Vol 40, No 1 (2019)

In this paper we describe all power-associative algebra structures on a two-dimensional vector space over algebraically closed fields and ℝ. The list of all two-dimensional left(right) unital power-associative algebras, along with their unit elements, is specified. Also we compare the result of the paper with that results obtained earlier.

The purpose of the present paper is to introduce the concepts of upper and lower *-continuous multifunctions. Several characterizations of upper and lower *-continuous multifunctions are investigated. The relationships between upper and lower *-continuous multifunctions and the other types of continuity are discussed.

The aim of this article is to develop a new linear model for count data. The main idea is in an application of a new generalized linear model framework, which we call the Negative Binomial-Sushila linear model. The Negative Binomial-Sushila distribution has been proposed recently and applied to count data. This distribution is constructed as a mixture of the Negative Binomial and Sushila distributions. The mixed distribution is a flexible alternative to the Poisson distribution when over-dispersed count data is analyzed. The parameters of this distribution are estimated using a Bayesian approach with R2jags package of the R language. The Negative Binomial-Sushila linear model is applied to fit two real data sets with an over-dispersion and its performance is compared with the performance of some traditional models. The results show that the Negative Binomial-Sushila generalized linear model fits the data sets better than the traditional generalized models for these data sets.

The notions of a comparative UP-filter and an allied UP-filter are introduced, and related properties are investigated. Relations between a UP-filter, an implicative UP-filter and a comparative UP-filter are discussed. Conditions for a UP-filter to be a comparative UP-filter are displayed. Conditions for a comparative UP-filter to be an implicative UP-filter are considered. We show that comparative UP-filters and implicative UP-filters coincide in a meet-commutative UP-algebra X satisfying the condition (∀x, y, z ∈ X) (x · (y · z) = y · (x · z)). Characterizations of a comparative UP-filter are stated. An extension property for comparative UP-filter is established. Conditions for a UP-filter to be an x-allied UP-filter for given x ∈ X are provided.

We study a uniform approximation to constant functions f(z) = const on compact subsets K of complex plane by logarithmic derivatives of rational functions with free poles. This problem can be treated in terms of electrostatics: we construct on K the constant electrostatic field due to electrons and positrons at points ∉ K. If K is a disk or an interval, we get the approximation, which close to the best. Also we get the new identity for generalized Laguerre polynomials. Our results related to the classical problem of rational approximation to the exponential function.

Denote by \(\mathbb{R}_+^\vee\) the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication. Let X be a topological space and C^∨(X) be the semiring of continuous nonnegative functions on X with pointwise operation max-addition and multiplication of functions. By a subalgebra we mean a nonempty subset A of C^∨(X) such that f ∨ g, fg, rf ∈ A for any f, g ∈ A, \(r \in \mathbb{R}_+^\vee\). We consider the lattice \(\mathbb{A}\)(C^∨(X)) of subalgebras of the semiring C^∨(X) and its sublattice \(\mathbb{A}_1\)(C^∨(X)) of subalgebras with unity. The main result of the paper is the proof of the definability of the semiring C^∨(X) both by the lattice \(\mathbb{A}\)(C^∨(X)) and by its sublattice \(\mathbb{A}_1\)(C^∨(X)).

A balanced social network is a social network where, for any member of the social network, the following two statements are true; a friend of my friend is my friend and an enemy of my enemy is my friend. In this paper we demonstrate a polynomial time greedy algorithm that balances any complete social network with n members by changing at most ⌈n²/4 − n/2⌉ of the initial relationships between the members of the network. We also demonstrate that the problem of determining the minimum number of relationships that needs to change so that a complete social network, where each member has at least as many friends as enemies, becomes balanced is still NP-Complete.

This article [1] has been retracted by the Editor-in-Chief, because, by some technical mistake, it has been published twice: originally in issue 1 of the same journal [2]. The Editor-in-Chief and the publisher apologize for any inconvenience. The authors agree to this retraction.