Nichtstandard Analysis

Author: Dieter Landers,Lothar Rogge
Publisher: Springer-Verlag
ISBN: 3642579159
Category: Mathematics
Page: 488
View: 7484

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Die Nichtstandard-Mathematik hat in den letzten Jahren einen gewaltigen Aufschwung erfahren und die Entwicklungen in den verschiedenartigsten Gebieten beeinflußt und befruchtet. Mit diesem Lehrbuch liegt nun die erste umfassende und leicht verständliche Einführung in dieses Thema in deutscher Sprache vor. An Vorkenntnissen braucht der Leser für ein gewinnbringendes Selbststudium nichts weiter als Grundkenntnisse in Linearer Algebra und Analysis, d.h. Kenntnisse des ersten Studienjahres. Ausführliche Beweise, viele Aufgaben mit Lösungen und eine gelungene didaktische Aufbereitung des Stoffes machen Methoden und Erkenntnisse durchsichtig und verständlich. Trotz der einfachen Lesbarkeit dieses Buches wird an mehreren Stellen bis zu neuesten Forschungsergebnissen vorgestoßen und viele Ergebnisse werden zum ersten Mal in Buchform vorgestellt. Mit diesem Lehrbuch wird der Leser in die Lage versetzt, schnell Nichtstandard-Methoden in den verschiedensten Bereichen selbständig anzuwenden. Es kann außerdem als Basis für ein- oder mehrsemestrige Vorlesungen verwendet werden. Aus dem Vorwort der Autoren: "Wir hoffen, daß unsere Leser beim Studium dieses Buches den Enthusiasmus der Autoren für die Schönheit, Eleganz und Wirksamkeit der Nichtstandard-Methoden teilen werden."

Nonstandard Analysis, Axiomatically

Author: Vladimir Kanovei,Michael Reeken
Publisher: Springer Science & Business Media
ISBN: 366208998X
Category: Mathematics
Page: 410
View: 6926

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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.

Nonstandard Analysis and Its Applications

Author: Nigel Cutland
Publisher: Cambridge University Press
ISBN: 052135109X
Category: Mathematics
Page: 346
View: 4057

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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.

Methods and Applications of Singular Perturbations

Boundary Layers and Multiple Timescale Dynamics
Author: Ferdinand Verhulst
Publisher: Springer Science & Business Media
ISBN: 0387229663
Category: Mathematics
Page: 328
View: 9415

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Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach

Nonstandard Asymptotic Analysis

Author: Imme van den Berg
Publisher: Springer
ISBN: 3540478108
Category: Mathematics
Page: 192
View: 5441

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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

Asymptotic Techniques for Use in Statistics

Author: O. E. Barndorff-Nielsen,D. R. Cox
Publisher: Chapman and Hall/CRC
Category: Mathematics
Page: 252
View: 2831

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The use in statistical theory of approximate arguments based on such methods as local linearization (the delta method) and approxi mate normality has a long history. Such ideas play at least three roles. First they may give simple approximate answers to distributional problems where an exact solution is known in principle but difficult to implement. The second role is to yield higher-order expansions from which the accuracy of simple approximations may be assessed and where necessary improved. Thirdly the systematic development of a theoretical approach to statistical inference that will apply to quite general families of statistical models demands an asymptotic formulation, as far as possible one that will recover 'exact' results where these are available. The approximate arguments are developed by supposing that some defining quantity, often a sample size but more generally an amount of information, becomes large: it must be stressed that this is a technical device for generating approximations whose adequacy always needs assessing, rather than a 'physical' limiting notion. Of the three roles outlined above, the first two are quite close to the traditional roles of asymptotic expansions in applied mathematics and much ofthe very extensive literature on the asymptotic expansion of integrals and of the special functions of mathematical physics is quite directly relevant, although the recasting of these methods into a probability mould is quite often enlightening.

Weak Convergence of Measures

Applications in Probability
Author: Patrick Billingsley
Publisher: SIAM
ISBN: 0898711762
Category: Mathematics
Page: 31
View: 5760

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A treatment of the convergence of probability measures from the foundations to applications in limit theory for dependent random variables. Mapping theorems are proved via Skorokhod's representation theorem; Prokhorov's theorem is proved by construction of a content. The limit theorems at the conclusion are proved under a new set of conditions that apply fairly broadly, but at the same time make possible relatively simple proofs.

Featured Reviews in Mathematical Reviews 1997-1999

With Selected Reviews of Classic Books and Papers from 1940-1969
Author: Donald G. Babbitt,Jane E. Kister
Publisher: American Mathematical Soc.
ISBN: 9780821896709
Category: Mathematics
Page: 541
View: 8288

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This second volume of Featured Reviews makes available special detailed reviews of some of the most important mathematical articles and books published from 1997 through 1999. Also included are excellent reviews of several classic books and articles published prior to 1970. Among those reviews, for example, are the following: Homological Algebra by Henri Cartan and Samuel Eilenberg, reviewed by G. Hochschild; Faisceaux algebriques coherents by Jean-Pierre Serre, reviewed by C. Chevalley; and On the Theory of General Partial Differential Operators by Lars Hormander, reviewed by J. L. Lions. In particular, those seeking information on current developments outside their own area of expertise will find the volume very useful. By identifying some of the best publications, papers, and books that have had or are expected to have a significant impact in applied and pure mathematics, this volume will serve as a comprehensive guide to important new research across all fields covered by MR.