We shall give a lower estimate of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Ya. Levin in a half plane.

We shall give a lower estimate of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Ya. Levin in a half plane.

In this paper, we study the extremal ranks and inertias of the Hermitian matrix expression $$ f(X,Y)=C_{4}-B_{4}Y-(B_{4}Y)^{*}-A_{4}XA_{4}^{*},$$ where $C_{4}$ is Hermitian, $*$ denotes the conjugate transpose, $X$ and $Y$ satisfy the following consistent system of matrix equations $A_{3}Y=C_{3}, A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2},X=X^{*}.$ As consequences, we get the necessary and sufficient conditions for the above expression $f(X,Y)$ to be (semi) positive, (semi) negative. The relations between the Hermitian part of the solution to the matrix equation $A_{3}Y=C_{3}$ and the Hermitian solution to the system of matrix equations $A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2}$ are also characterized. Moreover, we give the necessary and sufficient conditions for the solvability to the following system of matrix equations $A_{3}Y=C_{3},A_{1}X=C_{1},XB_{1}=D_{1}, A_{2}XA_{2}^{*}=C_{2},X=X^{*}, B_{4}Y+(B_{4}Y)^{*}+A_{4}XA_{4}^{*}=C_{4} $ and provide an expression of the general solution to this system when it is solvable.

In the present paper, we show the existence of a coupled fixed point for a non-decreasing mapping in partially ordered complete metric space using a partial order induced by an appropriate function $phi$. We also define the concept of weakly related mappings on an ordered space. Moreover common coupled fixed points for two and three weakly related mappings are also proved in the same space.

In this work, an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$ and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$ where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$, $mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$, $mathcal{G}$ is a linear self-conjugate involution operator and $Bin textsf{R}^{rtimes s}$. Numerical examples are given to verify the efficiency of the constructed method.

Suppose $G$ is a split connected reductive orthogonal or symplectic group over an infinite field $F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is the Lie algebra of the unipotent radical $N.$ Under the adjoint action of its stabilizer in $M,$ every maximal prehomogeneous subspaces of $frak{n}$ is determined.

Second derivative general linear methods (SGLMs) as an extension of general linear methods (GLMs) have been introduced to improve the stability and accuracy properties of GLMs. The coefficients of SGLMs are given by six matrices, instead of four matrices for GLMs, which are obtained by solving nonlinear systems of order and usually Runge--Kutta stability conditions. In this paper, we introduce a technique for construction of an special case of SGLMs which decreases the complexity of finding coefficients matrices.

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection methods onto matrix Krylov subspaces. Numerical examples are presented.

A consistency criteria is given for a certain class of finite positive measures on the surfaces of the finite dimensional unit balls in a real separable Hilbert space. It is proved, through a Kolmogorov type existence theorem, that the class induces a unique positive measure on the surface of the unit ball in the Hilbert space. As an application, this will naturally accomplish the work of Kanter (1973) on the existence and uniqueness of the spectral measures of finite dimensional stable random vectors to the infinite dimensional ones. The approach presented here is direct and different from the functional analysis approach in Kuelbs (1973), Linde (1983) and the indirect approach of Tortrat (1976) and Dettweiler (1976).

For a given measure space $(X,{mathscr B},mu)$ we construct all measure spaces $(Y,{mathscr C},lambda)$ in which $(X,{mathscr B},mu)$ is embeddable. The construction is modeled on the ultrafilter construction of the Stone--v{C}ech compactification of a completely regular topological space. Under certain conditions the construction simplifies. Examples are given when this simplification occurs.

Let $I$ be an ideal in a regular local ring $(R,n)$, we will find bounds on the first and the last Betti numbers of $(A,m)=(R/I,n/I)$. if $A$ is an Artinian ring of the embedding codimension $h$, $I$ has the initial degree $t$ and $mu(m^t)=1$, we call $A$ a {it $t-$extended stretched local ring}. This class of local rings is a natural generalization of the class of stretched local rings studied by Sally, Elias and Valla. For a $t-$extended stretched local ring, we show that ${h+t-2choose t-1}-h+1leq tau(A)leq {h+t-2choose t-1}$ and $ {h+t-1choose t}-1 leq mu(I) leq {h+t-1choose t}$. Moreover $tau(A)$ reaches the upper bound if and only if $mu(I)$ is the maximum value. Using these results, we show when $beta_i(A)=beta_i(gr_m(A))$ for each $igeq 0$. Beside, we will investigate the rigid behavior of the Betti numbers of $A$ in the case that $I$ has initial degree $t$ and $mu(m^t)=2$. This class is a natural generalization of {it almost stretched local rings} again studied by Elias and Valla. Our research extends several results of two papers by Rossi, Elias and Valla.

We propose to use a mathematical method based on stochastic comparisons of Markov chains in order to derive performance indice bounds. The main goal of this paper is to investigate various monotonicity properties of a single server retrial queue with first-come-first-served (FCFS) orbit and general retrial times using the stochastic ordering techniques.

The purpose of this paper is to establish some coupled coincidence point theorems for mappings having a mixed $g$-monotone property in partially ordered metric spaces. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend several well-known results in the literature.

We establish some relative volume comparison theorems for extremal volume forms of Finsler manifolds under suitable curvature bounds. As their applications, we obtain some results on curvature and topology of Finsler manifolds. Our results remove the usual assumption on S-curvature that is needed in the literature.

Based on an eigenvalue analysis, a new proof for the sufficient descent property of the modified Polak-Ribière-Polyak conjugate gradient method proposed by Yu et al. is presented.

In this paper some properties of weak regular injectivity for $S$-posets, where $S$ is a pomonoid, are studied. The behaviour of different kinds of weak regular injectivity with products, coproducts and direct sums is considered. Also, some characterizations of pomonoids over which all $S$-posets are of some kind of weakly regular injective are obtained. Further, we give some Baer conditions which state the relation among some kinds of weak regular injectivity.

In this paper, strong convergence theorems of Ishikawa type implicit iteration process with errors for a finite family of asymptotically nonexpansive in the intermediate sense and asymptotically quasi-pseudocontractive type mappings in normed linear spaces are established by using a new analytical method, which essentially improve and extend some recent results obtained by Yang [Convergence theorems of implicit iteration process for asymptotically pseudocontractive mappings, Bulletin of the Iranian Mathematical Society, Available Online from 12 April 2011] and others.

In this paper, we give a complete proof of Theorem 4.1(ii) and a new elementary proof of Theorem 4.1(i) in [Li and Shen, On the intersection of the normalizers of the derived subgroups of all subgroups of a finite group, J. Algebra, 323 (2010) 1349--1357]. In addition, we also give a generalization of Baer's Theorem.