The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as  increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives  a good agreement.

The group for the multiplication of closets is the set G|N of all closets of N in G, if G is a group and N is a normal subgroup of G. The term “G by N factor group” describes this set. In the quotient group G|N, N is the identity element. In this paper, we procure K(SL(2,125)) and K(SL(2,3125)) from the character table of rational representations for each group.

Background The Transportation Problem (TP) is a detailed model in operations study with applications in logistics, supply chain management, and resource allocation. The classical IBFS methods including North-West Corner, Least Cost and Vogel’s Approximation have competitive computational efficiency, but they are very sensitive to the structure of the problem and usually lead to a solution that is far from the global optimum. Classic enhancement strategies like the Generalized Distribution (MODI) and Stepping-Stone (SS) approaches have low computational complexity but may fall into a local optimum quickly, which makes them ineffective in large-scale or unbalanced problems. Methods We propose the first generic hybrid algorithm, calle

Topological indices provide important insights into the structural characteristics of molecular graphs. The present investigation proposes and explores a creative graph on a finite group G, which is known as the RIG. This graph is designated as ΓRS G2(4) indicating a simple undirected graph containing elements of G. Two distinct ertices are regarded as nearly the same if and only if their sum yields a non-trivial involution element in G. RIGs have been discovered in various finite groups. We examine several facets of the RIG by altering the graph through the conjugacy classes of G. Furthermore, we investigate the topological indices as applications in graph theory applying the distance matrix of the G2(4) group.

Statistical control charts are widely used in industry for process and measurement control . in this paper we study the use of markov chain approach in calculating the average run length (ARL) of cumulative sum (Cusum) control chart for defect the shifts in the mean of process , and exponentially weighted moving average (EWMA) control charts for defect the shifts for process mean and , the standard deviation . Also ,we used the EWMA charts based on the logarithm of the sample variance for monitoring a process standard deviation when the observations (products are selected from al_mamun factory ) are identically and independently distributed (iid) from normal distribution in continuous manufacturing .

Brainstorming is considered as one of the manners that develop learners' mental abilities. Besides, it can help learners get a lot of ideas and thoughts. And by following applied steps to answer the problem concerned, the researcher carried out this practical study aimed at:Developing the ideas of design of third year students/Institute of Fine Arts/Evening Studies- Baghdad/First Rusafa by employing Brainstorming mechanism to develop the ideas of design of institute students in designing the technical advertisement and to achieve the authenticity of the goal of the research, Department of Plastic Arts/Institute of Fine Arts/Evening Studies/Baghdad-First Rusafa was chosen as a case study for the research. It embraced (20) students who rep

This research aims to clarify the concept of doctrinal rules and adjust its basic terminologies. It further aims to lay down a map for the method of rooting this science by mentioning its rooted sources, in addition to drawing a miniature picture of its history, origin, formation and development. The paper ends with practical models to highlight its importance in rooting the science of nodal rules and facilitating the mentioning of its scattered discussions in a short and comprehensive phrase. The study further illustrates the pioneering role of doctrinal rules science in managing the doctrinal disputes, combining multiple sayings, and in bringing together opposing opinions. The study follows the inductive, descriptive and analytical app