Developments in Nonstandard Mathematics

Summary

This book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson's work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis.

This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson's death. This book will be of particular interest to students and researchers in nonstandard analysis, measure theory, generalised functions and mathematical physics.

Table of Contents

The infinitesimal rule of three
Nonstandard methods in the precalculus curruculum
Difference quotients and smoothness
Continuous maps with special properties
Some nonstandard methods in geometric topology
Delayed bifurcations in perturbed systems analysis of slow passage of Suhl-threshold
Functional analysis and NSA
Near-standard compact internal linear operators
Discrete Fredholm's equations
Nonstandard theory of generalized functions
Representing distributions by nonstandard polynomials
Contributions of nonstandard analysis to partial differential equations
Loeb measure theory
Unions of Loeb nullsets: the context
Gredient lines and distributions of functionals in infinite dimensional Euclidean spaces
Nonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interaction
Nonstandard analysis in selective unierses
Lattices and monads
A neometric survey
Long sequences and neocompact sets