Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

This research proposes the application of the dragonfly and fruit fly algorithms to enhance estimates generated by the Fama-MacBeth model and compares their performance in this context for the first time. To specifically improve the dragonfly algorithm's effectiveness, three parameter tuning approaches are investigated: manual parameter tuning (MPT), adaptive tuning by methodology (ATY), and a novel technique called adaptive tuning by performance (APT). Additionally, the study evaluates the estimation performance using kernel weighted regression (KWR) and explores how the dragonfly and fruit fly algorithms can be employed to enhance KWR. All methods are tested using data from the Iraq Stock Exchange, based on the Fama-French three-f

The plant licorice is considered important plants as nutritionally and medically and economically, as a rich in phytochemical, vitamins and minerals, and being widely available, Research indicated the presence of many nutrients such as (proteins, Carbohydrates, vitamins and minerals) as well as presence of Glycyrrhizin which responsible of sweet taste, that allowing the possibility to use it as natural intensity sweetener with few calories in Sweetening of many food. This research is aimed to study the Stability of Glycyrrhizin toward the various manufacturing conditions such as (thermal treatment, pH of foods and microwaves), so three factorial experiments was implemented to find out the Stability as following: 100C° - 121C° - Microwa

Maximum values of one particle radial electronic density distribution has been calculated by using Hartree-Fock (HF)wave function with data published by[A. Sarsa et al. Atomic Data and Nuclear Data Tables 88 (2004) 163–202] for K and L shells for some Be-like ions. The Results confirm that there is a linear behavior restricted the increasing of maximum points of one particle radial electronic density distribution for K and L shells throughout some Be-like ions. This linear behavior can be described by using the nth term formula of arithmetic sequence, that can be used to calculate the maximum radial electronic density distribution for any ion within Be like ions for Z<20.

        In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using  Mathcad 15.and graphic in Matlab R2015a.

Medicine is one of the fields where the advancement of computer science is making significant progress. Some diseases require an immediate diagnosis in order to improve patient outcomes. The usage of computers in medicine improves precision and accelerates data processing and diagnosis. In order to categorize biological images, hybrid machine learning, a combination of various deep learning approaches, was utilized, and a meta-heuristic algorithm was provided in this research. In addition, two different medical datasets were introduced, one covering the magnetic resonance imaging (MRI) of brain tumors and the other dealing with chest X-rays (CXRs) of COVID-19. These datasets were introduced to the combination network that contained deep lea

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 &

 Chapter One : the importance of research and the need for it .
       He has developed the concept of the curriculum has evolved as other educational concepts . Because the world has become a small village due to modern technology and are used in various aspects of life , and the rapid communication between the world can be accessed easily and conveniently . And that the purpose of education citizens who create social functions which ones to keep the culture , upgrade and repair flaws, and aims to develop the capacity of the individual and the preparations in the footsteps of scientific and technological development .
- The goal of research : The research aims to:
1 . What is the