In this paper, we survey some multiplier theorems and their applications to estimates for wave equation in Hardy spaces $H^p(\Bbb R^n)$. We also prove $H^p$ boundedness for Calder\'on-Zygmund operators of type $\sigma$ which can be considered as a generalization of classical singular integral operators. Using these results, we discuss some recent progress in $H^p$ regularity properties of the solving operators for hyperbolic equations.

A-ergodic nets and A-regularized approximation processes of operators are introduced and their convergence theorems are discussed. There are strong convergence theorems, uniform convergence theorems, theorems on optimal convergence, and theorems on non-optimal convergence and its sharpness. The general results provide unified approaches to investigation of convergence rates of ergodic limits and approximation of various operator families. In particular, we shall deduce some results for an r-times integrated resolvent family for a Volterra integral equation. The latter contains integrated semigroups and integrated cosine functions as special cases.

We present an overview of the MGMRES and LAN/MGMRES iterative methods for solving large sparse linear systems.

For positive integers $k$ and $d\geq 2$, a $k$-$S(d, 1)$-$labelling$ of a graph $G$ is a function on the vertex set of $G$, $f:V(G)\to\{0,1,2,\cdots,k-1\}$, such that $$ |f(u)-f(v)|_k\geq\left\{\begin{array}{ll} d &\mbox{ if }\ d_G(u,v)=1;\\ 1 &\mbox{ if }\ d_G(u,v)=2,\end{array}\right. $$ where $|x|_k=\min\{|x|,k-|x|\}$ is the {\em circular difference} modulo $k$. In general, this kind of labelling is called the $S(d,1)$-$labelling$. The $\sigma_d$-number of $G$, $\sigma_d(G)$, is the minimum $k$ of a $k$-$S(d,1)$-labelling of $G$. If the labelling is required to be injective, then we have analogous $k$-$S'(d,1)$-$labelling$, $S'(d,1)$-$labelling$ and $\sigma _d^\prime(G)$. If the circular difference in the definition above is replaced by the absolute difference, then $f$ is an $L(d,1)$-labelling of $G$. The $span$ of an $L(d,1)$-labelling is the difference of the maximum and the minimum labels used. The $\lambda_d$-number of $G$, $\lambda_d(G)$, is defined as the minimum span among all $L(d,1)$-labellings of $G$. In this case, we have the corresponding $L'(d,1)$-labelling and $\lambda_d'(G)$ for the labelling with injective condition. We will first study the relation between $\lambda_d$ and $\sigma_d$ as well $\lambda_d^\prime$ and $\sigma _d^\prime$. Then we consider these parameters on cycles and trees. Finally, we study the join of graphs and the multipartite graphs.

We give some inequalities of capacity in Gaussian channel with or without feedback. The nonfeedback capacity $C_{n,Z}(P)$ and the feedback capacity $C_{n,FB,Z}(P)$ are both concave functions of $P$. Though it is shown that $C_{n,Z}(P)$ is a convex function of $Z$ in some sense, $C_{n,FB,Z}(P)$ is a convex-like function of $Z$.

In recent years, several workers demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of familiar second-order differential equations associated (for example) with Gauss, Legendre, Jacobi, Chebyshev, Coulomb, Whittaker, Euler, Hermite, and Weber equations. The main object of this paper is to show how some of the most recent contributions on this subject, involving the Weber equations and their various generalized forms, can be obtained by suitably applying a general theorem on particular solutions of a certain family of fractional differintegral equations.

In this paper, earlier works of the present authors and a method due to Anosov for solving certain intego-functional equations are combined to show that the independence of the sample mean $\bar{X}_n$ and the $Z_n$-statistic characterizes the normal population, when the random samples are iid from a population having a continuous density function on ${\Bbb R}$, and the sample size $n\geq 3$; obviously the sample standard deviation is a $Z_n$-statistic. Further, an important subclass of $Z_n$-statistic with the form of a linear combination $\sum^n_{i=1} a_iX_{(i)}$ of order statistics is found, where $a_1 \leq \cdots \leq a_n$, not all equal and $\sum^n_{i=1}a_i=0$, which includes Gini$^\prime$s mean difference and the sample range but not the sample standard deviation. Similar approach can be applied to prove that the independence of $\bar{X}_n$ and $Z_n/\bar{X}_n$ characterizes the gamma distribution; obviously the independence of sample mean and sample coefficient of variation characterizes the gamma distribution. The study of identifying $Z_n$ to more known statistics will be the future work.

We present two convergence theorems for the H1-integral.

Hardy-type inequalities are proved for n-dimensional Hermite and special Hermite expansions. Paley-type theorems for these expansions are also deduced.

In this paper, we prove that an $n\times n~(n\geq 3)$ complex matrix $T$ is similar to an irreducible matrix if and only if $T$ is not quadratic and rank $(T-\lambda I)\geq n/2$ for every complex number $\lambda$. As an application, we prove that: for any integers $n$ and $k$ with $ 3 \leq k \lt n$, there exists an $n \times n$ irreducible nilpotent matrix of index $k$. This answers a question posed by P. R. Halmos

In this paper, we study the stability of similarity solutions for the problem $f'''+Q(Aff''-(f')^{2})=\beta$, with $Q\gt 0$, $A\geq 1$ and $\beta \in \Bbb R$. The given problem was derived from the symmetric reduction of similarity transformations from the Navier-Stokes equation for the planar flows. By imposing additional eigenvalue problems, our numerical studies show that the resultant steady flows are unstable as $Q$ becomes large for various $A\lt 2$. Furthermore, our analytical result gives that the steady flows are stable for small $Q$, when $1\leq A\lt 2$, or for any $Q\gt 0$ when $A\geq 2$. Moreover, the existence of asymmetric flows for various $A\lt 2$ is also found numerically.