The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

     The primary objective of the current paper is to suggest and implement effective computational methods (DECMs) to calculate analytic and approximate solutions to the nonlocal one-dimensional parabolic equation which is utilized to model specific real-world applications. The powerful and elegant methods that are used orthogonal basis functions to describe the solution as a double power series have been developed, namely the Bernstein, Legendre, Chebyshev, Hermite, and Bernoulli polynomials. Hence, a specified partial differential equation is reduced to a system of linear algebraic equations that can be solved by using Mathematica®12. The techniques of effective computational methods (DECMs) have been applied to solve some s

في هذا البحث نحاول تسليط الضوء على إحدى طرائق تقدير المعلمات الهيكلية لنماذج المعادلات الآنية الخطية والتي تزودنا بتقديرات متسقة تختلف أحيانا عن تلك التي نحصل عليها من أساليب الطرائق التقليدية الأخرى وفق الصيغة العامة لمقدرات K-CLASS. وهذه الطريقة تعرف بطريقة الإمكان الأعظم محدودة المعلومات "LIML" أو طريقة نسبة التباين الصغرى"LVR

The research aims to find approximate solutions for two dimensions Fredholm linear integral equation. Using the two-variables of the Bernstein polynomials we find a solution to the approximate linear integral equation of the type two dimensions. Two examples have been discussed in detail.

This article studies the nonlocal inverse boundary value problem for a rectangular domain, a second-order, elliptic equation and a two-dimensional equation. The main objective of the article is to find the unidentified coefficient and provide a solution to the problem. The two-dimensional second-order, convection equation is solved directly using the finite difference method (FDM). However, the inverse problem was successfully solved the MATLAB subroutine lsqnonlin from the optimization toolbox after reformulating it as a nonlinear regularized least-square optimization problem with a simple bound on the unknown quantity. Considering that the problem under study is often ill-posed and that even a small error in the input data can hav

يتعرض قانون الموازنة العامة الاتحادية للطعن بعدم الدستورية كغيره من القوانين، بل أن الطعن فيه يكاد يكون سنوياً حال نشره في الجريدة الرسمية ، وتوجه إليه المطاعن بعدم الدستورية إما عن إجراءات تشريعه أو لمضامينه المتعارضة مع الدستور نصاً أو روحاً ، ولكنّه إذا كانت مدة الطعن بعدم دستورية القوانين كافة متاحة دون قيد زمني محدد ولا تتطلب سوى إجراءات إقامة الدعوى العامة وأخصها قيام شرط المصلحة في حالة الدعوى ال

     This paper develops the work of Mary Florence et.al. on centralizer of semiprime semirings and presents reverse centralizer of semirings with several propositions and lemmas. Also introduces the notion of dependent element and free actions on semirings with some results of free action of centralizer and reverse centralizer on semiprime semirings and some another mappings.

Using a mathematical model to simulate the interaction between prey and predator was suggested and researched. It was believed that the model would entail predator cannibalism and constant refuge in the predator population, while the prey population would experience predation fear and need for a predator-dependent refuge. This study aimed to examine the proposed model's long-term behavior and explore the effects of the model's key parameters. The model's solution was demonstrated to be limited and positive. All potential equilibrium points' existence and stability were tested. When possible, the appropriate Lyapunov function was utilized to demonstrate the equilibrium points' overall stability. The system's persistence requirements were spe

The unprejudiced of this education is to gauge the ability of the retinoic acid to induce apoptotic cell death in hematological tumors through caspase dependent or independent apoptotic pathway, The cytotoxicity effects of retinoic acid of different concentrations (400,350,300,250,200,150,100,50,25,12.5 μg\ml) and exposure for all hematological malignancy cell lines (Human non-Hodgkin lymphoma SR and human multiple myeloma (COLO 677) and Human Monocytic Leukemia THP1 and Acute promyelocytic leukemia NB4) have been determined using a microtetrazolium (MTT) assay. Propodeum iodide and alcidine orange (AO/PI) paired discoloration was used to study the ability of retinoic acid to induce apoptosis in the infected cells and examined under fluore