Nonstandard Analysis for the Working Mathematician
Title | Nonstandard Analysis for the Working Mathematician PDF eBook |
Author | Peter A. Loeb |
Publisher | Springer |
Pages | 485 |
Release | 2015-08-26 |
Genre | Mathematics |
ISBN | 9401773270 |
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.
Non-standard Analysis
Title | Non-standard Analysis PDF eBook |
Author | Abraham Robinson |
Publisher | Princeton University Press |
Pages | 315 |
Release | 2016-08-11 |
Genre | Mathematics |
ISBN | 1400884225 |
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 Stochastic Analysis and Mathematical Physics
Title | Nonstandard Methods in Stochastic Analysis and Mathematical Physics PDF eBook |
Author | Sergio Albeverio |
Publisher | Courier Dover Publications |
Pages | 529 |
Release | 2009-02-26 |
Genre | Mathematics |
ISBN | 0486468992 |
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.
Radically Elementary Probability Theory
Title | Radically Elementary Probability Theory PDF eBook |
Author | Edward Nelson |
Publisher | Princeton University Press |
Pages | 112 |
Release | 1987 |
Genre | Mathematics |
ISBN | 9780691084749 |
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 Analysis
Title | Nonstandard Analysis PDF eBook |
Author | Leif O. Arkeryd |
Publisher | Springer Science & Business Media |
Pages | 392 |
Release | 1997-04-30 |
Genre | Mathematics |
ISBN | 9780792345862 |
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.
The Strength of Nonstandard Analysis
Title | The Strength of Nonstandard Analysis PDF eBook |
Author | Imme van den Berg |
Publisher | Springer Science & Business Media |
Pages | 415 |
Release | 2007-12-03 |
Genre | Mathematics |
ISBN | 3211499059 |
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.
Applied Nonstandard Analysis
Title | Applied Nonstandard Analysis PDF eBook |
Author | Martin Davis |
Publisher | Courier Corporation |
Pages | 210 |
Release | 2014-06-10 |
Genre | Mathematics |
ISBN | 0486152340 |
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.