An Introduction to Nonstandard Analysis

A New Approach to Nonstandard Analysis

In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing  a few applications of this approach with the former methods.

Nonstandard Analysis and an Application to the Symmetric Group on Natural Numbers

An introduction of nonstandard analysis in purely algebraic terms is presented. As an application, we give a nonstandard proof of a characterization theorem for compact subsets of Sym(N).

Nonstandard explicit third-order Runge-Kutta method with positivity property

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

FOUNDATIONS OF NONSTANDARD ANALYSIS A Gentle Introduction to Nonstandard Extensions

An Axiomatic Presentation of The Nonstandard Methods in Mathematics

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC. Introduction. For practitioners, the most popular foundational approach to nonstandard methods is the so-called super...