Nonstandard analysis and differentiable manifolds - Foundations

FOUNDATIONS OF NONSTANDARD ANALYSIS A Gentle Introduction to Nonstandard Extensions

Relaxed optimality conditions for mu-differentiable functions

We prove some fundamental properties of mu-differentiable functions. A new notion of local minimizer and maximizer is introduced and several extremum conditions are formulated using the language of nonstandard analysis.

Reverse Mathematics & Nonstandard Analysis: Why Some Theorems Are More Equal than Others

Reverse Mathematics is a program in foundations of mathematics initiated by Friedman ([1,2]) and developed extensively by Simpson ([4]). Its aim is to determine which minimal axioms prove theorems of ordinary mathematics. Nonstandard Analysis plays an important role in this program ([3, 5]). We consider Reverse Mathematics where equality is replaced by the predicate ≈, i.e. equality up to infin...

Banach Manifolds of Fiber Bundle Sections

In the past several years significant progress has been made in our understanding of infinite dimensional manifolds. Research in this area has split into two quite separate branches ; first a study of the theory of abstract Banach manifolds, and secondly a detailed study of the properties of certain classes of concrete manifolds that arise as spaces of differentiable maps or more generally as s...

Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions

It is well known that in four or more dimensions, there exist exotic manifolds; manifolds that are homeomorphic but not diffeomorphic to each other. More precisely, exotic manifolds are the same topological manifold but have inequivalent differentiable structures. This situation is in contrast to the uniqueness of the differentiable structure on topological manifolds in one, two and three dimen...