Nonstandard tools of nonsmooth analysis

Nonstandard Analysis of Graphs

Intuitively, it is tempting to view a cube in R as a discrete lattice with infinitesimal edges. A similar infinitesimal picture suggests itself with other fractal spaces such as the Sierpinski gasket or the Sierpinski sponge (see [13], [5].) The purpose of this paper is to show certain classes of metric spaces characterized by volume growth properties of balls ([4],[2]) can viewed as graphs wit...

Nonsmooth analysis of eigenvalues

The eigenvalues of a symmetric matrix depend on the matrix nons-moothly. This paper describes the nonsmooth analysis of these eigen-values. In particular, I present a simple formula for the approximate (limiting Fr echet) subdiierential of an arbitrary function of the eigen-values, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdiierential of the k't...

FOUNDATIONS OF NONSTANDARD ANALYSIS A Gentle Introduction to Nonstandard Extensions

Nonstandard Analysis in Point

We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given innnite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactiication and the nonstandard characterization of the sober spaces seem to be new in the...

Non-nonstandard Analysis

There is an ancient philosphical-religious divide over questions of infinity. This divide extends over mathematics and we see it in many forms. A perfect example is offered by the twin approaches to analysis: limits and infinitesimals. With Robinson’s discovery of nonstandard analysis, both approaches have achieved legitimacy. Today, there is a new approach, non-nonstandard analysis, which, in ...