Developments in Nonstandard Mathematics

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.

The infinitesimal rule of threeNonstandard methods in the precalculus curruculumDifference quotients and smoothnessContinuous maps with special propertiesSome nonstandard methods in geometric topologyDelayed bifurcations in perturbed systems analysis of slow passage of Suhl-thresholdFunctional analysis and NSANear-standard compact internal linear operatorsDiscrete Fredholm's equationsNonstandard theory of generalized functionsRepresenting distributions by nonstandard polynomialsContributions of nonstandard analysis to partial differential equationsLoeb measure theoryUnions of Loeb nullsets: the contextGredient lines and distributions of functionals in infinite dimensional Euclidean spacesNonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interactionNonstandard analysis in selective uniersesLattices and monadsA neometric surveyLong sequences and neocompact sets

Cutland\, Nigel J; Neves\, Vitor; Oliveira\, A F; Sousa-Pinto\, Jose