Open Journal of Mathematical Sciences (OMS)

Let \(k>0\) an integer. \(F\), \(\tau \), \(N\), \(N_{k}\), \(N_{k}^{(2)}\) and \(A\) denote the classes of finite, torsion, nilpotent, nilpotent of class at most \(k\), group which every two generator subgroup is \(N_{k}\) and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively is \(FA\)-group). Secondly, we prove that a finitely generated \(\tau N\)-group (respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)\) (respectively \(((FN_{k})F,\infty)\)) is a \(\tau N_{k}^{(2)}\)-group (respectively \(FN_{k}^{(2)}\)-group). Thirdly we prove that a finitely generated \(\tau N\)-group ( respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)^{\ast}\) (respectively \(((FN_{k})F,\infty)^{\ast}\)) is a \(\tau N_{c}\)-group (respectively \(FN_{c}\)-group) for certain integer \(c\) and we extend this results to the class of \(NF\)-groups.

The generalized hierarchical product of graphs was introduced by L. Barrière et al. in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.

Most of molecular structures have symmetrical characteristics. It inspires us to calculate the topological indices by means of group theory. In this paper, we present the formulations for computing the several distance-based topological indices using group theory. We solve some examples as applications of our results.

The main goal of this article is to study the oscillation criteria of the second-order neutral differential equations on time scales. We give several theorems and related examples to illustrate the applicability of these theorems. Our results extend some recent work in the literature.

Locally harmonious coloring is a relax version of standard harmonious coloring which only needs that the color pairs for adjacent edges are different. In this remark, we introduce the concept of fractional locally harmonious coloring, and present some basic facts for this coloring.

In the present article, a time fractional diffusion problem is formulated with special boundary conditions, specifically the nonlocal boundary conditions. This new problem is then solved by utilizing the Laplace transform method coupled to the well-known Adomian decomposition method after employing the modified version of Beilin’s lemma featuring fractional derivative in time. The Caputo fractional derivative is used. Some test problems are included.

In this paper, we introduce and study the classes \(S_{n,\mu}(\gamma,\alpha,\beta,\) \(\lambda,\nu,\varrho,\mho)\) and \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) of functions \(f\in A(n)\) with \((\mu)z(D^{\mho+2}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{‘} \) \(+(1-\mu)z(D^{\mho+1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{‘}\neq0\), where \(\nu>0,\varrho,\omega,\lambda,\alpha,\mu \geq0, \mho\in N_{0}, z\in U\) and \(D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z):A(n)\longrightarrow A(n),\) is the linear differential operator, newly defined as
\( D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=z-\sum_{k=n}^{\infty}\left( \dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho} a_{k+1}z^{k+1}. \)
Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion relation for the functions included in the classes \(S_{n,\mu} (\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) are given.

This article presents, effects of fractional order derivative and magnetic field on double convection flow of viscous fluid over a moving vertical plate with constant temperature and general concentration. The model is fractionalized by using Caputo-Fabrizio derivative operator. Closed form solutions of the fluid velocity, concentration and temperature are obtained by means of the Laplace transform. Numerical computations and graphical illustrations are used in order to study the effects of the Caputo-Fabrizio time-fractional parameter , magnetic parameter , Prandtl and Grashof numbers on velocity field.

We shall present new oscillation criteria of second order nonlinear difference equations with a non-positive neutral term of the form \(\Delta(a(t)(\Delta(x(t)-p(t)x(t-k)))^{\gamma})+q(t)x^{\beta}(t+1-m)=0,\) with positive coefficients. Examples are given to illustrate the main results.

In this work we develop the weighted square integral estimates for the second derivatives of weak subsolution of forth order Laplace equation. It is natural generalization of inequalities develop for the Superharmonic functions in [1].