Journal of Applied Sciences and Nanotechnology

This paper is concerned with the concept of modern Roman domination in graphs. A Modern Roman dominating function on a graph is labeling such that every vertex with label 0 is adjacent to two vertices; one of them of label 2 and the other of label 3 and every vertex with label 1 is adjacent to a vertex with label 2 or label 3. The weight of a Roman dominating function is the value . The minimum weight of all possible Roman dominating functions is called the "Roman Domination Number" of a graph. This dominance can be used in many aspects of life, for example in computer networks, transmission lines, and many others. In this paper, the modern Roman domination of the fan graph and the double fan graph with their complement are determined. Also, it has been determined the the number of modern Roman dominations of  the corona of two specific graphs like the corone of two fan graph, two double fan graph ,fan graph and double fan graph and the oppisit of them.

The idea of generalizing quasi injective by employing a new term is introduced in this paper. The introduction of principally self-injective modules, which are principally self-injective modules. A number of characteristics and characterizations of such modules have been established. In addition, the idea of strongly mainly self-pure sub-modules was added, which is similar to strongly primarily self-injective sub-modules. Some characteristics of injective, quasi-injective, principally self-injective, principally injective, absolutely self-pure, absolutely pure, and finitely R-injective modules being lengthened to strongly principally self-injective modules. So, in the present work, some properties are added to the concept in a manner similar to the absolutely self-neatness. The fundamental features of these concepts and their interrelationships are linked to the conceptions of some rings. (Von Neumann) regular, left SF-ring, and left pp-ring rings are described via such concept. For instance, the homomorphic picture of every principally injective module be strongly principally self-injective if R being left pp-ring, and another example for a commutative ring R of every strongly principally self-injective module be flat if R being (Von Neumann) regular. Also, a ring R be (Von Neumann) regular if and only if each R-module being strongly principally self-injective module.

In the context of the multiple-criteria decision aid (MCDA), several fuzzy integrals concerning capacities (non-additive measures) have been introduced by various researchers in the last sixty years.  Recently, Lehrer has proposed a new integral for capacities known as concave integral. The concave integral is based on the decomposition of random variables into simple ingredients. The concave integral concerning capacity is defined as the maximum value obtained among all its decompositions. The paper aims to model a new integration based on the decomposition of random variables into simple ingredients for multi-criteria decision making support when underlying scales are bipolar.  This paper proposes a generalization of the concave integral in terms of decompositions of the integrated function to be suitable for bipolar scales. We show that the random variable is analyzed as a combination of indicators, where each allowed decomposition has a value determined by the bi-capacity. Lastly, we illustrate our framework by a practical example.

Chaos theory has attracted much attention because it fully reflects the complexity of the system, which is an essential property in many applications, especially in the optimization problem.  In this paper, the possibility of improving research by means of evolutionary algorithms (genetic algorithms) will be discussed which used to solve non- linear programming problems. This improvement and development are carried out using a highly quality chaotic map, which was proposed to be used for generating real values (keys) that are used as reference values for the genetic algorithm. A comparison between the results without using chaotic systems and the results after generating the keys is performed. It shows that the results after the chaotic local search (CLS) are improved and congregate with the optimum value of the solutions obtained by the projected process before the CLS. Moreover, the differences between the proposed systems for improvement are also compared. The evaluation parameters for the proposed chaotic function are developed using the Mathematica 11.2 program.