In Chapter 8 of the book, “Graphs and Networks: Transfinite and Nonstandard,” (published by Birkhauser-Boston in 2004), nonstandard versions of transfinite graphs and of electrical networks having such graphs were defined and examined but only for the first two ranks, 0 and 1, of transfiniteness. In the present work, these results are extended to higher ranks of transfiniteness. Such is done in...

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...

a nonstandard finite difference (nsfd) scheme has been constructed and analyzed for a mathematical model that describes lotka–volterra food web model. this new discrete system has the same stability properties as the continuous model and,on the whole, it preservesthe same local asymptotic stability properties. linearized stability theory and schur–cohn criteria are used for local asymptoticstab...

Social media language contains huge amount and wide variety of nonstandard tokens, created both intentionally and unintentionally by the users. It is of crucial importance to normalize the noisy nonstandard tokens before applying other NLP techniques. A major challenge facing this task is the system coverage, i.e., for any user-created nonstandard term, the system should be able to restore the ...

Let (X,Γ) be a uniform space with its uniformity generated by a set of pseudo-metrics Γ. Let the symbol “ ' ” denote the usual infinitesimal relation on ∗X, and define a new infinitesimal relation “ ≈ ” on ∗X by writing x ≈ y whenever ∗ρ(x, p) ' ∗ρ(y, p) for each ρ ∈ Γ and each p ∈ X. We call (X,Γ) an S-space if the relations ' and ≈ coincide on fin(∗X). S-spaces are interesting because their n...

This article focuses on how maternal employment in nonstandard schedules at night, on the weekends, or that rotate on a weekly basis influence preschoolers’ behavioral outcomes. Examining low-income working mothers and their children aged 2 – 4 years from the Welfare, Children, and Families: A Three-City Study (N 1⁄4 206), we find that maternal nonstandard schedules are associated with negative...

A notion of ideal value of N -sequences is axiomatized through elementary properties. The resulting theory ZFC[α] provides nonstandard analysis with a general foundational framework. 1991 MSC. Primary: 26E35, 03E70, 03H05. Secondary: 03C20, 03E35. In this paper, the axiomatic system ZFC[α] is presented. It is a generalization in a set theoretic context of an approach to nonstandard analysis rec...