The Strength of Nonstandard Analysis

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1 The Strength of Nonstandard Analysis Bearbeitet von Imme van den Berg, Vitor Neves 1. Auflage Buch. xx, 401 S. Hardcover ISBN Format (B x L): 17 x 24,2 cm Gewicht: 930 g Weitere Fachgebiete > Philosophie, Wissenschaftstheorie, Informationswissenschaft > Wissenschaften: Allgemeines > Geschichte der Naturwissenschaften, Formalen Wissenschaften & Technik schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 Contents I Foundations 1 1 The strength of nonstandard analysis 3 by H. Jerome Keisler 1.1 Introduction The theory PRA ω The theory NPRA ω The theory WNA Bounded minima and overspill Standard parts Liftings of formulas Choice principles in L(PRA ω ) Saturation principles Saturation and choice Second order standard parts Functional choice and ( 2 ) Conclusion The virtue of simplicity 27 by Edward Nelson Part I. Technical Part II. General Analysis of various practices of referring in classical or non standard mathematics 33 by Yves Péraire 3.1 Introduction Généralités sur la référentiation Le calcul de Dirac. L égalité de Dirac Calcul de Heaviside sans transformée de Laplace. L égalité de Laplace Exemples xi

3 xii Contents 4 Stratified analysis? 47 by Karel Hrbacek 4.1 The Robinsonian framework Stratified analysis An axiomatic system for stratified set theory ERNA at work 64 by C. Impens and S. Sanders 5.1 Introduction The system The language The axioms The Sousa Pinto approach to nonstandard generalised functions 76 by R. F. Hoskins 6.1 Introduction Generalised functions and N.S.A Distributions, ultradistributions and hyperfunctions Schwartz distributions The Silva axioms Fourier transforms and ultradistributions Sato hyperfunctions Harmonic representation of hyperfunctions Prehyperfunctions and predistributions The differential algebra A(Ω ε ) Predistributions of finite order Predistributions of local finite order Predistributions of infinite order Conclusion Neutrices in more dimensions 92 by Imme van den Berg 7.1 Introduction Motivation and objective Setting Structure of this article The decomposition theorem Geometry of neutrices in R 2 and proof of the decomposition theorem Thickness, width and length of neutrices On the division of neutrices Proof of the decomposition theorem

4 Contents xiii II Number theory Nonstandard methods for additive and combinatorial number theory. A survey 119 by Renling Jin 8.1 The beginning Duality between null ideal and meager ideal Buy-one-get-one-free scheme From Kneser to Banach Inverse problem for upper asymptotic density Freiman s 3k 3 + b conjecture Nonstandard methods and the Erdős-Turán conjecture 133 by Steven C. Leth 9.1 Introduction Near arithmetic progressions The interval-measure property III Statistics, probability and measures Nonstandard likelihood ratio test in exponential families 145 by Jacques Bosgiraud 10.1 Introduction A most powerful nonstandard test Some basic concepts of statistics Main definitions Tests Exponential families Basic concepts Kullback-Leibler information number The nonstandard test Large deviations for X n-regular sets n-regular sets defined by Kullback-Leibler information The nonstandard likelihood ratio test ln α n infinitesimal n ln α c n ln α c Comparison with nonstandard tests based on X Regular nonstandard tests Case when Θ 0 is convex

5 xiv Contents 11 A finitary approach for the representation of the infinitesimal generator of a markovian semigroup 170 by Schérazade Benhabib 11.1 Introduction Construction of the least upper bound of sums in IST The global part of the infinitesimal generator Remarks On two recent applications of nonstandard analysis to the theory of financial markets 177 by Frederik S. Herzberg 12.1 Introduction A fair price for a multiply traded asset Fairness-enhancing effects of a currency transaction tax How to minimize unfairness Quantum Bernoulli experiments and quantum stochastic processes 189 by Manfred Wolff 13.1 Introduction Abstract quantum probability spaces Quantum Bernoulli experiments The internal quantum processes From the internal to the standard world Brownian motion The nonstandard hulls of the basic internal processes The symmetric Fock space and its embedding into L Applications of rich measure spaces formed from nonstandard models 206 by Peter Loeb 14.1 Introduction Recent work of Yeneng Sun Purification of measure-valued maps More on S-measures 217 by David A. Ross 15.1 Introduction Loeb measures and S-measures Egoroff s Theorem A Theorem of Riesz Conditional expectation

6 Contents xv 16 A Radon-Nikodým theorem for a vector-valued reference measure 227 by G. Beate Zimmer 16.1 Introduction and notation The existing literature The nonstandard approach A nonstandard vector-vector integral Uniform convexity Vector-vector derivatives without uniform convexity Remarks Differentiability of Loeb measures 238 by Eva Aigner 17.1 Introduction S-differentiability of internal measures Differentiability of Loeb measures IV Differential systems and equations The power of Gâteaux differentiability 253 by Vítor Neves 18.1 Preliminaries Smoothness Smoothness and finite points Smoothness and the nonstandard hull Strong uniform differentiability The non-standard hull Nonstandard Palais-Smale conditions 271 by Natália Martins and Vítor Neves 19.1 Preliminaries The Palais-Smale condition Nonstandard Palais-Smale conditions Palais-Smale conditions per level Nonstandard variants of Palais-Smale conditions per level Mountain Pass Theorems Averaging for ordinary differential equations and functional differential equations 286 by Tewfik Sari 20.1 Introduction

7 xvi Contents 20.2 Deformations and perturbations Deformations Perturbations Averaging in ordinary differential equations KBM vector fields Almost solutions The stroboscopic method for ODEs Proof of Theorem 2 for almost periodic vector fields Proof of Theorem 2 for KBM vector fields Functional differential equations Averaging for FDEs in the form z (τ) = εf (τ, z τ ) The stroboscopic method for ODEs revisited Averaging for FDEs in the form ẋ(t) = f (t/ε, x t ) The stroboscopic method for FDEs Path-space measure for stochastic differential equation with a coefficient of polynomial growth 306 by Toru Nakamura 21.1 Heuristic arguments and definitions Bounds for the -measure and the -Green function Solution to the Fokker-Planck equation Optimal control for Navier-Stokes equations 317 by Nigel J. Cutland and Katarzyna Grzesiak 22.1 Introduction Preliminaries Nonstandard analysis The stochastic Navier-Stokes equations Controls Optimal control for d = Controls with no feedback Costs Solutions for internal controls Optimal controls Hölder continuous feedback controls (d = 2) Controls based on digital observations (d = 2) The space H The observations Ordinary and relaxed feedback controls for digital observations Costs for digitally observed controls

8 Contents xvii Solution of the equations Optimal control Optimal control for d = Existence of solutions for any control The control problem for 3D stochastic Navier-Stokes equations The space Ω Approximate solutions Optimal control Hölder continuous feedback controls (d = 3) Approximate solutions for Hölder continuous controls. 344 Appendix: Nonstandard representations of the spaces H r Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations 349 by João Paulo Teixeira 23.1 Introduction A discretization for the diffusion-advection equations in the torus Some standard estimates for the solution of the discrete problem Main estimates on the hyperfinite discrete problem Existence and uniqueness of solution V Infinitesimals and education Calculus with infinitesimals 369 by Keith D. Stroyan 24.1 Intuitive proofs with small quantities Continuity & extreme values Microscopic tangency in one variable The Fundamental Theorem of Integral Calculus Telescoping sums & derivatives Continuity of the derivative Trig, polar coordinates & Holditch s formula The polar area differential Leibniz s formula for radius of curvature Changes Small changes The natural exponential

9 xviii Contents Concerning the history of the calculus Keisler s axioms Small, medium, and large hyperreal numbers Keisler s algebra axiom The uniform derivative of x Keisler s function extension axiom Logical real expressions Logical real formulas Logical real statements Continuity & extreme values Microscopic tangency in one variable The Fundamental Theorem of Integral Calculus The Local Inverse Function Theorem Second differences and higher order smoothness Pre-University Analysis 395 by Richard O Donovan 25.1 Introduction Standard part Stratified analysis Derivative Transfer and closure