Optimization and nonstandard analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a]
Dekker
1994
|
| Schriftenreihe: | Pure and applied mathematics
184 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 356 S. |
| ISBN: | 0824792815 |
Internformat
MARC
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| 245 | 1 | 0 | |a Optimization and nonstandard analysis |c J. E. Rubio |
| 264 | 1 | |a New York [u.a] |b Dekker |c 1994 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface Hi
1 Optimization and Nonstandard Analysis 1
1.1 Optimization 1
1.1.1 Minimizers 1
1.1.2 Existence of minimizers 6
1.2 The role of nonstandard analysis 9
1.3 A set theoretical general framework for mathematics 15
1.3.1 Mathematics as operation on sets 15
1.3.2 A large set—an Aleph 18
1.4 A language for the Aleph 23
1.4.1 Syntax 23
1.4.2 Semantics 26
1.5 Nonstandard analysis 30
1.5.1 A story of two Alephs 30
1.5.2 Transfer 35
1.6 Appendix: Filters, ultrafilters 41
1.7 Notes and references 47
2 Further Concepts and Applications 51
2.1 New concepts and properties 51
2.1.1 Properties of the * mapping 51
2.1.2 Internal elements 56
2.1.3 Enlargements 60
2.2 Saturation 63
2.2.1 Cardinal numbers 63
2.2.2 Saturated Alephs 68
2.3 Topology 72
vii
viii Contents
2.3.1 Topology and filters 72
2.3.2 The cardinalities of a topological space .... 76
2.3.3 Nonstandard relations 79
2.3.4 Nonstandard characterizations 81
2.4 Metric spaces 86
2.5 The hyperreals 89
2.6 Notes and references 92
3 Measure Theory and Infinite Dimensional Linear Pro¬
gramming 95
3.1 A summary of standard measure theory 96
3.1.1 Measures 96
3.1.2 Integrals 97
3.1.3 The extension of measures 100
3.2 A summary of nonstandard measure theory 103
3.2.1 Extensions and Loeb measures 103
3.2.2 Integrals and liftings 105
3.2.3 The standard part map 108
3.2.4 Hyperfinite measure spaces. Elementary in¬
tegration 110
3.3 Measures and optimization 113
3.4 Hyperfinite approximation 119
3.4.1 Introduction 119
3.4.2 The nonstandard framework and hyperfinite
approximation 121
3.4.3 Minimizers 125
3.4.4 Approximation 127
3.4.5 A nonstandard characterization of realcom
pactness 128
3.4.6 Appendix 129
3.5 Notes and references 130
4 Linear Spaces, a Variational Principle and Penalties 133
4.1 Linear normed spaces, standard
and nonstandard 134
4.2 The variational principle 137
Contents ix
4.2.1 Elementary derivatives and a necessary con¬
dition for optimality 139
4.3 The new variational principle 141
4.4 Calculus of variations 145
4.5 Penalties and optimal control theory 151
4.6 Notes and references 160
5 The Control of Homogenized Systems 163
5.1 S continuity and some further results on integration 163
5.2 A homogenized model of the diffusion equation . . . 168
5.2.1 The standard problem 169
5.3 The homogenized model 170
5.3.1 The nonstandard equations 171
5.3.2 The microlocal parameters 172
5.3.3 The homogenized model 173
5.4 The homogenized control problem 178
5.5 Warning 180
5.6 Introduction to the control treatment 180
5.7 Metamorphosis 181
5.8 Existence and approximation 183
5.9 Numerical results 191
5.10 Notes and references 194
6 Distributions 195
6.1 Standard distributions 195
6.1.1 Spaces of functions 195
6.1.2 Distributions 199
6.1.3 Spaces of distributions 201
6.2 Nonstandard distributions 205
6.2.1 Representation of linear mappings by hyper
finite operators 206
6.2.2 Nonstandard representation of distributions . 213
6.2.3 Functions of distributions 221
6.3 Optimization 225
6.3.1 The problem of moments in L 225
6.3.2 Optimal control problems with impulsive con¬
trols and nonlinear control laws 228
x Contents
6.4 Notes and references 232
References 235
Appendix A: Exercises 253
A.I Chapter 1 253
A.2 Chapter 2 256
A.3 Chapter 3 258
A.4 Chapter 4 260
A.5 Chapter 5 261
A.6 Chapter 6 262
Appendix B: Bibliography 265
Index 352
|
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| id | DE-604.BV009977433 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:44:16Z |
| institution | BVB |
| isbn | 0824792815 |
| language | English |
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| physical | X, 356 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
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| publisher | Dekker |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Rubio, Julio Enrique Verfasser aut Optimization and nonstandard analysis J. E. Rubio New York [u.a] Dekker 1994 X, 356 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 184 Optimierung (DE-588)4043664-0 gnd rswk-swf Nonstandard-Analysis (DE-588)4137021-1 gnd rswk-swf Optimierung (DE-588)4043664-0 s Nonstandard-Analysis (DE-588)4137021-1 s DE-604 Pure and applied mathematics 184 (DE-604)BV000001885 184 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006612332&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rubio, Julio Enrique Optimization and nonstandard analysis Pure and applied mathematics Optimierung (DE-588)4043664-0 gnd Nonstandard-Analysis (DE-588)4137021-1 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4137021-1 |
| title | Optimization and nonstandard analysis |
| title_auth | Optimization and nonstandard analysis |
| title_exact_search | Optimization and nonstandard analysis |
| title_full | Optimization and nonstandard analysis J. E. Rubio |
| title_fullStr | Optimization and nonstandard analysis J. E. Rubio |
| title_full_unstemmed | Optimization and nonstandard analysis J. E. Rubio |
| title_short | Optimization and nonstandard analysis |
| title_sort | optimization and nonstandard analysis |
| topic | Optimierung (DE-588)4043664-0 gnd Nonstandard-Analysis (DE-588)4137021-1 gnd |
| topic_facet | Optimierung Nonstandard-Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006612332&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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