Nonstandard Criteria for Borel-Measurability

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire sp...

Verma Modules Induced from Nonstandard Borel Subalgebras

In this paper we study composition series and em-beddings of Verma modules induced from "nonstandard" Borel subalgebras. This article can be viewed as a generalization of Futorny's work on imaginary Verma modules for Λ\ ' where the center of the Kac-Moody algebra acts nontrivially. Introduction. Let A be an indecomposable symmetrizable generalized Cartan matrix, fl(A) = n_ © f) 0 n+ the triangu...

Borel and countably determined reducibility in nonstandard domain

We consider, in a nonstandard domain, reducibility of equivalence relations in terms of the Borel reducibility ≤B and the countably determined (CD, for brevity) reducibility ≤CD. This reveals phenomena partially analogous to those discovered in modern “standard” descriptive set theory. The ≤CD-structure of CD sets (partially) and the ≤B-structure of Borel sets (completely) in N is described. We...

An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability

Nonstandard finite difference schemes for differential equations

In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with ...