Let R be a 2-torision free prime ring and ?, ?? Aut(R). Furthermore, G: R×R?R is a symmetric generalized (?, ?)-Biderivation associated with a nonzero (?, ?)-Biderivation D. In this paper some certain identities are presented satisfying by the traces of G and D on an ideal of R which forces R to be commutative

This study aims at recognizing Pesticides and how the process of pesticides biodegradation by microbiology took place, and the effect of environmental condition on this process. And how the research uncovered the efficiency of microbiology in the biodegradation process of pesticides, as the perfect temperature for the biodegradation process is 40 °C and humidity effect on pesticides efficiency, when high humidity reduces pesticide efficiency and the perfect acidity to increase bacteria efficiency is 7, for the incubation period, it was found during the previous studies that the best incubation period is 5-7 days, in this period the bacteria imprint on pesticides and increase biodegradation of it.

   This paper aims to prove an existence theorem for Voltera-type equation in a generalized G- metric space, called the -metric space, where the fixed-point theorem in - metric space is discussed and its application.  First, a new contraction of Hardy-Rogess type is presented and also then fixed point theorem is established for these contractions in the setup of -metric spaces. As application, an existence result for Voltera integral equation is obtained.  

Recently, complementary perfect corona domination in graphs was introduced. A dominating set S of a graph G is said to be a complementary perfect corona dominating set (CPCD – set) if each vertex in  is either a pendent vertex or a support vertex and  has a perfect matching. The minimum cardinality of a complementary perfect corona dominating set is called the complementary perfect corona domination number and is denoted by . In this paper, our parameter hasbeen discussed for power graphs of path and cycle.

In this paper we define and study new generalizations of continuous functions namely, -weakly (resp., w-closure, w-strongly) continuous and the main properties are studies: (a) If f : X®Y is w-weakly (resp., w-closure, w-strongly) continuous, then for any AÌX and any BÌY the restrictions fïA : A®Y and fB : f -1(B)®B are w-weakly (resp., w-closure, w-strongly) continuous. (b) Comparison between deferent forms of generalizations of continuous functions. (c) Relationship between compositions of deferent forms of generalizations of continuous functions. Moreover, we expanded the above generalizations and namely almost w-weakly (resp., w-closure, w-strongly) continuous functions and we state and prove several results concerning it.

The main idea of this paper is to define other types of a fuzzy local function and study the advantages and differences between them in addition to discussing some definitions of finding new fuzzy topologies. Also in this research, a new type of fuzzy closure has been defined, where the relation between the new type and different types of fuzzy local function has been studied

The main purpose of this work is to introduce some types of fuzzy convergence sequences of operators defined on a standard fuzzy normed space (SFN-spaces) and investigate some properties and relationships between these concepts. Firstly, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional is given. Then the notions of weakly and strongly fuzzy convergence sequences of operators  are introduced and essential theorems related to these concepts are proved. In particular, if ( ) is a strongly fuzzy convergent sequence with a limit  where linear operator from complete standard fuzzy normed space  into a standard fuzzy normed space  then  belongs to the set of all fuzzy bounded linear operators

Heterocyclic compounds are crucial for medicinal chemistry and the development of therapeutic agents like broad-spectrum antibiotics. This study devised a facile procedure to synthesize novel antimicrobial bicyclic heterocycles from 2-mercapto-3-phenylquinazolin-4(3H)-one. Advanced analytical techniques including 1 H and 13C NMR, elemental analysis, and FT-IR spectroscopy characterized the intricate chemical structures of the products. In vitro assays tested the heterocycles against aerobic and anaerobic bacterial strains using fluconazole and ciprofloxacin as antifungal and antibacterial controls. Results demonstrated the formidable broad-spectrum antibacterial and antifungal activities of the synthesized compounds, with growth inhibition

         Continuous functions are novel concepts in topology. Many topologists contributed to the theory of continuous functions in topology. The present authors continued the study on continuous functions by utilizing the concept of gpα-closed sets in topology and introduced the concepts of weakly, subweakly and almost continuous functions. Further, the properties of these functions are established.

R. Vasuki [1] proved fixed point theorems for expansive mappings in Menger spaces. R. Gujetiya and et al [2] presented an extension of the main result of Vasuki, for four expansive mappings in Menger space. In this article, an important lemma is given to prove that the iteration sequence is Cauchy under suitable condition in Menger probabilistic G-metric space (shortly, MPGM-space). And then, used to obtain three common fixed point theorems for expansive type mappings.