Nonstandard Methods Of Analysis

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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Mauro Di Nasso,Isaac Goldbring,Martino Lupini
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory Book PDF

Author : Mauro Di Nasso,Isaac Goldbring,Martino Lupini
Publisher : Springer
Page : 206 pages
File Size : 55,8 Mb
Release : 2019-05-23
Category : Mathematics
ISBN : 9783030179564

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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory by Mauro Di Nasso,Isaac Goldbring,Martino Lupini Pdf

The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.

Nonstandard Analysis in Practice

Francine Diener,Marc Diener
Nonstandard Analysis in Practice Book PDF

Author : Francine Diener,Marc Diener
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642577581

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Nonstandard Analysis in Practice by Francine Diener,Marc Diener Pdf

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.

Radically Elementary Probability Theory

Edward Nelson
Radically Elementary Probability Theory Book PDF

Author : Edward Nelson
Publisher : Princeton University Press
Page : 112 pages
File Size : 41,6 Mb
Release : 1987
Category : Mathematics
ISBN : 0691084742

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Radically Elementary Probability Theory by Edward Nelson Pdf

Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

Nonstandard Methods in Functional Analysis

Siu-Ah Ng
Nonstandard Methods in Functional Analysis Book PDF

Author : Siu-Ah Ng
Publisher : World Scientific
Page : 339 pages
File Size : 41,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814287555

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Nonstandard Methods in Functional Analysis by Siu-Ah Ng Pdf

In the early 1960s, by using techniques from the model theory of first-order logic, Robinson gave a rigorous formulation and extension of Leibniz'' infinitesimal calculus. Since then, the methodology has found applications in a wide spectrum of areas in mathematics, with particular success in the probability theory and functional analysis. In the latter, fruitful results were produced with Luxemburg''s invention of the nonstandard hull construction. However, there is still no publication of a coherent and self-contained treatment of functional analysis using methods from nonstandard analysis. This publication aims to fill this gap.

Non-standard Analysis

Abraham Robinson
Non standard Analysis Book PDF

Author : Abraham Robinson
Publisher : Princeton University Press
Page : 308 pages
File Size : 53,8 Mb
Release : 2016-08-11
Category : Mathematics
ISBN : 9781400884223

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Non-standard Analysis by Abraham Robinson Pdf

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 Analysis for the Working Mathematician

Peter A. Loeb,Manfred P. H. Wolff
Nonstandard Analysis for the Working Mathematician Book PDF

Author : Peter A. Loeb,Manfred P. H. Wolff
Publisher : Springer
Page : 481 pages
File Size : 43,6 Mb
Release : 2015-08-26
Category : Mathematics
ISBN : 9789401773270

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Nonstandard Analysis for the Working Mathematician by Peter A. Loeb,Manfred P. H. Wolff Pdf

Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a ‘secret weapon’ by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler’s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics.

Nonstandard Methods in Stochastic Analysis and Mathematical Physics

Sergio Albeverio,Jens Erik Fenstad,Raphael Høegh-Krohn,Tom Lindstrøm
Nonstandard Methods in Stochastic Analysis and Mathematical Physics Book PDF

Author : Sergio Albeverio,Jens Erik Fenstad,Raphael Høegh-Krohn,Tom Lindstrøm
Publisher : Courier Dover Publications
Page : 529 pages
File Size : 41,5 Mb
Release : 2009-02-26
Category : Mathematics
ISBN : 9780486468990

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Nonstandard Methods in Stochastic Analysis and Mathematical Physics by Sergio Albeverio,Jens Erik Fenstad,Raphael Høegh-Krohn,Tom Lindstrøm Pdf

Two-part treatment begins with a self-contained introduction to the subject, followed by applications to stochastic analysis and mathematical physics. "A welcome addition." — Bulletin of the American Mathematical Society. 1986 edition.

Nonstandard Methods in Fixed Point Theory

Asuman G. Aksoy,Mohamed A. Khamsi
Nonstandard Methods in Fixed Point Theory Book PDF

Author : Asuman G. Aksoy,Mohamed A. Khamsi
Publisher : Springer Science & Business Media
Page : 149 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461234449

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Nonstandard Methods in Fixed Point Theory by Asuman G. Aksoy,Mohamed A. Khamsi Pdf

A unified account of the major new developments inspired by Maurey's application of Banach space ultraproducts to the fixed point theory for non-expansive mappings is given in this text. The first third of the book is devoted to laying a careful foundation for the actual fixed point theoretic results which follow. Set theoretic and Banach space ultraproducts constructions are studied in detail in the second part of the book, while the remainder of the book gives an introduction to the classical fixed point theory in addition to a discussion of normal structure. This is the first book which studies classical fixed point theory for non-expansive maps in the view of non-standard methods.

Nonstandard Analysis

Martin Väth
Nonstandard Analysis Book PDF

Author : Martin Väth
Publisher : Springer Science & Business Media
Page : 255 pages
File Size : 40,7 Mb
Release : 2007
Category : Mathematics
ISBN : 9783764377731

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Nonstandard Analysis by Martin Väth Pdf

This book introduces Robinson's nonstandard analysis, an application of model theory in analysis. Unlike some texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but points to some applications in more advanced analysis. The contents proceed from a discussion of the preliminaries to Nonstandard Models; Nonstandard Real Analysis; Enlargements and Saturated Models; Functionals, Generalized Limits, and Additive Measures; and finally Nonstandard Topology and Functional Analysis. No background in model theory is required, although some familiarity with analysis, topology, or functional analysis is useful. This self-contained book can be understood after a basic calculus course.

Nonstandard Methods of Analysis

A.G. Kusraev,Semën Samsonovich Kutateladze
Nonstandard Methods of Analysis Book PDF

Author : A.G. Kusraev,Semën Samsonovich Kutateladze
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 43,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401111362

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Nonstandard Methods of Analysis by A.G. Kusraev,Semën Samsonovich Kutateladze Pdf

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.

Nonstandard Analysis

Alain Robert
Nonstandard Analysis Book PDF

Author : Alain Robert
Publisher : Courier Corporation
Page : 184 pages
File Size : 43,5 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 0486432793

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Nonstandard Analysis by Alain Robert Pdf

This concise text is based on the axiomatic internal set theory approach. Theoretical topics include idealization, standardization, and transfer, real numbers and numerical functions, continuity, differentiability, and integration. Applications cover invariant means, approximation of functions, differential equations, more. Exercises, hints, and solutions. "Mathematics teaching at its best." — European Journal of Physics. 1988 edition.

Nonstandard Analysis

Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson
Nonstandard Analysis Book PDF

Author : Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson
Publisher : Springer Science & Business Media
Page : 392 pages
File Size : 47,5 Mb
Release : 1997-04-30
Category : Mathematics
ISBN : 079234586X

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Nonstandard Analysis by Leif O. Arkeryd,Nigel J. Cutland,C. Ward Henson Pdf

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.

Lectures on the Hyperreals

Robert Goldblatt
Lectures on the Hyperreals Book PDF

Author : Robert Goldblatt
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461206156

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Lectures on the Hyperreals by Robert Goldblatt Pdf

An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.

Nonstandard Methods for Stochastic Fluid Mechanics

Marek Capi?ski,Nigel Cutland
Nonstandard Methods for Stochastic Fluid Mechanics Book PDF

Author : Marek Capi?ski,Nigel Cutland
Publisher : World Scientific
Page : 256 pages
File Size : 49,8 Mb
Release : 1995
Category : Science
ISBN : 9810217102

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Nonstandard Methods for Stochastic Fluid Mechanics by Marek Capi?ski,Nigel Cutland Pdf

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 Asymptotic Analysis

Imme van den Berg
Nonstandard Asymptotic Analysis Book PDF

Author : Imme van den Berg
Publisher : Springer
Page : 192 pages
File Size : 50,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540478102

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Nonstandard Asymptotic Analysis by Imme van den Berg Pdf

This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes topics of general interest, like the remarkable behaviour of Taylor polynomials of elementary functions. Noting that within nonstandard analysis, "small", "large", and "domain of validity of asymptotic behaviour" have a precise meaning, a nonstandard alternative to classical asymptotics is developed. Special emphasis is given to applications in numerical approximation by convergent and divergent expansions: in the latter case a clear asymptotic answer is given to the problem of optimal approximation, which is valid for a large class of functions including many special functions. The author's approach is didactical. The book opens with a large introductory chapter which can be read without much knowledge of nonstandard analysis. Here the main features of the theory are presented via concrete examples, with many numerical and graphic illustrations. N