This paper introduces a relation between resultant and the Jacobian determinant
by generalizing Sakkalis theorem from two polynomials in two variables to the case of (n) polynomials in (n) variables. This leads us to study the results of the type:  ,            and use this relation to attack the Jacobian problem. The last section shows our contribution to proving the conjecture.

In this research, the Williamson-Hall method and of size-strain plot method was employed to analyze X- ray lines for evaluating the crystallite size and lattice strain and of cadmium oxide nanoparticles. the crystallite size value is (15.2 nm) and (93.1 nm) and lattice strain (4.2 x10−4 ) and (21x10−4) respectively. Also, other methods have been employed to evaluate the crystallite size. The current methods are (Sherrer and modified Sherrer methods ) and their results are (14.8 nm) and (13.9nm) respectively. Each method of analysis has a different result because the alteration in the crystallite size and lattice strain calculated according to the Williamson-Hall and size-strain plot methods shows that the non-uniform strain in nan

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.

     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

In our article, three iterative methods are performed to solve the nonlinear differential equations that represent the straight and radial fins affected by thermal conductivity. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. For comparison purposes, the numerical solutions were further achieved by using the fourth Runge-Kutta (RK4) method, Euler method and previous analytical methods that available in the literature. Moreover, the convergence of the proposed methods was discussed and proved. In addition, the maximum error remainder values are also evaluated which indicates that the propo

Background: The aim of this study was for estimation and comparism of masticatory efficiency in patient wearing heat cured acrylic and flexible base partial denture, finding out the role of peanuts and carrots on the measurement of chewing efficiency, and to find out whether the types of P.D. (being single or two opposing each other) has any effect on the masticatory performance. Materials and methods: Twenty partially edentulous patients were selected. Five of these patients were selected having Kennedy class I with no modification against natural dentition, other 5patients having Cl.I against Cl.I . The other 5 patients having Cl.III against natural dentition and the last 5 patients were with Cl.III against Cl.III. several ways were used

This study was design to investigate the dimensional stability of heat-activated acrylic resin with different methods of flask cooling (15 minutes rapid cooling, one hour bench cooling, four hours delayed deflasking, and 24 hours delayed deflasking) at different time intervals (immediately, two days, seven days, 30 days) after deflasking. Heat-activated acrylic resin was used to prepare acrylic samples. Then, measurement of the distances where achieved between the centers of selected marks in the acrylic samples. They were measured at different time intervals for different methods of flask cooling. The results showed that the group samples of the four hours and 24 hours of delayed deflasking was insignificantly different from the control an

In this work, the effect of aluminum (Al) dust particles on the DC discharge plasma properties in argon was investigated. A magnetron is placed behind the cathode at different pressures and with varying amounts of Al. The plasma temperature (Te) and density (ne) were calculated using the Boltzmann equation and Stark broadening phenomena, which are considered the most important plasma variables through which the other plasma parameters were calculated. The measurements showed that the emission intensity decreases with increasing pressure from 0.06 to 0.4 Torr, and it slightly decreases with the addition of the NPs. The calculations showed that the ne increased and Te decreased with pressure. Both Te and ne were reduced by increasing