In this paper, we deal with meromorphic functions and its $k$th derivative. We consider a set of roots of unity, which shares the functions of the form ${(f^{n}P(f))}^{(k)}$ and ${(g^{n}P(g))}^{(k)}$ and obtain uniqueness results which improves the results of V. H. An and H. H. Khoai. Also, by introducing the concept of weighted sharing, we obtain analogous results from these authors.

In this paper, we establish some new Hermite-Hadamard integral inequalities for $m$-convex functions via $(p,q)$-calculus. Also, some known results are recaptured as special cases from our results. At the end, a briefly conclusion is given.

In this paper we have discussed about second Hankel determinant of Ma-Minda starlike bi-univalent and Ma-Minda convex bi-univalent functions in the open unit disc $\mathrm{\Delta }$ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis involving the Fekete-Szegö parameter $\mathrm{\lambda }$.

The nonlinear conformable time-fractional modified Camassa-Holm (MCH) equation plays an important role in physics. It is an interesting model to define change waves with weak nonlinearity. The aim of this study is to present the new exact solutions of conformable time-fractional MCH equation. For this purpose, an effective method which is the Improved Bernoulli Sub-Equation Function Method (IBSEFM) has been used. The 2D and 3D graphs and contour surfaces acquired from the values of the solutions are plotted by the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional partial differential equations arising in mathematical physics.

Using projection (submersion) of the cotangent bundle T*M over a manifold M, we define a semi-tensor (pull-back) bundle tM of type (p,q). The purpose of this paper is to investigate horizontal lifts of vector fields in a special class of semi-tensor (pull-back) bundle tM of type (p,q).

The aim of this paper is to establish some new inequalities of Hermite-Hadamard and Fejér type using generalized $h-$convex and generalized $(h_1,h_2)$–convex functions on the co-ordinates. Some known results are recaptured from our results as special cases.

In this paper, we investigate the uniqueness of meromorphic functions sharing a set with counting multiplicity and also with weight 1 in an angular domain.

In this paper, we introduce $C^{*}$-algebra valued modular $S$-metric spaces. Some common fixed point results for $S$-weak operator pair on $C^{*}$-algebra valued modular $S$-metric spaces is given with suitable examples. Stability of the fixed point and application for a system of integral equations are also discussed here.

In this work, we use the Banach fixed point theorem to establish new results on the stability and asymptotic stability of the zero solution for a neutral integro-dynamic equation on time scales. The results obtained here extend those due to Yankson. We end by giving two examples to illustrate our work.

The purpose of this study is to present a new modification of finite difference method (FDM) for approximating the solution of the two-interval boundary value problems for second order differential equations, whose main feature is the nature of the imposed conditions. Namely, the investigated problems contains not only boundary conditions at the points of the considered interval, but also an additional conditions at one interior point of interaction, so-called transmission conditions. Naturally, the analysis of two-interval boundary-value problems is more complicated and it is not clear how to extend the classical FDM to such type problems. The proposed modification of FDM tested on two model problems with known exact solutions. The obtained result are illustrate the applicability and efficiency of our own algoritm, which can be readily extended to all many-interval problems.

Using holomorphic extensions of functions, we introduce so-called deformed lifts in the tangent bundle of two order being defined as the set of all higher order 2-jets and investigate some properties of these lifts.

In this research, we intend to show that the nonlocal fractional Cauchy problem $D^{\alpha }u(t)=A(t)u(t)+f(t,u(t))$, $t\in J=[0,1]$ with integral initial condition $u(0)={\displaystyle {\int }_0^{1}g(s,u(s))ds}$, in the Banach space $X$, where $A$ is a generator of $\alpha$-resolvent operator function ${\{T(t)\}}_{t\ge 0}$ and $f$, $g$ are given functions satisfying some assumptions, has an almost periodic solution.

Schur proved that if $G$ is a group such that $G/Z(G)$ is finite, then $G^{\prime }$ is finite. Several authors studied the converse of Schur’s theorem. In this paper, we extend the converse of Schur’s theorem to a pair of groups. Also, we determine the structure of a pairs of groups under some conditions.

In this paper we aim to establish some results depending on the comparative growth properties of composite transcendental entire or meromorphic functions and some special type of differential polynomials generated by one of the factors on the basis of generalized order $\left(\alpha ,\beta \right)$ and generalized lower order $\left(\alpha ,\beta \right)$ where $\alpha$ and $\beta$ are continuous non-negative functions defined on $\left(-\infty ,+\infty \right)$.

We consider a coupled viscoelastic plate nonlinear equations with degenerate damping terms on a bounded domain in $R^n$. We establish the global existence results under different conditions.

In 1940, P. Hall introduced the concept of isoclinism of groups. Heidarian et al. studied the concept of $n$-isoclinism of pairs of groups. In this paper, we investigate the concept of $n$-isoclinism between two pairs of groups and give some relations between the notions of $n$-isoclinism of groups and $n$-isoclinism of pairs of groups.

Let ${\mathbb{F}}_{3^d}$ is the finite field of order $3^d$ with $d$ be a positive integer, we consider $A_4:={\mathbb{F}}_{3^d}[\epsilon ]={\mathbb{F}}_{3^d}[X]/(X^4)$ is a finite quotient ring, where ${\epsilon }^4=0$. In this paper, we will show an example of encryption and decryption. Firstly, we will present the elliptic curve over this ring. In addition, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. Precisely, we give a numerical example of cryptography (encryption and decryption) by using two methods with a secret key.

Let $R$ be an associative ring with identity and $M$ be a left $R$-module. In this paper, we introduce $M$-Gorenstein injective modules as a generalization of Gorenstein injective modules. We verify some properties of $M$-Gorenstein injective modules analogous to those holding for Gorenstein injective modules. There is an interesting theorem in classical homological algebra which asserts that $R$ is a Noetherian ring if and only if the class of injective modules over $R$ is closed under arbitrary direct sum. Our goal in this paper is to investigate the $M$-Gorenstein injective counterpart of this fact. If the class of $M$-Gorenstein injective modules over $R$ is closed under arbitrary direct sum, then $R$ will be a Noetherian ring. Also, it has been proved that in the special case $M=R$, when $R$ is a commutative Noetherian ring with a dualizing complex, then the class of $R$-Gorenstein injective modules is closed under arbitrary direct sum. In the main theorem of this paper, we prove the general case of this result. More precisely, we show that for any left $R$-module $M$ over a Noetherian ring $R$ in which every $R$-module has finite $M$-Gorenstein injective dimension, the class of $M$-Gorenstein injective modules is closed under arbitrary direct sum.