A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity...

This document is an assessment of the value of optimal linear interpolation enclosures and of nonstandard intervals, especially with respect to applications in computer graphics, and of the extent a future IEEE interval standard should support these. It turns out that essentially all present applications of nonstandard intervals to practical problems can be matched by similarly efficient approa...

This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the vertice...

It is well known that standard peptides, which comprise proteinogenic amino acids, can act as specific chemical probes to target proteins with high affinity. Despite this fact, a number of peptide drug leads have been abandoned because of their poor cell permeability and protease instability. On the other hand, nonstandard peptides isolated as natural products often exhibit remarkable pharmaco-...

‎In this paper we propose a numerical scheme to solve the one dimensional nonlinear Klein-Gorden equation‎. ‎We describe the mathematical formulation procedure in details‎. ‎The scheme is three level explicit and based on nonstandard finite difference‎. ‎It has nonlinear denominator function of the step sizes‎. ‎Stability analysis of the method has been given and we prove that the proposed meth...

This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducin...

Interpretation with Infinitesimals Towards Scalability in Nonstandard Static Analysis Kengo Kido, Swarat Chaudhuri, and Ichiro Hasuo 1 University of Tokyo, Japan 2 Rice University, USA Abstract. Towards the goal of correctness and reliability of hybrid systems, we continue the nonstandard static analysis program where hybrid dynamics is turned into purely discrete one with explicit use of infin...

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. The theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of “nonstandard analysis”. The paper begins with a short exposition of the construction of the hyperreal number system and the fundamental results of nonstandard analysis which are us...

Nonstandard digraphs and transfinite digraphs have been defined and examined in two prior technical reports. The present work examines digraphs that are both nonstandard and transfinite. This requires a combination in certain ways of the techniques used in the prior two works. We first construct herein nonstandard transfinite digraphs for all the naturalnumber ranks. Then, a special kind of non...