In this literature survey, we analyze on the uniqueness problems of meromoprhic functions with its reduced linear $c$-shift operator $L_{c}^{r}f$ taking into consideration of partial sharing and using the notion of weighted sharing of sets with complex constants $CM$.

The goal of this paper is to demonstrate that incorporating the fractional form of the exponential function allows for an extension of special polynomials, such as the Hermite-Kampé de Fériet (H-KdF) polynomials with two variables, and the Mittag-Leffler-Gould-Hopper polynomials. This extension can be achieved by defining expansions using fractional powers, while still preserving the essential properties of the corresponding polynomial versions. Specifically, a fractional version of the classical Hermite polynomials is derived. The main properties are based on expansions in integer powers.

In this paper, we consider $f$ be a nonconstant entire function and $L(f)$ be a small meromorphic functions of $f$. We deal with unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1. The example is exhibited to validate certain claims of the main result.

In this paper, we introduce the concept of $k-$directional Darboux curves associated with a given spacelike curve $\alpha $ lying on a lightlike surface in Minkowski $3$-space $\mathbb{E}_{1}^{3}$. These $k-$directional Darboux curves are curves generated by specific vector fields derived from the Darboux frame along the curve. We explore the connections between these associated curves and their curvatures, and we also propose a new method for classifying special curves, such as helices and slant helices, using these ideas. In addition, we give the related examples and their graphics.

In this paper, the structure of the second relative homology and the relative stem cover of the direct sum of two pairs of Leibniz algebras are determined by means of the non-abelian tensor product of Leibniz algebras. We also characterize all pairs of finite-dimensional nilpotent Leibniz algebras such that $\dim({\frak n})=n$, $\dim({\frak g}/{\frak n})=m$ and $\dim(HL_2({\frak g},{\frak n}))=n(n+2m)-3$.

In this study, we consider the one-dimensional system of piezoelectric beams with thermal and magnetic effects in the presence of a nonlinear damping term and with infinite history acting on the mechanical equation. Under suitable assumptions on the nonlinear damping term and the kernel, we demonstrate the existence and uniqueness of the solution applying the semigroup theory more precisely by Hille-Yosida theorem. And by introducing a suitable Lyapunov functional and using some properties of convex functions, we show the general stability for the system which the exponential and polynomial decays are merely special cases. Furthermore, our results are independent of any relationship between system parameters.