Title | Nonstandard Analysis in Practice [electronic resource] / edited by Francine Diener, Marc Diener |
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Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1995 |
Connect to | http://dx.doi.org/10.1007/978-3-642-57758-1 |
Descript | XIV, 250 p. 14 illus. online resource |
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 Airyโs 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 Itoโs calculus โ Girsanovโs theorem -- 4.7 The โdensityโ 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 Stirlingโs 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