In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible–Infected–Chronic–AIDS) model and analyze dynamical properties of discretized system. We prove that is dynamically consistent with continuous, maintaining essential standard model, namely, positivity boundedness solutions, equilibrium points, their local global stability.

OBJECTIVES The effect of employment insecurity on employee health is an important public health issue due to the recent effects of neoliberalism and the global financial crisis (2007-2008) on labor markets. This study aims to evaluate the differences in cardiovascular health status and the use of preventive screening services between standard and nonstandard workers. METHODS Waged employees (...

In contrast with Tennenbaum’s theorem [8] that says Peano Arithmetic (PA) has no nonstandard computable model, Shepherdson [6] constructed a computable nonstandard model for a very weak fragment of PA, called Open Induction (Iopen) in which the induction scheme is only allowed to be applied for quantifier-free formulas (with parameters). Since then several attempts have been made from both side...

This article outlines ways in which nonstandard methods can be applied to continuum theory in the plane. 1. Nonstandard Models Nonstandard analysis, first developed by Abraham Robinson in the early 1960’s [10], makes use of the fact that there are other models besides the usual ones that satisfy all the same mathematical statements that can be made in First Order Logic. The basic idea is to exp...

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC.

A weak theory nonstandard analysis, with types at all finite levels over both the integers and hyperintegers, is developed as a possible framework for reverse mathematics. In this weak theory, we investigate the strength of standard part principles and saturation principles which are often used in practice along with first order reasoning about the hyperintegers to obtain second order conclusio...

We show that the transitivity of pointwise Turing reducibility on the recursively enumerable sets of integers cannot be proven in P− + IΣ1, first order arithmetic with induction limited to Σ1 predicates. We produce a example of intransitivity in a nonstandard model of P+IΣ1 by a finite injury priority construction.

We design and analyze multigrid methods for the saddle point problems resulting from Raviart-Thomas-Nédélec mixed finite element methods (of order at least 1) for the Darcy system in porous media flow. Uniform convergence of the W -cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed.

We show how to interpret weak König’s lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Harvey Friedman about weak König’s lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in h...