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 simulation of passively Q-switching is four non – linear first order differential equations. The optimization of passively Q-switching simulation was carried out using the constrained Rosenbrock technique. The maximization option in this technique was utilized to the fourth equation as an objective function; the parameters, γa, γc and β as were dealt with as decision variables. A FORTRAN program was written to determine the optimum values of the decision variables through the simulation of the four coupled equations, for ruby laser Q–switched by Dy +2: CaF2.For different Dy +2:CaF2 molecules number, the values of decision variables was predicted using our written program. The relaxation time of Dy +2: CaF2, used with ruby was

In this paper we prove the boundedness of the solutions and their derivatives of the second order ordinary differential equation x ?+f(x) x ?+g(x)=u(t), under certain conditions on f,g and u. Our results are generalization of those given in [1].

This paper considers approximate solution of the hyperbolic one-dimensional wave equation with nonlocal mixed boundary conditions by improved methods based on the assumption that the solution is a double power series based on orthogonal polynomials, such as Bernstein, Legendre, and Chebyshev. The solution is ultimately compared with the original method that is based on standard polynomials by calculating the absolute error to verify the validity and accuracy of the performance.

Stubbornness is of the nature of the mean, we do not see it in the character of a person whose heart God filled with faith, wisdom and knowledge. And stubbornness does not fall into it except hearts that are arrogant, envious and objectionable, hearts that do not admit mistakes, and are not satisfied with the truth. Rather, you see humiliation,humiliation and shortcomings in returning to Him. He loves corruption * Except in humiliation, humiliation and shortcomings, the stubborn fall, even if they are proud when it is said to him, "Beware of God", pride takes him for sin, so his count is Hell, and his misery.

traces of stubbornness    :key wor

Radiological assessment for the East Baghdad oilfield-southern part was conducted in the current study. 10 samples (scale, soil, sludge, water, and oil) from the different stages of oil production were collected. 232Th, 226Ra, and 40K in the samples were analyzed with 40% efficiency for Gamma spectrometry. system based on HPGe. The findings indicated that the examined sites exhibit comparatively lower levels of NORM contamination, in contrast to other global oilfields. Nevertheless, certain areas, particularly those within separation stages, demonstrate relatively elevated NORM concentrations exceeding the global average in soil and sludge. The maximum value of 226Ra, 232Th, was found in sludge sample the findings indicated that ove

         Cataract, which is the opacity inside clear ocular lens of eye, result in the scattering of visible light as it passes via the lens and consequently deterioration in optical image. The aim of the present study was to investigate whether safranal, an active constituent of Crocus sativus L. stigmas, has a protective effect on the cataract in the rat's pups. The animals were randomly divided into five groups, each of which consisted of 7 rat pups. Group I served as normal control (vehicle administration). For testing cataract induction, animals of Groups II, III, and IV were administered a single subcutaneous injection of sodium selenite on postpartum day 12. After sodium selenite intoxicatio

Experts have given much attention on the use of waste in asphalt paving because of its significance from a sustainability perspective. This paper evaluated the performance properties of asphalt concrete mixes modified with Crumb Rubber (CR) as a partial replacement for two grade sizes of fine aggregate (2.36, and 0.3 mm) at six replacement rates: 0%, 2%, 4%, 6%, 8%, and 10% by weight. Asphalt concrete mixes were prepared at their Optimum Asphalt Content (OAC) and then tested for their engineering properties. Marshall properties, fatigue, rutting, ideal CT index test, Scanning Electron Microscopy (SEM), and Energy-Dispersive X-ray (EDX) spectroscopy were deployed to examine the crystalline structure and elemental composition of the C