Abstract

Let $R$ be a prime ring with extended centroid $C$, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed integer. If $R$ admits the generalized derivations $H$ and $G$ such that $H(u^2)^n=G(u)^{2n}$ for all $u\in L$, then one of the following holds: {(1)} $H(x)=ax$ and $G(x)=bx$ for all $x\in R$, with $a,b\in C$ and $a^n=b^{2n}$; {(2)} char$(R)\neq 2$, $R$ satisfies $s_4$, $H(x)=ax+[p,x]$ and $G(x)=bx$ for all $x\in R$, with $b\in C$ and $a^n=b^{2n}$; {(3)} char$(R)=2$ and $R$ satisfies $s_4$. As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

Keywords: Prime ring; generalized derivation; extended centroid; Utumi quotient ring; Banach algebra.

MSC: 16W25, 16N60, 16R50, 16D60

Abstract

{$2$-rainbow dominating function} (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V(G)$ with $f(v)=\emptyset$ the condition $\cup_{u\in N(v)}f(u)=\{1,2\}$ is fulfilled. The {weight} of a 2RDF $f$ is the value $\omega(f)=\Sigma_{v\in V}|f (v)|$. The {$2$-rainbow domination number} of a graph $G$, denoted by $\gamma_{r2}(G)$, is the minimum weight of a 2RDF of G. The {$2$-rainbow domination subdivision number} $\tx{\rm sd}_{\gamma_{r2}}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the $2$-rainbow domination number. In this paper, we initiate the study of $2$-rainbow domination subdivision number in graphs.

Keywords: domination number; $2$-rainbow domination number; $2$-rainbow domination subdivision number.

MSC: 05C69

Abstract

We prove common fixed point theorems in metric spaces for expansive mappings satisfying an implicit relation without non-decreasing assumption and surjectivity using the concept of weak compatibility which generalize some theorems appearing in the recent literature.

Keywords: Weakly compatible mappings; common fixed point; metric space.

MSC: 47H10, 54H25

Abstract

A function is said to be bi-univalent in the open unit disk $\Bbb{D}$ if both the function and its inverse are univalent in $\Bbb{D} $. By the same token, a function is said to be bi-Bazilevič in $\Bbb{D}$ if both the function and its inverse are Bazilevič there. The behavior of these types of functions are unpredictable and not much is known about their coefficients. In this paper we use the Faber polynomial expansions to find upper bounds for the coefficients of classes of bi-Bazilevič functions. The coefficients bounds presented in this paper are better than those so far appeared in the literature. The technique used in this paper is also new and we hope that this will trigger further interest in applying our approach to other related problems.

Keywords: Faber polynomials; bi-Bazilevič functions; univalent functions.

MSC: 30C45, 30C50

Abstract

In this paper, we present a new approach to proximity structures based on the recognition of many of the entities important in the theory of ideals. So, we give a characterization of the basic proximity using ideals. Also, we introduce the concept of $g$-proximities and we show that for different choice of ``$g$'' one can obtain many of the known types of generalized proximities. Also, characterizations of some types of these proximities -- ($g_0,h_0$) -- are obtained.

Keywords: ideals; basic proximity; proximity space; g-proximities; nearness; topological space.

MSC: 54E05

Abstract

The basic properties of the generalized relative lower order of entire functions are discussed in this paper. In fact, we improve here some results of Datta, Biswas and Biswas [Casp. J. Appl. Math. Ecol. Econ., {\bf1}, 2 (2013), 3--18].

Keywords: Entire function; generalized relative lower order; Property~(A).

MSC: 30D20, 30D30, 30D35