Nonstandard analysis in practice:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1995
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 250 S. graph. Darst. |
| ISBN: | 3540602976 0387602976 |
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MARC
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| 245 | 1 | 0 | |a Nonstandard analysis in practice |c Francine Diener ... Editors |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1995 | |
| 300 | |a XIV, 250 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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| adam_text | Table of Contents
1. Tutorial
F. Diener and M. Diener 1
1.1 A new view of old sets 1
1.1.1 Standard and infinitesimal real numbers, and the Leib¬
niz rules 2
1.1.2 To be or not to be standard 4
1.1.3 Internal statements (standard or not) and external
statements 5
1.1.4 External sets 5
1.2 Using the extended language 7
1.2.1 The axioms 8
1.2.2 Application to standard objects 11
1.3 Shadows and S properties 14
1.3.1 Shadow of a set 14
1.3.2 S continuity at a point 15
1.3.3 Shadow of a function 15
1.3.4 S differentiability 17
1.3.5 Notion of S theorem 18
1.4 Permanence principles 18
1.4.1 The Cauchy principle 18
1.4.2 Fehrele principle 19
2. Complex analysis
A. Fruchard 23
2.1 Introduction 23
2.2 Tutorial 28
2.2.1 Proof of the Robinson Callot theorem 28
2.2.2 Applications 30
2.2.3 Exercises with answers 31
2.2.4 Periodic functions 31
2.3 Complex iteration 34
2.4 Airy s equation 42
2.4.1 The distinguished solutions 46
2.5 Answers to exercises 50
X Table of Contents
3. The Vibrating String
Pierre Delfini and Claude Lobry 51
3.1 Introduction 51
3.2 Fourier analysis of (DEN) 53
3.2.1 Diagonalisation of A 53
3.2.2 Interpretation of N i large 54
3.2.3 Resolution of (DEN) 57
3.3 An interesting example 58
3.4 Solutions of limited energy 63
3.4.1 A preliminary theorem 63
3.4.2 Limited energy: S continuity of solution 65
3.4.3 Limited energy: propagation and reflexion 67
3.4.4 A particular case: comparison with classical model.... 69
3.5 Conclusion 70
4. Random walks and stochastic differential equations
Eric Benoit 71
4.1 Introduction 71
4.2 The Wiener walk with infinitesimal steps 71
4.2.1 The law of wt for a fixed t 72
4.2.2 Law of w 74
4.3 Equivalent processes 77
4.3.1 The notion 77
4.3.2 Macroscopic properties 79
4.3.3 The brownian process 79
4.4 Diffusions. Stochastic differential equations 81
4.4.1 Definitions 81
4.4.2 Theorems 82
4.4.3 Change of variable 83
4.5 Probability law of a diffusion 83
4.6 Ito s calculus Girsanov s theorem 85
4.7 The density of a diffusion 87
4.8 Conclusion 89
5. Infinitesimal algebra and geometry
Michel Goze 91
5.1 A natural algebraic calculus 91
5.1.1 The Leibniz rules 91
5.1.2 The algebraic geometric calculus underlying a point of
the plane 91
5.2 A decomposition theorem for a limited point 92
5.2.1 The decomposition theorem 93
5.2.2 Geometrical approach 95
5.2.3 Algebraic approach 96
5.3 Infinitesimal riemannian geometry 97
Table of Contents XI
5.3.1 Orthonormal decomposition of a point 97
5.3.2 The Serret Frenet frame of a differentiable curve in R3 97
5.3.3 The curvature and the torsion 98
5.4 The theory of moving frames 100
5.4.1 The theory of moving frames 100
5.4.2 The moving frame: an infinitesimal approach 102
5.4.3 The Serret Frenet fibre bundle 104
5.5 Infinitesimal linear algebra 104
5.5.1 Nonstandard vector spaces 104
5.5.2 Perturbation of linear operators 105
5.5.3 The Jordan reduction of a complex linear operator ... 107
6. General topology
Tewfik Sari 109
6.1 Halos in topological spaces 109
6.1.1 Topological proximity 109
6.1.2 The halo of a point 110
6.1.3 The shadow of a subset Ill
6.1.4 The halo of a subset 112
6.2 What purpose do halos serve ? 113
6.2.1 Comparison of topologies 114
6.2.2 Continuity 114
6.2.3 Neighbourhoods, open sets and closed sets 114
6.2.4 Separation and compactness 115
6.3 The external definition of a topology 116
6.3.1 Halic preorders and P halos 117
6.3.2 The ball of centre x and radius a 119
6.3.3 Product spaces and function spaces 121
6.4 The power set of a topological space 123
6.4.1 The Vietoris topology 123
6.4.2 The Choquet topology 124
6.5 Set valued mappings and limits of sets 125
6.5.1 Semicontinuous set valued mappings 125
6.5.2 The topologization of semi continuities 126
6.5.3 Limits of Sets 127
6.5.4 The topologization of the notion of the limit of sets ... 129
6.6 Uniform spaces 130
6.6.1 Uniform proximity 131
6.6.2 Limited, accessible and nearstandard points 132
6.6.3 The external definition of a uniformity 133
6.7 Answers to the exercises 134
XII Table of Contents
7. Neutrices, external numbers, and external calculus
Found Koudjeti and Imme van den Berg 145
7.1 Introduction 145
7.2 Conventions; an example 146
7.3 Neutrices and external numbers 149
7.4 Basic algebraic properties 150
7.4.1 Elementary operations 151
7.4.2 On the shape of a neutrix 153
7.4.3 On the product of neutrices 156
7.5 Basic analytic properties 158
7.5.1 External distance and extrema of a collection of exter¬
nal numbers 159
7.5.2 External integration 162
7.5.3 External functions 165
7.6 Stirling s formula 168
7.7 Conclusion 169
8. An external probability order theorem with applications
I.P. van den Berg 171
8.1 Introduction 171
8.2 External probabilities 173
8.2.1 Possible values 174
8.2.2 Externally measurable sets 174
8.2.3 Monotony 175
8.2.4 Additivity 175
8.2.5 Almost certain and negligible 176
8.3 External probability order theorems 176
8.4 Weierstrass, Stirling, De Moivre Laplace 178
8.4.1 The Weierstrass theorem 178
8.4.2 Stirling s formula revisited 180
8.4.3 A central limit theorem 180
9. Integration over finite sets
Pierre Cartier and Yvette Perrin 185
9.1 Introduction 185
9.2 S integration 186
9.2.1 Measure on a finite set 186
9.2.2 Rare sets 186
9.2.3 S integrable functions 187
9.3 Convergence in SLl(F) 189
9.3.1 Strong convergence 189
9.3.2 Convergence almost everywhere 190
9.3.3 Averaging 191
9.3.4 Martingales relative to a function 192
9.3.5 Commentary 194
Table of Contents XIII
9.3.6 Definitions 195
9.3.7 Quadrable sets 196
9.3.8 Partitions in quadrable subsets 197
9.3.9 Lebesgue integrable functions 199
9.3.10 Average of L integrable functions 200
9.3.11 Decomposition of S integrable functions 202
9.3.12 Commentary 203
9.4 Conclusion 204
10. Ducks and rivers: three existence results
Francine Diener and Marc Diener 205
10.1 The ducks of the Van der Pol equation 205
10.1.1 Definition and existence 205
10.1.2 Application to the Van der Pol equation 208
10.1.3 Duck cycles: the missing link 211
10.2 Slow fast vector fields 212
10.2.1 The fast dynamic 212
10.2.2 Application of the fast dynamic 213
10.2.3 The slow dynamic 214
10.2.4 Application of the slow dynamic 215
10.3 Robust ducks 216
10.3.1 Robust ducks, buffer points, and hills and dales 217
10.3.2 An other approach to the hills and dales method 219
10.4 Rivers 220
10.4.1 The rivers of the Liouville equation 220
10.4.2 Existence of an attracting river 223
10.4.3 Existence of a repelling river 223
11. Teaching with infinitesimals
Andri Deledicq 225
11.1 Meaning rediscovered 226
11.1.1 Continuity having continuity troubles 226
11.1.2 The wooden language of limits 227
11.1.3 A second marriage between intuition and formalism... 227
11.2 the evidence of orders of magnitude 228
11.2.1 Minimal rules and the vocabulary of calculus 228
11.2.2 Colour numbers 229
11.2.3 The algebraic game of huge 230
11.3 Completeness and the shadows concept 234
11.3.1 The geometrical game of almost 234
11.3.2 Examples 235
11.3.3 Brave new numbers 238
XIV Table of Contents
References 239
List of contributors 245
Index 247
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| indexdate | 2024-07-09T17:52:38Z |
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| isbn | 3540602976 0387602976 |
| language | English |
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| publishDate | 1995 |
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| spelling | Nonstandard analysis in practice Francine Diener ... Editors Berlin [u.a.] Springer 1995 XIV, 250 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Nonstandard-Analysis (DE-588)4137021-1 s Diener, Francine Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006959649&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nonstandard analysis in practice Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4137021-1 (DE-588)4001865-9 |
| title | Nonstandard analysis in practice |
| title_auth | Nonstandard analysis in practice |
| title_exact_search | Nonstandard analysis in practice |
| title_full | Nonstandard analysis in practice Francine Diener ... Editors |
| title_fullStr | Nonstandard analysis in practice Francine Diener ... Editors |
| title_full_unstemmed | Nonstandard analysis in practice Francine Diener ... Editors |
| title_short | Nonstandard analysis in practice |
| title_sort | nonstandard analysis in practice |
| topic | Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Nonstandard mathematical analysis Nonstandard-Analysis Analysis |
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