The research discussed the topic of the functional role of responsive materials in being elements of a functional transformation in the design of industrial products, based on the study of the structures of smart materials and their performance capabilities at the level of action and self-reaction that characterize this type of materials.

Basic features of responsive materials have been identified to be elements of self-functional insertion into the industrial product design, which contributes to raising the efficiency and functional capacity of the industrial product and enhancing the ability of products to perform self-acting interactions in the structural structure of the material structure of the product and its ability to res

The current research dealt with the rapid development of industrial product design in recent times, and this development in the field of design led to the emergence of modern trends in many terms and theories to direct greater interest in the cognitive foundations of design and its relationship with the components of other natural sciences, and despite the impressive technological development, nature remains With its content of formative values and structural dimensions, it is the first source of inspiration and the source of all modern mathematical sciences and theories, as God made them tend towards organization to continue to provide us with endless inspiration. Hence, the fractional one, which is an important part of dedicating the d

The local resolving neighborhood  of a pair of vertices  for  and  is if there is a vertex  in a connected graph  where the distance from  to  is not equal to the distance from  to , or defined by . A local resolving function  of  is a real valued function   such that  for  and . The local fractional metric dimension of graph  denoted by , defined by  In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph  and graph , where graph  is a connected graphs and graph  is a complate graph &

    Many designs have been suggested for unipolar magnetic lenses based on changing the width of the inner bore and fixing the other geometrical parameters of the lens to improve the performance of unipolar magnetic lenses. The investigation of a study of each design included the calculation of its axial magnetic field the magnetization of the lens in addition to the magnetic flux density using the Finite Element Method (FEM) the Magnetic Electron Lenses Operation (MELOP) program version 1 at three different values of current density (6,4,2 A/mm²). As a result, the clearest values and behaviors were obtained at current density (2 A/mm²). it was found that the best magnetizing properties, the high

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptio

The metric dimension and dominating set are the concept of graph theory that can be developed in terms of the concept and its application in graph operations. One of some concepts in graph theory that combine these two concepts is resolving dominating number. In this paper, the definition of resolving dominating number is presented again as the term dominant metric dimension. The aims of this paper are to find the dominant metric dimension of some special graphs and corona product graphs of the connected graphs  and , for some special graphs  . The dominant metric dimension of  is denoted by  and the dominant metric dimension of corona product graph G and H is denoted by .

The main aim of this paper is to introduce the concept of a Fuzzy Internal Direct Product of fuzzy subgroups of group . We study some properties and prove some theorems about this concept ,which is very important and interesting of fuzzy groups and very useful in applications of fuzzy mathematics in general and especially in fuzzy groups.