In this paper, a method based on binary-coded genetic algorithm is proposed to explore an optimization method, for obtaining an optimal elliptical tank. This optimization method enhances the rollover threshold of a tank vehicle, especially under partial filling conditions. Minimizing the overturning moment imposed on the vehicle due to c.g. height of the liquid load, lateral acceleration and cargo load shift are properly applied. In the process, the width and height of tanker are assumed as constant parameters. Additionally, considering the constant cross-sectional area, an optimum elliptical tanker of each filling condition is presented to provide more roll stability. Moreover, the magnitudes of lateral and vertical translation of the cargo within the proposed optimal cross section under a constant lateral acceleration field are compared with those of conventional elliptical tank to demonstrate the performance potentials of the optimal shapes.Comparing the vehicle rollover threshold of proposed optimal tank with that of currently used elliptical and circular tank reveals that the optimal tank is improved approximately 18% higher than conventional one.

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation, which allow e cient implementations of ECC. In this paper, we improve e cient algorithm for exponentiation on elliptic curves de ned over Fp in terms of a ne coordinates. The algorithm computes 2n2(2n1P +Q) directly from random points P and Q on an elliptic curve, without computing the intermediate points. Moreover, we apply the algorithm to exponentiation on elliptic curves with width{w Mutual Opposite Form (wMOF) and analyze their computational complexity. This algorithm can speed up the wMOF exponentiation of elliptic curves of size 160{bit about (21. 7%) as a result of its implementation with respect to a ne coordinates.

In this work, we investigate the existence of weak solutions for the following semi-linear elliptic system\begin{equation*}\left\{\begin{array}{c}-\Delta u+p(x)u=\alpha u+\phi \left( x,v\right) \ \ \ \ \text{in }\Omega ,\\-\Delta v+q(x)v=\beta v+\psi \left( x,u\right)  \ \ \ \  \text{in }\Omega ,%\end{array}\right.\end{equation*}with Dirichlet boundary condition, where $\Omega $ is a bounded open set of $\mathbb{R}^{N}$ $\left( N\geq 2\right) ,$ $\alpha ,\beta $ two real parameters, $\left( p(x),q(x)\right) \in \left( L^{\infty }\left( \Omega \right) \right) ^{2}$ and $p(x),q(x)\geq 0.$ using the Leray-Schauder's topological degree and under some suitable conditions for the non linearities $\phi $ and $\psi$, we show the existence of nontrivial solutions.

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The elliptic Sombor matrix of $G$, denoted by $A_{ESO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_i+d_j)\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases.Let the eigenvalues of the elliptic Sombor matrix $A_{ESO}(G)$ be $\rho_1\geq \rho_2\geq \ldots\geq \rho_n$ which are the roots of the elliptic Sombor characteristic polynomial $\prod_{i=1}^n (\rho-\rho_i)$. The elliptic Sombor energy ${E_{ESO}}$ of $G$ is the sum of absolute values of the eigenvalues of $A_{ESO}(G)$. In this paper, we compute the elliptic Sombor characteristic polynomial and the elliptic Sombor energy for some graph classes. We compute the elliptic Sombor energy of cubic graphs of order $10$ and as a consequence, we see that two $k$-regular graphs of the same order may have different elliptic Sombor energy.