Nonstandard Analysis in Practice: Universitext

1. Tutorial.- 1.1 A new view of old sets.- 1.2 Using the extended language.- 1.3 Shadows and S-properties.- 1.4 Permanence principles.- 2. Complex analysis.- 2.1 Introduction.- 2.2 Tutorial.- 2.3 Complex iteration.- 2.4 Airy’s equation.- 2.5 Answers to exercises.- 3. The Vibrating String.- 3.1 Introduction.- 3.2 Fourier analysis of (DEN).- 3.3 An interesting example.- 3.4 Solutions of limited energy.- 3.5 Conclusion.- 4. Random walks and stochastic differential equations.- 4.1 Introduction.- 4.2 The Wiener walk with infinitesimal steps.- 4.3 Equivalent processes.- 4.4 Diffusions. Stochastic differential equations.- 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.- 5.2 A decomposition theorem for a limited point.- 5.3 Infinitesimal riemannian geometry.- 5.4 The theory of moving frames.- 5.5 Infinitesimal linear algebra.- 6. General topology.- 6.1 Halos in topological spaces.- 6.2 What purpose do halos serve ?.- 6.3 The external definition of a topology.- 6.4 The power set of a topological space.- 6.5 Set-valued mappings and limits of sets.- 6.6 Uniform spaces.- 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.- 7.5 Basic analytic properties.- 7.6 Stirling’s formula.- 7.7 Conclusion.- 8. An external probability order theorem with applications.- 8.1 Introduction.- 8.2 External probabilities.- 8.3 External probability order theorems.- 8.4 Weierstrass, Stirling, De Moivre-Laplace.- 9. Integration over finite sets.- 9.1 Introduction.- 9.2 S-integration.-9.3 Convergence in SL1(F).- 9.4 Conclusion.- 10. Ducks and rivers: three existence results.- 10.1 The ducks of the Van der Pol equation.- 10.2 Slow-fast vector fields.- 10.3 Robust ducks.- 10.4 Rivers.- 11. Teaching with infinitesimals.- 11.1 Meaning rediscovered.- 11.2 the evidence of orders of magnitude.- 11.3 Completeness and the shadows concept.- References.- List of contributors.

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.