Nonstandard analysis and its applications

Nonstandard Kernels and their Applications

Besides using standard radial basis functions, there are a few good reasons to look for kernels with special properties. This survey will provide several examples, starting from an introduction into kernel construction techniques. After providing the “missing” Wendland functions, we focus on kernels based on series expansions. These have some very interesting special cases, namely polynomial an...

Applications of Nonstandard Analysis in Additive Number Theory (a) = Inf

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density. 1 Prologue In this paper we use N for the set of all natural numbers. For any two integers a and b, a; b] denotes the interval of all integers x between a and b (a 6 x 6 b). If S is an internal set, jSj denotes the internal cardinality of S. The l...

- Algebras and Nonstandard Analysis

Suppose B is an ultraproduct of finite dimensional C∗-algebras. We consider mapping and injectability properties for separable C∗algebras into B. In the case of approximately finite C∗-algebras, we obtain a classification of these mappings up to inner conjugacy. Using a Theorem of Voiculescu, we show that for nuclear C ∗-algebras injectability into an ultraproduct of finite dimensional C ∗-alge...

Applications of nonstandard analysis in additive number theory

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

Better Nonstandard Universes with Applications