Nonstandard Analysis in Practice

The purpose of this book is to provide an effective introduction to nonstandard methods. A short tutorial giving the necessary background, is followed by applications to various domains, independent from each other. These include complex dynamical systems, stochastic differential equations, smooth and algebraic curves, measure theory, the external calculus, with some applications to probability. The authors have been using Nonstandard Analysis for many years in their research. They all belong to the growing nonstandard school founded by G. Reeb, which is attracting international and interdisciplinary interest.

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This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals.

Contenu

1. Tutorial.- 1.1 A new view of old sets.- 1.1.1 Standard and infinitesimal real numbers, and the Leibniz rules.- 1.1.2 To be or not to be standard.- 1.1.3 Internal statements (standard or not) and external statements.- 1.1.4 External sets.- 1.2 Using the extended language.- 1.2.1 The axioms.- 1.2.2 Application to standard objects.- 1.3 Shadows and S-properties.- 1.3.1 Shadow of a set.- 1.3.2 S-continuity at a point.- 1.3.3 Shadow of a function.- 1.3.4 S-differentiability.- 1.3.5 Notion of S-theorem.- 1.4 Permanence principles.- 1.4.1 The Cauchy principle.- 1.4.2 Fehrele principle.- 2. Complex analysis.- 2.1 Introduction.- 2.2 Tutorial.- 2.2.1 Proof of the Robinson-Callot theorem.- 2.2.2 Applications.- 2.2.3 Exercises with answers.- 2.2.4 Periodic functions.- 2.3 Complex iteration.- 2.4 Airy's equation.- 2.4.1 The distinguished solutions.- 2.5 Answers to exercises.- 3. The Vibrating String.- 3.1 Introduction.- 3.2 Fourier analysis of (DEN).- 3.2.1 Diagonalisation of A.- 3.2.2 Interpretation of N i-large.- 3.2.3 Resolution of (DEN).- 3.3 An interesting example.- 3.4 Solutions of limited energy.- 3.4.1 A preliminary theorem.- 3.4.2 Limited energy: S-continuity of solution.- 3.4.3 Limited energy: propagation and reflexion.- 3.4.4 A particular case: comparison with classical model.- 3.5 Conclusion.- 4. Random walks and stochastic differential equations.- 4.1 Introduction.- 4.2 The Wiener walk with infinitesimal steps.- 4.2.1 The law of wt for a fixed t.- 4.2.2 Law of w.- 4.3 Equivalent processes.- 4.3.1 The notion.- 4.3.2 Macroscopic properties.- 4.3.3 The brownian process.- 4.4 Diffusions. Stochastic differential equations.- 4.4.1 Definitions.- 4.4.2 Theorems.- 4.4.3 Change of variable.- 4.5 Probability law of a diffusion.- 4.6 Ito's calculus - Girsanov's theorem.- 4.7 The "density" of a diffusion.- 4.8 Conclusion.- 5. Infinitesimal algebra and geometry.- 5.1 A natural algebraic calculus.- 5.1.1 The Leibniz rules.- 5.1.2 The algebraic-geometric calculus underlying a point of the plane.- 5.2 A decomposition theorem for a limited point.- 5.2.1 The decomposition theorem.- 5.2.2 Geometrical approach.- 5.2.3 Algebraic approach.- 5.3 Infinitesimal riemannian geometry.- 5.3.1 Orthonormal decomposition of a point.- 5.3.2 The Serret-Prenet frame of a differentiable curve in ?3.- 5.3.3 The curvature and the torsion.- 5.4 The theory of moving frames.- 5.4.1 The theory of moving frames.- 5.4.2 The moving frame: an infinitesimal approach.- 5.4.3 The Serret-Frenet fibre bundle.- 5.5 Infinitesimal linear algebra.- 5.5.1 Nonstandard vector spaces.- 5.5.2 Perturbation of linear operators.- 5.5.3 The Jordan reduction of a complex linear operator.- 6. General topology.- 6.1 Halos in topological spaces.- 6.1.1 Topological proximity.- 6.1.2 The halo of a point.- 6.1.3 The shadow of a subset.- 6.1.4 The halo of a subset.- 6.2 What purpose do halos serve ?.- 6.2.1 Comparison of topologies.- 6.2.2 Continuity.- 6.2.3 Neighbourhoods, open sets and closed sets.- 6.2.4 Separation and compactness.- 6.3 The external definition of a topology.- 6.3.1 Halic preorders and P-halos.- 6.3.2 The ball of centre x and radius a.- 6.3.3 Product spaces and function spaces.- 6.4 The power set of a topological space.- 6.4.1 The Vietoris topology.- 6.4.2 The Choquet topology.- 6.5 Set-valued mappings and limits of sets.- 6.5.1 Semicontinuous set-valued mappings.- 6.5.2 The topologization of semi-continuities.- 6.5.3 Limits of Sets.- 6.5.4 The topologization of the notion of the limit of sets.- 6.6 Uniform spaces.- 6.6.1 Uniform proximity.- 6.6.2 Limited, accessible and nearstandard points.- 6.6.3 The external definition of a uniformity.- 6.7 Answers to the exercises.- 7. Neutrices, external numbers, and external calculus.- 7.1 Introduction.- 7.2 Conventions; an example.- 7.3 Neutrices and external numbers.- 7.4 Basic algebraic properties.- 7.4.1 Elementary operations.- 7.4.2 On the shape of a neutrix.- 7.4.3 On the product of neutrices.- 7.5 Basic analytic properties.- 7.5.1 External distance and extrema of a collection of external numbers.- 7.5.2 External integration.- 7.5.3 External functions.- 7.6 Stirling's formula.- 7.7 Conclusion.- 8. An external probability order theorem with applications.- 8.1 Introduction.- 8.2 External probabilities.- 8.2.1 Possible values.- 8.2.2 Externally measurable sets.- 8.2.3 Monotony.- 8.2.4 Additivity.- 8.2.5 Almost certain and negligible.- 8.3 External probability order theorems.- 8.4 Weierstrass, Stirling, De Moivre-Laplace.- 8.4.1 The Weierstrass theorem.- 8.4.2 Stirling's formula revisited.- 8.4.3 A central limit theorem.- 9. Integration over finite sets.- 9.1 Introduction.- 9.2 S-integration.- 9.2.1 Measure on a finite set.- 9.2.2 Rare sets.- 9.2.3 S-integrable functions.- 9.3 Convergence in SL1(F).- 9.3.1 Strong convergence.- 9.3.2 Convergence almost everywhere.- 9.3.3 Averaging.- 9.3.4 Martingales relative to a function.- 9.3.5 Commentary.- 9.3.6 Definitions.- 9.3.7 Quadrable sets.- 9.3.8 Partitions in quadrable subsets.- 9.3.9 Lebesgue integrable functions.- 9.3.10 Average of L-integrable functions.- 9.3.11 Decomposition of S-integrable functions.- 9.3.12 Commentary.- 9.4 Conclusion.- 10. Ducks and rivers: three existence results.- 10.1 The ducks of the Van der Pol equation.- 10.1.1 Definition and existence.- 10.1.2 Application to the Van der Pol equation.- 10.1.3 Duck cycles: the missing link.- 10.2 Slow-fast vector fields.- 10.2.1 The fast dynamic.- 10.2.2 Application of the fast dynamic.- 10.2.3 The slow dynamic.- 10.2.4 Application of the slow dynamic.- 10.3 Robust ducks.- 10.3.1 Robust ducks, buffer points, and hills-and-dales.- 10.3.2 An other approach to the hills-and-dales method.- 10.4 Rivers.- 10.4.1 The rivers of the Liouville equation.- 10.4.2 Existence of an attracting river.- 10.4.3 Existence of a repelling river.- 11. Teaching with infinitesimals.- 11.1 Meaning rediscovered.- 11.1.1 Continuity having continuity troubles.- 11.1.2 The "wooden language" of limits.- 11.1.3 A second marriage between intuition and formalism.- 11.2 the evidence of orders of magnitude.- 11.2.1 Minimal rules and the vocabulary of calculus.- 11.2.2 Colour numbers.- 11.2.3 The algebraic game of huge.- 11.3 Completeness and the shadows concept.- 11.3.1 The geometrical game of almost.- 11.3.2 Examples.- 11.3.3 Brave new numbers.- References.- List of contributors.