Open Access Original Research Article

Xiaobin Yao

We study the asymptotic behavior of solutions to the non-autonomous stochastic extensible plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence of a random attractor.

Open Access Original Research Article

Liangying Miao

In this paper, we consider the following fully parabolic two-competing-species chemotaxis model

$$

\left\{\begin{array}{ll}

\displaystyle u_{1t}=\Delta{u_{1}}-\chi \nabla\cdot(u_{1}\nabla{v_{1}})+\mu_{1}u_{1}(1-u_{1}-e_{1}u_{2}),&x\in\Omega,~ t>0,\\

\displaystyle u_{2t}=\Delta{u_{2}}-\xi\nabla\cdot(u_{2}\nabla{v_{2}})+\mu_{2}u_{2}(1-e_{2}u_{1}-u_{2}),&x\in\Omega,~t>0,\\

\displaystyle v_{1t}=\Delta{v_{1}}+u_{1}- v_{1},&x\in\Omega,~ t>0, \\

\displaystyle v_{2t}=\Delta{v_{2}}+u_{2}- v_{2},&x\in\Omega,~ t>0

\end{array}\right.

$$

under homogeneous Neumann boundary conditions, where Ω ⊂ ℝ^{n } (n≥3) is a convex bounded domain with smooth boundary. Relying on a comparison principle, we show that the problem possesses a unique

global bounded solution if ^{μ}_{1} and ^{μ}_{2} are large enough.

Open Access Original Research Article

Thoraya Abdelwhab, Xiaoyan Yang

In this paper, we study modules whose endomorphism rings are Baer, which we call endoBaer modules. We provide some characterizations of endoBaer modules and investigate their properties. Some classes of rings R are characterized in terms of endoBaer R-modules. It is shown that a direct summand of an endoBaer modules inherits the property, while a direct sum of endoBaer modules does not. Necessary and sucient conditions for a nite direct sum of endoBaer modules to be an endoBaer module are provided.

Open Access Original Research Article

Yaremenko Mikola (Nikolay) Ivanovich

The hypersurface y^{n–1} in y^{n} space is studied for this piece of work. We established the correlation between tensors of hypersurface y^{n–1} and tensors of embedding space y^{n} . The second non-symmetrical tensor of hypersurface has been introduced, which have been obtained from the analog of Peterson-Codazzi equation in nonsymmetrical case.Also we have introduced the tensor that is associated with square of angle between normal and adjacent normal and it is represented in terms of metric and second tensors of hypersurface. The geodesics on hypersurface have been studied, and nontrivial example of geodesics on hypersurface with torsion and Euclid metric was constructed.

Open Access Review Article

Thoraya Abdelwhab, Xiaoyan Yang

In this paper, we study modules whose endomorphism rings are right Rickart (or right p.p.) rings, which we call R-endoRickart modules. We provide some characterizations of R-endoRickart modules. Some classes of rings are characterized in terms of R-endoRickart modules. We prove that an R-endoRickart module with no innite set of nonzero orthogonal idempotents in its endomorphism ring is precisely an endoBaer module. We show that a direct summand of an R-endoRickart modules inherits the property, while a direct sum of R-endoRickart modules does not. Necessary and sucient conditions for a nite direct sum of R-endoRickart modules to be an R-endoRickart module are provided.