We show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions. The generalized functions under consideration are constructed as families of functions modulo a free filter, as it is usually done in applied analysis. In contrast with models of nonstandard analysis, we do not require the filter to be an ultrafilter. The principles are intended to be used...

Users of non-standard methods in mathematics have always been interested in the following question: what does non-standard analysis offer the mathematical community? The issues raised by this question are neverending and there is a whole spectrum of possible answers. The paper [HK] offered an explanation from the point of view of logic, explaining how the principles used in the superstructure a...

In this paper two new combinatorial principles in nonstandard analysis are isolated and applications are given. The second principle provides an equivalent formulation of Henson’s isomorphism property.

In this chapter, we develop the general framework of nonstandard analysis and the necessary logic for the transfer principle. We will begin each section with a brief summary for readers who want to postpone the technical details until a later reading. The summary will note any important definitions and results of the section that the reader should know before going on. For example, Definition 2...

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a ...

‎In this paper‎, ‎the fractional--order form of three dimensional chemostat model with variable yields is introduced‎. ‎The stability analysis of this fractional system is discussed in detail‎. ‎In order to study the dynamic behaviours of the mentioned fractional system‎, ‎the well known nonstandard (NSFD) scheme is implemented‎. ‎The proposed NSFD scheme is compared with the forward Euler and ...

Interpretation with Infinitesimals∗ Towards Scalability in Nonstandard Static Analysis (Extended Abstract) Kengo Kido, Swarat Chaudhuri, and Ichiro Hasuo 1 University of Tokyo, Japan 2 JSPS Research Fellow 3 Rice University, USA Abstract Towards the goal of correctness and reliability of hybrid systems, we continue our nonstandard static analysis program (with Suenaga and Sekine) where hybrid d...

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