## Nonstandard Analysis in PracticeFrancine Diener, Marc Diener 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 R³ | 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 SL¹F | 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 |

239 | |

List of contributors | 245 |

247 | |

### Common terms and phrases

algebraic analytic appreciable asymptotic axioms Cauchy filter classical compact complex consider contains continuous function converges decomposition definition denote differential equation domain dotted-line ducks equivalent example external formulas external interval external numbers external probability external set f is S-continuous f\dm function f graph hal(x halic halo Hence i-close i-large i-small idempotent infinitely close infinitesimal internal function internal set intersection Julia set lemma Let f limited number linear mapping metric space near-standard neighbourhood system neutrix nonstandard analysis notion open set orders of magnitude partition polynomial PR(A Pr{x Proof properties Proposition prove quadrable random variable real number S-integrable satisfies sequence shadow slow curve slow-fast solution standard elements standard open set standard point standard set standard subset standard topological space Stirling's formula stochastic differential equation suppose theorem theory trajectories uniform space unlimited integer values vector field Vstx Wiener walk zero