Synopses & Reviews
In 1961 Robinson introduced an entirely new version of the theory of infinitesimals, which he called `Nonstandard analysis'. `Nonstandard' here refers to the nature of new fields of numbers as defined by nonstandard models of the first-order theory of the reals. This system of numbers was closely related to the ring of Schmieden and Laugwitz, developed independently a few years earlier. During the last thirty years the use of nonstandard models in mathematics has taken its rightful place among the various methods employed by mathematicians. The contributions in this volume have been selected to present a panoramic view of the various directions in which nonstandard analysis is advancing, thus serving as a source of inspiration for future research. Papers have been grouped in sections dealing with analysis, topology and topological groups; probability theory; and mathematical physics. This volume can be used as a complementary text to courses in nonstandard analysis, and will be of interest to graduate students and researchers in both pure and applied mathematics and physics.
Table of Contents
Preface.
Part I: Analysis. Singular traces and nonstandard analysis;
S. Albeverio, D. Guido, A. Ponosov, S. Scarlatti. Navier--Stokes equations;
M. Capinski, N. Cutland. Hyperfinite approximations of commutative topological groups;
E.I. Gordon. A note on the myope topology;
T. Norberg. Nonlinear theories of generalized functions;
M. Oberguggenberger. A nonstandard approach to the Pettis integral;
H. Osswald. A counterexample to the spectral mapping theorem revisited from a nonstandard point of view;
H. Ploss. Nonstandard polynomials in several variables;
H. Render. An existence result for a class of partial differential equations with smooth coefficients;
T. Todorov. On the generation of topology by external equivalence relations;
B. Wietschorke. Nonstandard hulls of Lebesgue--Bochner spaces;
B. Zimmer. Part II: Probability Theory. A nonstandard approach to diffusions on manifolds and nonstandard heat kernels;
H. Akiyama. A nonstandard approach to the Malliavin Calculus;
N.J. Cutland, Siu-Ah Ng. Ergodic transformations in AST;
M. Kalina. Nonstandard characterization for a general invariance principle;
D. Landers, L. Rogge. Andersons's Brownian motion and the infinite dimensional Ornstein--Uhlenbeck process;
T. Lindstrøm. Two applications of NSA in the theory of stochastic dynamical systems;
A. Ponosov. Nonstandard methods and the space of experiments;
D.A. Ross. Part III: Mathematical Physics. Infinite range forces and strong L
^{1}-asymptotics for the space--homogeneous Boltzmann equation;
L. Arkeryd. A nonstandard analysis approach to the theory of quantum meanfield systems;
M.P.H. Wolff. Subject Index.