The Strength of Nonstandard Analysis (eBook)

Foreword 5
Acknowledgements 11
Contents 13
Part I Foundations 21
The strength of nonstandard analysis 22
The virtue of simplicity 46
Analysis of various practices of referring in classical or non standard mathematics 52
Stratified analysis? 66
ERNA at work 83
The Sousa Pinto approach to nonstandard generahsed functions 95
Neutrices in more dimensions 111
Part II Number theory 136
Nonstandard methods for additive and combinatorial number theory. A survey 137
Nonstandard methods and the Erdos- Turan conjecture 151
Part III Statistics, probability and measures 161
Nonstandard likelihood ratio test in exponential families 162
A finitary approach for the representation of the infinitesimal generator of a markovian semigroup 187
On two recent applications of nonstandard analysis to the theory of financial markets 194
Quantum Bernoulli experiments and quantum stochastic processes 206
Applications of rich measure spaces formed from nonstandard models 223
More on S- measures 234
A Radon- Nikodym theorem for a vector- valued reference measure 244
Differentiability of Loeb measures 255
Differential systems and equations 267
The power of Gateaux differentiability 268

Nonstandard Palais- Smale conditions 286
Averaging for ordinary differential equations and functional differential equations 301
Path- space measure for stochastic differential equation with a coefficient of polynomial growth 321
Optimal control for Navier- Stokes equations 332
Local- in- time existence of strong solutions of the n- dimensional Burgers equation via discretizations 364
Infinitesimals and education 382
Calculus with infinitesimals 383
Pre- University Analysis 409