We present identities on alternating sums of powers of integers and sums of powers of integers, and results on these sums as polynomials including new proofs of known results.

The term “Rough Set”(RS) in mathematics describes a distinct method of handling ambiguity and uncertainty. It is significant in both pure and applied mathematics. RS allows us to eliminate redundant or irrelevant attributes and it remains effective in the fields of knowledge acquisition, industrial control, medical diagnosis, expert systems, and data mining. Algebraic structure is a key component of RS theory. Combining the RS with abstract algebra is one method for generalizing it. Several studies proposed and examined the concepts of RS with groups and rings in approximation spaces. In general, a homomorphism is a structure preserving map between two algebraic structures of the same type. This study establishes some theorems, discusses $\mathrm{\Gamma }$ homomorphism and epimorphism in the $\mathrm{\Gamma }$- ring structure, and also demonstrates some of its characteristics.

In this article, we investigate the problem of transcendental meromorphic functions that share weak weights with one of their polynomials. Let $F$ and $G$ represents a transcendental meromorphic function. If $f^{n}P\left(f\right)-a_1$ and ${\left(f^{n}P\left(f\right)\right)}^{\left(k\right)}-a_2$ share “(0,1)” and $n,k$ and $m$ represent three positive integers such that $n\ge k+m+1$ then ${\left(f^{n}P\left(f\right)\right)}^{\left(k\right)}\equiv f^{n}P\left(f\right)a_2{a}_1^{-1}$. Furthermore, if $a_1\equiv a_2$, then $\mathcal{F}\left(z\right)=c{\mathrm{e}}^{z\lambda /n}$, where $c$ and $\lambda$ are nonzero constants such that ${\lambda }^{(k+m)}=1$. This result supports the Conjecture advanced by I. Lahiri, S. Majumder.

We develop an approximate identity to study the convergence of cone-adapted continuous shearlet transform in $L^p$-sense, $1\le p\le \infty$, in higher dimensions. The point wise convergence and reconstruction formula also will be discussed.

The invariants of the modular and integral cohomology rings of $\mathrm{\mathbb{Z}}/p\times \mathrm{\mathbb{Z}}/p$ under the action of $SL(2,p)$, where $p$ is an odd prime, are described. These invariants are related to the well-known Dickson invariants and are an ingredient in recent results (see [16]) concerning the $Qd\left(p\right)=(\mathrm{\mathbb{Z}}/p{)}^2⋊SL(2,p)$ groups which arise in the study of a well-known conjecture by Benson and Carlson [4] relating to free actions on finite complexes homotopy equivalent to products of spheres.

A module $M$ is called co-Hopfian if every injective $R$-endomorphism of $M$ is an isomorphism. This article introduces the concept s-cH module which is a generalized co-Hopfian. If $M$ is an $R$-module and every injective $R$-endomorphism of $M$ has a strongly essential image, then $M$ is said to be s-cH. We provide some properties of these module and we provide an example of a module which is not s-cH. Also we provide an example of a module which is s-cH, but not co-Hopfian. We show that if an $R$-module $M$ has a descending chain on non strongly essential modules, then $M$ is s-cH. Also we investigate the behavior of s-cH modules under a direct product.

In this Manuscript, we introduce Wijsman’s $p$-lacunary summability of order $\alpha$, Wijsman’s lacunary statistical convergence of order $\alpha$, Hausdorff’s lacunary statistical convergence of order $\alpha$, and Wijsman’s statistical Cauchy for triple set sequences. We embark on a comprehensive exploration of these groundbreaking concepts, delving into their inherent properties, potential interconnections, and ties to established literary ideas. Our examination aims to bring clarity to the characteristics of these innovative notions, uncovering their potential associations. Additionally, we investigate the correlations between these recently introduced concepts and pre-existing notions in the literature.

In this paper we prove Schur convexity, Schur geometric convexity and Schur harmonic convexity of the Q-type means with respect to arithmetic mean and geometric mean for two variables .

In this article, we derive representations for the arctangent function and for several related functions in terms of infinite series of rational expressions. The computational advantages of such representations are discussed, along with the pedagogical benefits of covering such approaches in undergraduate courses.

One of the generalizations of the idea of rings is the idea of gamma rings. The current research reviews a number of established findings on the gamma rings. Many new examples are given to illustrate this structure. We specifically demonstrate the existence of isomorphism theorems for gamma rings. Finally, the gamma ring example is included in the well-known prime avoidance theorem for commutative rings.

We introduce the notion of a 2-absorbing strong primary fuzzy ideal of a lattice which is a generalization of primary fuzzy ideal. If $\mu$ is a 2-absorbing strong primary fuzzy ideal of a lattice, then $\mu \left(0\right)$ is not necessarily equal to 1. Hence we find condition under which a fuzzy ideal $\mu$ of $L$ will be a 2-absorbing strong primary fuzzy ideal when $\mu \left(0\right)\ne 1$ as well as $\mu \left(0\right)=1$. In fact, we have shown that if $\mu$ is a fuzzy ideal of a lattice and if $\mu \left(0\right)\ne 1$, then $\mu$ is a 2-absorbing strong primary fuzzy ideal of $L$ if and only if $\left|Im\right(\mu \left)\right|=2$ and ${\mu }_{*}=\{r\in L|\mu \left(r\right)=\mu \left(0\right)\}$ is a primary ideal of $L$.

Let $R$ be a $\mathrm{\Gamma }$-ring and $A$ be an $R_{\mathrm{\Gamma }}$-module. $A$ is called $F$-regular $R_{\mathrm{\Gamma }}$-module if every $R_{\mathrm{\Gamma }}$-submodule of $A$ is pure. The aim of this paper is to generalize results of $F$-regular $R$-modules and introduce theorems concerning $F$-regular $R_{\mathrm{\Gamma }}$-modules. We also explore the relationship between these modules and $R_{\mathrm{\Gamma }}$-submodules, which are a generalization of $F$-regular $R$-module semiprime submodule.