American Institute of Mathematical Sciences

Let \begin{document}$ \mathcal{G}(\mathcal{X}) $\end{document} and \begin{document}$ \mathcal{G}(\mathcal{Y}) $\end{document} be Gorenstein subcategories induced by an admissible balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document} in an abelian category \begin{document}$ \mathcal{A} $\end{document}. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of \begin{document}$ \mathcal{A} $\end{document} induced by the balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document}. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring \begin{document}$ R $\end{document}.

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.