Let $G$ be a finite group and $x ∈ G.$ The nilpotentiser of $x$ in $G$ is defined to be the subset $Nil_G(x) =\{y∈ G :\langle x,y \rangle \ is\ nilpotent\}.$ $G$ is called an $\mathcal{N}$-group ($n$-group) if $Nil_G(x)$ is a subgroup (nilpotent subgroup) of $G$ for all $x ∈ G\setminus Z^∗(G)$ where $Z^∗(G)$ is the hypercenter of $G$. In the present paper, we determine finite $\mathcal{N}$-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite $n$-groups.

In this paper we study the non-relativistic and low Mach number limits of strong solutions to a full compressible MHD-$P1$ approximate model arising in radiation magnetohydrodynamics. We prove that, as the parameters go to zero, the solutions of the primitive system converge to that of the classical incompressible magnetohydrodynamic equations.

In this paper, we investigate global stability of complex-valued periodic solution of a delayed discontinuous neural networks. By employing discontinuous, nondecreasing and bounded properties of activation, we analyzed exponential stability of state trajectory and $L^1$−exponential convergence of output solution for complex-valued delayed networks. Meanwhile, we applied to complex-valued discontinuous neural networks with periodic coefficients. The new results depend on $M$−matrices of real and imaginary parts and hence can include ones of real-valued neural networks. An illustrative example is given to show the effectiveness of our theoretical results.

In using the structure-preserving algorithm (SDA) [Linear Algebra Appl., 2005, vol. 396, pp. 55–80] to solve a continuous-time algebraic Riccati equation, a parameter-dependent linear fractional transformation $z→(z−\gamma)/(z+\gamma)$ is first performed in order to bring all the eigenvalues of the associated Hamiltonian matrix in the open left half-plane into the open unit disk. The closer the eigenvalues are brought to the origin by the transformation via judiciously selected parameter $\gamma,$ the faster the convergence of the doubling iteration will be later on. As the first goal of this paper, we consider several common regions that contain the eigenvalues of interest and derive the best $\gamma$ so that the images of the regions under the transform are closest to the origin. For our second goal, we investigate the same problem arising in solving an $M$-matrix algebraic Riccati equation by the alternating-directional doubling algorithm (ADDA) [SIAM J. Matrix Anal. Appl., 2012, vol. 33, pp. 170–194] which uses the product of two linear fractional transformations $(z_1,z_2)→[(z_2−\gamma_2)/(z_2+\gamma_1)][(z_1−\gamma_1)/$$(z_1+\gamma_2)]$ that involves two parameters. Illustrative examples are presented to demonstrate the efficiency of our parameter selection strategies.

For the iteration solution of singular generalized saddle point problems, a fast shift-splitting iteration method based on shift-splitting technique and symmetric and skew-symmetric splitting with respect to the upper-left block of the system matrix is proposed in this paper. Semi-convergence of the proposed method is carefully studied for singular case, and the conditions guaranteeing the semi-convergence are derived. Numerical experiments of a class of linearized Navier-Stokes equations are implemented to demonstrate the feasibility and effectiveness of the proposed method.

In this paper, we study a new two-step iteration scheme of mixed type for two total asymptotically nonexpansive self mappings and two total asymptotically nonexpansive non-self mappings and establish some weak convergence theorems in the framework of uniformly convex Banach spaces. Our results extend and generalize several results from the current existing literature.