Journal of Mathematical Extension
Volume:17 Issue: 3, Mar 2023

In this paper, some graph parameters of the zero-divisor graph $\Gamma(R)$ of a finite commutative ring $R$ for $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{2p}$ where $p>2$ a prime, are investigated. The graph $\Gamma(R)$ is a simple graph whose vertex set is the set of non-zero zero-divisors of a commutative ring $R$ with non-zero identity and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$.
In this paper, we study some of the topological indices such as graph energies, the Zagreb indices and the domination parameters of graphs $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{p^2} \big)$ and $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{2p}\big)$.

Fractional calculus is a branch of classical mathematics, which deals with the generalization of fractional order derivative and integral operator. Recently, a great deal of research has been carried out on the use of fractional calculus to study the phenomena associated with fractal structures and processes. Fractals have a fractional dimension and occur naturally in non-linear and imbalanced phenomena in various forms and contexts. In recent years, various types of derivatives and fractional and fractal calculus have been proposed by many scientists and have been extensively utilized. Measurements are localized in physical processes, and local fractional calculus is a useful tool for solving some type of physical and engineering problems. In this article, we applied the local fractional Yang-Laplace variational for solving the local fractional linear and nonlinear KdV equation on a Cantor set within local fractional derivative. we emphasize on the LFYLVM method which is a combination form of local fractional variational iteration method and Yang-Laplace transform. The non-differentiable exact and approximate solutions are obtained for kind of local fractional linear and nonlinear KdV equations. Most of the solutions obtained from this method are obtained in series form that converge rapidly in physical problems. Illustrative examples are included to demonstrate the high accuracy and convergence of this algorithm It is shown that the used method is an efficient and easy method to implement for linear and nonlinear problems arising in science and engineering.

In this paper, we first discuss the class of modified two-stage procedure for estimation of the autoregressive parameter in a first-order autoregressive model ( AR(1)). 
We prove the significant properties of the modified two-stage procedure, including asymptotic efficiency, asymptotic risk efficiency, and asymptotic consistency for the point and the interval estimation based on least-squares estimators.
Then, the introduced class is generalized to the p-order autoregressive model ( AR(p)) and is checked their asymptotic properties of which.
Also, we conduct comprehensive Monte Carlo simulation studies to test the properties of the proposed procedure in practice.
Finally, a real-time series is provided to investigate the applicability of the class of modified two-stage variables.

In this paper, we prove an equality for the case of twice-differentiable convex functions with respect to the conformable fractional integrals. With the help of this equality, we establish several Simpson-type inequalities for twice-differentiable convex functions by using conformable fractional integrals. Sundry significant inequalities are obtained by taking advantage of the convexity, the H\"{o}lder inequality, and the power mean inequality. By using the specific selection of our results, we give several new and well-known results in the literature.

‎An open subset of a space is said to be e-open if its closure is also open and if a space has a base consisting of e-open sets‎, ‎we call it an e-space‎. ‎In this paper we first introduce e-spaces and compare them with relative spaces such as extremally disconnected and zero-dimensional spaces‎. ‎Subspaces of e-spaces and product of e-spaces are investigated and we define the concept of e-compactness and characterize e-compact spaces via $e$-convergence of nets and filters.We introduce e-separation axioms $T_1^e-T_4^e$ and investigate the counterparts of results in the literature of topology concerning separation axioms‎. ‎It is shown that a space is a $T_3-e$-space if and only if it is zerodimensional and a space is a $T_4^e$-space if and only if it is strongly zerodimensional‎. ‎In contrast to extremally disconnected spaces whose product is not necessarily an extremally disconnected space‎, ‎we observe that any product of $e$-spaces is an $e$-space‎. ‎Also we see that the $e$-closure of a set need not be $e$-closed‎, ‎contrary to closure of a set which is closed‎.

Data Envelopment Analysis (DEA) can be applied to a net- work structure systems. Considering two-stage network, this paper im- proves a leader-follower structure. Likewise, it is not possible to reduce the production of undesirable outputs without any cost. The motiva- tion of this study is handling a fair treatment of the undesirable outputs in leader-follower structure. In the presence of the additional inputs in the second stage, this approach searches the proportional abatement in desirable and undesirable factors. Besides, it focuses on the less sacri- ce of the desirable products. Moreover, the theoretical contributions of the proposed model have been illustrated in 13 poultry industry and 25 power plants in Iran.