In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

Till now, different constitutive models have been applied to model the hot deformation flow curves of different materials. In this research, the hot deformation flow stress of API X65 pipeline steel was modeled using the power law equation with strain dependent constants. The results was compared with the results of the other previously examined constitutive equations including the Arrhenius eq...

This work focuses on the formulation of a four-equation model for simulating unsteady two-phase mixtures with phase transition and strong discontinuities. The main assumption consists in homogeneous temperature, pressure velocity fields between two phases. Specifically, we present extension residual distribution scheme to solve system written non-conservative form, i.e. terms internal energy in...

Given the linear matrix equation AXB=C, we partition it into form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- post-multiply both sides of by four orthogonal projectors generated from coefficient matrices A1, B1, B2 to obtain reduced equations. In this situation, each equations involves just one unknown submatrices X11, X12, X21, X22, respectively. paper, study relationships between general...

The observation of ideal four-wave mixing dynamics is notoriously difficult to implement experimentally due the generation higher-order sidebands and optical loss, which limit potential interaction distance. Here, we overcome this problem with an experimental technique that uses programmable phase amplitude shaping iterate wave initial conditions injected into fiber. This extends effective prop...