Nonstandard Analysis in Practice

Front Cover
Francine Diener, Marc Diener
Springer Science & Business Media, Dec 14, 1995 - Mathematics - 250 pages
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In the early seventies, Georges Reeb learnt about Robinson's Nonstandard Analysis (NSA). It became quickly obvious to him that a kind of revolution had happened in mathematics: the old dream of actual infinitesimals had been realized. This seemed to him a great event. He was a mechanics-minded topologist, in the dynamical tradition of Painleve, Poincare, Cartan, and the topological tradition ofhis master Ehres mann. He got convinced that NSA is exactly the right framework within which to study dynamics with small parameters, including problems of asymptotics and bifurcations. This was the starting point ofthe second school he created in Strasbourg, the topological school (which was mainly oriented towards foliation theory and dynamical systems) being the first. This new school grew up with the philosophy of using NSA in everyday mathematics, as a tool to get simple and natural proofs and to detect new mathematical phenomena. The first works were focused on differential equations, but other topics, including perturbation problems in algebra, quickly became of interest. The axiomatic presentation of NSA by E. Nelson within Internal Set Theory in 1977 gave a second impulse to the Alsatian school; this was partly because this formal settingwas in agreementwith Reeb's philosophicalconvictionthat infinitesimals were an unexpected benefit of the impossibmty of formalising the intuitive feeling that all natural numbers are of the same kind.
  

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Contents

Tutorial
1
111 Standard and infinitesimal real numbers and the Leibniz rules
2
112 To be or not to be standard
4
113 Internal statements standard or not and external statements
5
12 Using the extended language
7
121 The axioms
8
122 Application to standard objects
11
13 Shadows and Sproperties
14
654 The topologization of the notion of the limit of sets
129
66 Uniform spaces
130
661 Uniform proximity
131
662 Limited accessible and nearstandard points
132
663 The external definition of a uniformity
133
67 Answers to the exercises
134
Neutrices external numbers and external calculus
145
72 Conventions an example
146

132 Scontinuity at a point
15
134 Sdifferentiability
17
135 Notion of Stheorem
18
142 Fehrele principle
19
Complex analysis
23
22 Tutorial
28
222 Applications
30
223 Exercises with answers
31
23 Complex iteration
34
24 Airys equation
42
241 The distinguished solutions
46
25 Answers to exercises
50
Pierre Delfini and Claude Lobry
51
32 Fourier analysis of DEN
53
322 Interpretation of N ilarge
54
323 Resolution of DEN
57
33 An interesting example
58
34 Solutions of limited energy
63
Scontinuity of solution
65
propagation and reflexion
67
comparison with classical model
69
35 Conclusion
70
Random walks and stochastic differential equations
71
421 The law of wt for a fixed t
72
422 Law of w
74
43 Equivalent processes
77
432 Macroscopic properties
79
44 Diffusions Stochastic differential equations
81
442 Theorems
82
443 Change of variable
83
46 Itos calculus Girsanovs theorem
85
47 The density of a diffusion
87
48 Conclusion
89
Infinitesimal algebra and geometry
91
52 A decomposition theorem for a limited point
92
521 The decomposition theorem
93
522 Geometrical approach
95
523 Algebraic approach
96
the case of curves in R3
97
533 The curvature and the torsion
98
the case of surfaces in R
100
an infinitesimal approach
102
543 The SerretFrenet fibre bundle
104
552 Perturbation of linear operators
105
General topology
109
612 The halo of a point
110
613 The shadow of a subset
111
614 The halo of a subset
112
62 What purpose do halos serve?
113
621 Comparison of topologies
114
624 Separation and compactness
115
63 The external definition of a topology
116
631 Halic preorders and fcPhalos
117
632 The ball of centre x and radius a
119
633 Product spaces and function spaces
121
64 The power set of a topological space
123
642 The Choquet topology
124
65 Setvalued mappings and limits of sets
125
652 The topologization of semicontinuities
126
653 Limits of Sets
127
73 Neutrices and external numbers
149
74 Basic algebraic properties
150
741 Elementary operations
151
742 On the shape of a neutrix
153
743 On the product of neutrices
156
75 Basic analytic properties
158
numbers
159
752 External integration
162
753 External functions
165
76 Stirlings formula
168
77 Conclusion
169
An external probability order theorem with applications
171
82 External probabilities
173
821 Possible values
174
823 Monotony
175
825 Almost certain and negligible
176
84 Weierstrass Stirling De MoivreLaplace
178
842 Stirlings formula revisited
180
Integration over finite sets
185
92 Sintegration
186
923 Sintegrable functions
187
93 Convergence in SLF
189
932 Convergence almost everywhere
190
933 Averaging
191
934 Martingales relative to a function
192
935 Commentary
194
936 Definitions
195
937 Quadrable sets
196
938 Partitions in quadrable subsets
197
939 Lebesgue integrable functions
199
9310 Average of Lintegrable functions
200
9311 Decomposition of Sintegrable functions
202
9312 Commentary
203
94 Conclusion
204
Ducks and rivers three existence results
205
1012 Application to the Van der Pol equation
208
the missing link
211
102 Slowfast vector fields
212
1022 Application of the fast dynamic
213
1023 The slow dynamic
214
1024 Application of the slow dynamic
215
103 Robust ducks
216
1031 Robust ducks buffer points and hillsanddales
217
1032 An other approach to the hillsanddales method
219
104 Rivers
220
1042 Existence of an attracting river
223
Teaching with infinitesimals
225
111 Meaning rediscovered
226
1112 The wooden language of limits
227
the evidence of orders of magnitude
228
1122 Colour numbers
229
1123 The algebraic game of huge
230
Completeness and the shadows concept
234
1132 Examples
235
1133 Brave new numbers
238
References
239
List of contributors
245
Index
247
Copyright

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