UID:
almafu_9959238161502883
Format:
1 online resource (xiii, 346 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08688-7
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1-107-09414-3
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1-107-10008-9
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1-107-08793-7
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1-139-17211-5
Series Statement:
London Mathematical Society student texts ;
Content:
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.
Note:
Papers presented at a conference held at the University of Hull in 1986.
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Cover; Series Page; Title; Copyright; CONTENTS; PREFACE; CONTRIBUTORS; AN INVITATION TO NONSTANDARD ANALYSIS; INTRODUCTION; I. A SET OF HYPERREALS; I.1 CONSTRUCTION OF *R; I.1.1 Example; I.1.2 Definition; I.1.3 Definition; I.1.4 Example; I.1.5 Definition; I.1.6 Proposition; I.1.7 Definition; I.1.8 Lemma; I.2 INTERNAL SETS AND FUNCTIONS; I.2.1 Definition; I. 2.2 Example; I.2.3 Proposition; I.2.4 Corollary; I.2.5 Theorem (x1-saturation); I.2.6 Corollary; I.2.7 Proposition; I.2.8 Definition; I.2.9 Proposition; I.2.10 Definition; I.2.11 Exaaple; I.2.12 Proposition; I.3 INFINITESIMAL CALCULUS
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I.3.1 PropositionI.3.2 Proposition; I.3.3 Proposition; I.3.4 Corollary; I.3.5 Proposition; I.3.6 Corollary; I.3.7 Theorem; II. SUPERSTRUCTURES AND LOEB MEASURES; II.1 SUPERSTRUCTURES; II.1.1 Definition; II.1.2 Definition; II.1.3 LeMMA; II.1.4 Proposition; II.2 LOEB MEASURES; II. 2.1 Exaaple; II. 2.2 Definition; II .2.3 Lemma; II. 2.4 Lemma; II .2.5 Theorem; II. 2.6 Exaaple; II. 2.7 Example; II. 2.8 Lemma; II. 2.10 Theorem; II.2.11 Theorem; II.2.12 Corollary; II.3 BROWNIAN MOTION; II.3.1 Definition; II.3.2 Lemma; II.3.3 Lemma; II.3.4 Lemma; II.3.4 Lemma; II.3.6 Theorem
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III. SATURATION AND TOPOLOGYIII.1 BEYOND x1-SATURATION; III.1.1 Definition; III.1.2 Theorem; III.1.3 TheoreM; III.1.4 Lemma; III.2 GENERAL TOPOLOGY; III.2.1 Proposition; III.2.2 Proposition; III.2.3 Proposition; III.2.4 Proposition; III.2.5 Example; III.2.6 Proposition; III.2.7 Tychonov's Theorem; III.2.8 Alaoglu's Theorea; III.2.9 Ascoli's Theorea; III.2.10 Example; III.3 COMPLETIONS, COMPACTIFICATIONS. AND NONSTANDARD HULLS; III.3.1 Proposition; III.3.2 Corollary; III.3.3 Proposition; III.3.4 Example; III.3.5 Example; III.3.6 Proposition; III.3.7 Corollary; III.3.8 Example
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III.3.9 PropositionIV. THE TRANSFER PRINCIPLE; IV.1 THE LANGUAGES L(V(S) AND L*(V(S)); IV.1.1 Definition; IV.I .2 Example; IV.2 LOS' THEOREM AND THE TRANSFER PRINCIPLE; IV.2.1 Definition; IV.2.2. Lemma; IV.2.3 Los' Theorem; IV.2.4 Transfer Principle; IV.2.5 Internal Definition Principle; IV.3 AXIOMATIC NONSTANDARD ANALYSIS; APPENDIX. ULTRAFILTERS; A.1 Proposition; A.2 Lemma; A.3 Lemma; A.4 Theorem; NOTES; REFERENCES; INFINITESIMALS IN PROBABILITY THEORY; 1. THE HYPERFINITE TIME LINE; Definition; 1.2 Proposition; 1.3 Corollary; 1.4 Theorem (Anderson (1982))
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2. UNIVERSAL AND HOMOGENEOUS PROBABILITY SPACES2.1 Proposition; 2.2 Proposition; Definition; Definition; 2.3 Theorem (Keisler (1984)); 3. STOCHASTIC PROCESSES; 3.1 Lemma; 3.2 Proposition; 3.3 Proposition; 4. PRODUCTS OF LOEB SPACES; 4.1 ExampIe; 4.2 Fubini Theorem for Loeb Measures (Keisler(1984)); 4.3 Theorem (Keisler (1984)); 5. LIFTINGS OF STOCHASTIC PROCESSES; Definition; 5.1 Proposition; Definition; 5.2 Lemma; Definition; 5.3 Proposition; 5.4 Example; 5.5 Example; 5.6 Example; 6. ADAPTED PROBABILITY SPACES; 6.1 Proposition; Definition; Definition; 6.2 Theorem; 7. ADAPTED DISTRIBUTIONS
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Definition
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English
Additional Edition:
ISBN 0-521-35947-3
Additional Edition:
ISBN 0-521-35109-X
Language:
English
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