This paper investigates the normality problem of the connected 13-valent symmetric Cayley graphs $\Ga$ of finite nonabelian simple groups $G$, where the vertex stabilizer $\A_v$ is soluble for $\A=\Aut\Ga$ and $v\in V\Ga$. It is shown that $\Ga$ is either normal or $G= \A_n$ with $n = 12, 38, 116, 207, 311, 935$, or $1871$. Further, 13-valent symmetric non-normal Cayley graphs of $\A_{38}$, $\A_{116}$ and $\A_{207}$ are constructed. This provides some more examples of non-normal 13-valent symmetric Cayley graphs of finite nonabelian simple groups since such graph (of valency 13) was first constructed by Fang, Ma and Wang in [6].

A new generalized differential operator $\mathcal{F}_{\ell ,\alpha ,\beta ,\lambda }^{m,p,q,s}(\alpha _{1})$ $(\alpha ,\beta ,\lambda \in \mathbb{C},\ell \in \mathbb{N}$ and $m\in \mathbb{N}_{0})$ for functions of the form $f(z)=z^{-p}+\sum _{k=n}^{\infty }a_{k}z^{k},$ which are meromorphic in the punctured unit disc $\nabla ^{\ast }=\{z\in \mathbb{C}:0<|z|<1\}=\nabla \setminus \{0\}$ was introduced by Ali and Bulboaca [1]. The primary objective of this study is to provide an investigation argument results of multivalent meromorphic functions defined by this new generalized differential operator.

In this paper, we intended to extend some well-known results for the torsion-theoretical generalization of supplementing modules. A module that has a $\tau$-supplement in every extension is called $\tau$-supplementing modules, where $\tau$ is an idempotent radical. We study these modules by generalizing several results both on supplementing modules and injective modules. We study generalizations and characterizations of $SI$-ring, semiartinian rings and perfect rings by $\tau$-supplementing modules.

This paper mainly consists of two parts. Firstly, for any transcendent entire function $f$ with a Borel exceptional value, we investigate the uniqueness of $f$ concerning it's differential and difference. Secondly, we study the 4 $IM$ theorem of the modified third $Painlev\acute{e}$ transcendents. Some examples are used to illustrate our results.

Let $R$ be a commutative ring with unity. The total graph of commutative ring $R$, denoted by $T_{\Gamma}(R)$ is the graph with $R$ as the set of vertices and for $u,v \in R$, $u$ is adjacent to $v$ if and only if $u+v \in Z(R)$, where $Z(R)$ is the set of zero divisors of $R$. In this paper, we examine and determine the spectrum, the Laplacian spectrum, the energy and the Laplacian energy of $T_{\Gamma} (\mathbb{Z}_{m}\times\mathbb{Z}_{n})$ for $(m,n)=(2,q),(2^{k},q),(2^{k},2^{k}),(2^{k},2)$.

Let $D$ be a strict directed graph and $A(D)$ be the adjacency matrix of $D$. The directed graph $D$ is said to have the reciprocal eigenvalue property $(R)$ if $A(D) $ is nonsingular and $\frac{1}{\lambda}$ is an eigenvalue of $A(D) $ whenever $\lambda$ is an eigenvalue of $A(D)$. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then $D$ is said to have the strong reciprocal eigenvalue property $(SR)$. In this article, we construct some strict directed graphs with property $(SR)$, and some necessary conditions for a directed graph satisfying property $(R)$ are obtained.

This paper extends the method of successive approximations to an $\rm {n}$th-order initial value problem without converting it to a first-order system. We obtain a bound in a closed form for the difference between two successive iterates. Finally, an error estimate for the solutions has been calculated and it is shown to be increasingly accurate as n increases. Suitable illustrations to support the result are provided.

Let $R=\mathbb{F}_q+u\mathbb{F}_q+u^2\mathbb{F}_q$, where $q$ is power of a prime and $u^3=0$. In this paper, we study triple cyclic codes over $R.$ These codes can be identified as $R[x]$-submodules of the $R[x]$-module $R_{r,s,t}=R[x]/\langle x^r-1 \rangle \times R[x]/\langle x^s-1 \rangle \times R[x]/\langle x^t-1 \rangle.$ The generator polynomials and minimal generating sets of this family of codes as $R[x]$-submodule of $R_{r,s,t}$ are determined. We show that the dual of triple cyclic codes over $R$ is again triple cyclic codes over $R$. Further, we determine the relationship of generators between the triple cyclic codes and their duals.

The traditional study of reproducing systems involves the Gabor and wavelet systems. Gabor systems are generated by the action of translations and modulations on a single or finite family of functions in $ \ltr$ or $\ltrd$. The wavelet systems are reproduced by the actions of dilations and translations. Improving and extending these notions, it is natural to construct a systems combining the actions of translations, modulations and dilations on a single or finite family of functions in $ \ltr$ or $\ltrd$. This systems known as \textit{Wave Packet Systems}. In this paper after introducing these systems in details and reviewing the frameness of theses systems, we present a condition under which the wave packet system $ \{D_{a^{j}}T_{kb}E_{mc}\varphi\}_{j,m,k\in\mz}$ is a frame for $\ltr$ and we show that there is a large class of parameters and functions that cannot generate a Bessel sequence.

In this study, we define and describe the concept of a weakly 1-absorbing prime filter and that weakly 1-absorbing prime filter can only become a weakly 2-absorbing filter under certain conditions, which are explored. It is noted that the product of 1-absorbing prime filter to become weakly (1-absorbing prime filter). Finally, a discussion is made about the lattice epimorphic images and pre image of weakly (1-absorbing prime filter) in an ADL.

In this paper, the $\tau_q$-weak global dimension $\tau_q$-\cwd$(R)$ of a commutative ring $R$ is introduced. Rings with $\tau_q$-weak global dimension equal to $0$ are studied in terms of homologies, direct products, polynomial extensions and amalgamations. Besides, we investigate the $\tau_q$-weak global dimensions of polynomial rings.