: 232

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The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstandard analysis (12,37,42,46) is the first chapter, which eases the reader into the subject with an elementary model suitable for the calculus, and the fourth chapter on measure theory in nonstandard models.

Principles of Infinitesimal Stochastic and Financial Analysis

Author: Imme van den Berg

Publisher: World Scientific

ISBN: 9789810243586

Category: Mathematics

Page: 136

View: 3301

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There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.

Introduction to Asymptotics

A Treatment Using Nonstandard Analysis

Author: D S Jones

Publisher: World Scientific

ISBN: 9814497967


Page: 176

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Many branches of science and engineering involve applications of mathematical analysis. An important part of applied analysis is asymptotic approximation which is, therefore, an active area of research with new methods and publications being found constantly. This book gives an introduction to the subject sufficient for scientists and engineers to grasp the fundamental techniques, both those which have been known for some time and those which have been discovered more recently. The asymptotic approximation of both integrals and differential equations is discussed and the discussion includes hyperasymptotics as well as uniform asymptotics. There are many numerical examples to illustrate the relation between theory and practice. Exercises in the chapters enable the book to be used as a text for an introductory course. Contents:Basic TheoryIntegralsSeriesUniform AsymptoticsHyperasymptoticsDifferential EquationsIntroduction to Nonstandard AnalysisReferencesIndex Readership: General. keywords:Asymptotics;Bessel Functions;Differential Equations;Fourier Integrals;Hyperasymptotics;Nonstandared Analysis;Series;Stationary Phase;Stieltjes Transforms;Stokes' Phenomenon “A very attractive feature of the book is the numerous examples illustrating the methods. A fine collection of exercises enriches each chapter, challenging the reader to check his progress in understanding the methods.” Mathematical Reviews “As an introductory book to asymptotics, with chapters on uniform asymptotics and exponential asymptotics, this book clearly fills a gap … it has a friendly size and contains many convincing numerical examples and interesting exercises. Hence, I recommend the book to everyone who works in asymptotics.” SIAM “… it is an excellent book that contains interesting results and methods for the researchers. It will be useful for the students interested in analysis and lectures on asymptotic methods … The reviewer recommends the book to everyone who is interested in analysis, engineers and specialists in ODE-s” Acta Sci. Math. (Szeged)

Nonstandard Methods for Stochastic Fluid Mechanics

Author: Marek Capi?ski,Nigel Cutland

Publisher: World Scientific

ISBN: 9789810217105

Category: Science

Page: 227

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This book is an exposition of a new approach to the Navier-Stokes equations, using powerful techniques provided by nonstandard analysis, as developed by the authors. The topics studied include the existence and uniqueness of weak solutions, statistical solutions and the solution of general stochastic equations.The authors provide a self-contained introduction to nonstandard analysis, designed with applied mathematicians in mind and concentrated specifically on techniques applicable to the Navier-Stokes equations. The subsequent exposition shows how these new techniques allow a quick and intuitive entrance into the mathematical theory of hydrodynamics, as well as provide a research tool that has proven useful in solving open problems concerning stochastic equations.

Nonstandard Methods of Analysis

Author: A.G. Kusraev,Semën Samsonovich Kutateladze

Publisher: Springer Science & Business Media

ISBN: 9401111367

Category: Mathematics

Page: 439

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Nonstandard Methods of Analysis is concerned with the main trends in this field; infinitesimal analysis and Boolean-valued analysis. The methods that have been developed in the last twenty-five years are explained in detail, and are collected in book form for the first time. Special attention is paid to general principles and fundamentals of formalisms for infinitesimals as well as to the technique of descents and ascents in a Boolean-valued universe. The book also includes various novel applications of nonstandard methods to ordered algebraic systems, vector lattices, subdifferentials, convex programming etc. that have been developed in recent years. For graduate students, postgraduates and all researchers interested in applying nonstandard methods in their work.