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Nonstandard analysis and it...

Nonstandard analysis and its applications

Cutland, Nigel

Cambridge University Press 1988

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

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monografia Rebiun39425801 https://catalogo.rebiun.org/rebiun/record/Rebiun39425801 m o d cr cnu---unuuu 121115s1988 enk ob 101 0 eng d 846492891 9781139172110 electronic bk.) 1139172115 electronic bk.) 9781107087934 electronic bk.) 1107087937 electronic bk.) 052135109X 9780521351096 0521359473 9780521359474 DEBBG BV043057283 DEBSZ 446450235 CAMBR eng pn CAMBR N$T IDEBK OCLCF YDXCP OCL OCLCQ AGLDB YDX OCLCO OCLCQ UAB OCLCQ VTS REC STF AU@ M8D UKAHL OCLCQ OCL AJS SFB OCLCQ OCLCO OCLCQ OCLCO OCLCL UKKRT OCLCL OCLCA MAT 003000 bisacsh 519.4 22 https://id.oclc.org/worldcat/ddc/E3Rj3h344RHQJCtGHDPCRXTtw8 31.40 bcl 31.46 bcl *00B25 msc 03-06 msc SK 130 rvk Nonstandard analysis and its applications edited by Nigel Cutland Cambridge [England] New York Cambridge University Press 1988 Cambridge [England] New York Cambridge [England] New York Cambridge University Press 1 online resource (xiii, 346 pages) 1 online resource (xiii, 346 pages) Text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society student texts 10 Papers presented at a conference held at the University of Hull in 1986 Includes bibliographical references and index 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 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 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 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)) 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 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 mathematical analysis- Congresses Nonstandard mathematical analysis- Congresses Analyse mathématique non standard- Congrès Analyse mathématique non standard- Congrès MATHEMATICS- Applied. Nonstandard mathematical analysis. Nonstandard-Analysis. Analyse mathématique non standard. Conference papers and proceedings. Cutland, Nigel Print version Nonstandard analysis and its applications. Cambridge, [England] ; New York : Cambridge University Press, 1988 052135109X (DLC) 88016194 (OCoLC)18013779 London Mathematical Society student texts 10

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monografia Rebiun39425801 https://catalogo.rebiun.org/rebiun/record/Rebiun39425801 m o d cr cnu---unuuu 121115s1988 enk ob 101 0 eng d 846492891 9781139172110 electronic bk.) 1139172115 electronic bk.) 9781107087934 electronic bk.) 1107087937 electronic bk.) 052135109X 9780521351096 0521359473 9780521359474 DEBBG BV043057283 DEBSZ 446450235 CAMBR eng pn CAMBR N$T IDEBK OCLCF YDXCP OCL OCLCQ AGLDB YDX OCLCO OCLCQ UAB OCLCQ VTS REC STF AU@ M8D UKAHL OCLCQ OCL AJS SFB OCLCQ OCLCO OCLCQ OCLCO OCLCL UKKRT OCLCL OCLCA MAT 003000 bisacsh 519.4 22 https://id.oclc.org/worldcat/ddc/E3Rj3h344RHQJCtGHDPCRXTtw8 31.40 bcl 31.46 bcl *00B25 msc 03-06 msc SK 130 rvk Nonstandard analysis and its applications edited by Nigel Cutland Cambridge [England] New York Cambridge University Press 1988 Cambridge [England] New York Cambridge [England] New York Cambridge University Press 1 online resource (xiii, 346 pages) 1 online resource (xiii, 346 pages) Text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society student texts 10 Papers presented at a conference held at the University of Hull in 1986 Includes bibliographical references and index 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 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 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 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)) 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 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 mathematical analysis- Congresses Nonstandard mathematical analysis- Congresses Analyse mathématique non standard- Congrès Analyse mathématique non standard- Congrès MATHEMATICS- Applied. Nonstandard mathematical analysis. Nonstandard-Analysis. Analyse mathématique non standard. Conference papers and proceedings. Cutland, Nigel Print version Nonstandard analysis and its applications. Cambridge, [England] ; New York : Cambridge University Press, 1988 052135109X (DLC) 88016194 (OCoLC)18013779 London Mathematical Society student texts 10