A fractional Cattaneo model from the generalized Cattaneo model with two fractional derivatives of different orders is considered for studying the thermoelastic response for a multilayer elliptic ring membrane with source function. The solution is obtained by applying an integral transform technique analogous to Vodicka's approach considering series expansion functions in terms of an eigenfunction to the generalized fractional Cattaneo-type heat conduction equation within an elliptic coordinates system. The analytical expressions of displacement and stress components employing Airy's stress function approach are investigated. The results are obtained as a series solution in terms of Mathieu functions and hold convergence test. The effects of fractional parameters on the temperature fields and their thermal stresses are also discussed. The findings are depicted graphically for different kinds of surface temperature gradients, and it is distinguished that the higher the fractional-order parameter, the higher the thermal response. Lastly, the generalized theory of thermoelasticity predicts an instantaneous response, but the fractional theory, which is currently under consideration, predicts a delayed response to physical stimuli, which is something that can be seen occurring in nature. This delayed response can be explained by the fact that fractional theories are currently being considered. This gives credibility to the motivation behind this topic of study in the research.

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.

Let G be a simple graph. The elliptic Sombor index of G is defined as $$    ESO(G) = \sum_{uv} \left(d_{u}+ d_{v} \right)\sqrt{d^{2}_{u}+d^{2}_{v}}, $$  where d_{u} denotes the degree of the vertex u, and the sum runs over the set of edges of G. In this paper we solve the extremal value problem of ESO over the set of (connected) chemical graphs and over the set of chemical trees, with equal number of vertices.

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.

In this paper, we consider the boundary value problem of a quasilinear elliptic system in degenerate form with data belongs to the dual of Sobolev Spaces. The existence result is proved by means of Young measures and mild monotonicity assumptions.