Let $E$ be a Riesz space with lattice ordered norm $\|\cdot\|$. Amemiya proved that $E$ is complete under this norm if $E$ has weak Fatou property for monotone sequence (: $E$ is monotone complete) with respect to the norm $\|\cdot\|$. This is a generalization of the Riesz-Fisher's and the Nakano's theorem. In the cases of non normed Riesz space or non lattice ordered norm, this theorem is not true in general. We shall investigate in this paper a necessary and sufficient condition for Amemiya's theorem to be valid in a partially ordered normed linear space.

In this note, we shall give a description of ($\alpha ,\beta $)-derivations $\delta ,g$ and $h$ of a prime ring $R$ satisfying $\delta (x) = ag(x) + h(x)b$ for all $x\in U$, where $a$ and $b$ are some fixed noncentral elements of $R$ and $U$ a nonzero ideal of $R$. This result generalizes some known results.

Let $R$ be a prime ring with center ${\cal Z}$ and let $f(X_1,...,X_n)$ be a multilinear polynomial which is not central-valued on $R$. Suppose that $d$ and $\delta $ are derivations on $R$ such that $d(f(x_1,..., x_n))f(x_1,..., x_n)-f(x_1,...,x_n)$ $\delta (f(x_1,...,x_n)) \in {\cal Z}$ for all $x_1,...,x_n$ in some nonzero ideal of $R$. Then either $d =\delta = 0$ or $\delta =-d$ and $f(X_1,...,X_n)^2$ is central-valued on $R$, except when char $R = 2$ and $R$ satisfies the standard identity $s_4$ in 4 variables.

In the list of optimal regular graph designs (RGDs) provided by J. A. John and Mitchell, the E-optimality of some designs is still unknown. We have found the answers for some parameter sets including type of $(v,~v,~2),~(v,~v,~v-2)$. These E-optimal regular graph designs added by BIBDs are again E-optimal under certain domain.

Let X be a random variable with negative binomial density $$ f(x|\theta)=\displaystyle{\Gamma (x+r)\over \Gamma (x+1)\Gamma (r)}\theta^x(1-\theta)^r, $$ where $x=0, 1, 2, \cdots , 0 \lt \theta \lt 1,~r \gt 0$. For the hypothesis testing problem $$ H_0 : \theta \leq \theta_0~~~~{\rm versus}~~~~H_1 : \theta \gt \theta_0 $$ based on observing X$=x$, where $\theta_0$ is specified, we consider it as an estimation problem within a decision-theoretic framework. We prove the admissibility of estimator $p(x)= P_{\theta_0}(X \geq x)$, the $p$-value, for estimating the accuracy of the test, $1_{(0,\theta_0)}(\theta)$, under the squared error loss.

A particular type of branching processes in varying environments is considered. It is assumed that all individuals of the same generation produce, given that the preceding generation is not extinct, randomly and independently of the past generations the same number of children. We show that the number of children in the nth generation normed by its expectation converges almost surely to a limit whose expectation is 0 or 1. We give a sufficient condition for convergence in quadratic mean to a limit whose mean is one. A nonclassical norming sequence of constants is defined so that the almost sure limit is finite greater than zero with probability 1. We also show, under certain circumstances, that the almost sure limit has infinite mean.

This paper is concerned with generation theorems for exponentially equicontinuous $n$-times integrated $C$-semigroups of linear operators on a sequentially complete locally convex space (SCLCS). The generator of a nondegenerate $n$-times integrated $C$-semigroup is characterized. The proofs will base on a SCLCS-version of the Widder-Arendt theorem about the Laplace transforms of Lipschitz continuous functions, and on some properties of a $C$-pseudoresolvent. We also discuss the existence and uniqueness of solutions of the abstract Cauchy problem: $u'=Au+f,~u(0)=x$, for $x\in C(D(A^{n+1}))$ and suitable function $f$.

We first prove that for any graph $G$ with a positive vertex weight function $w$, there exists a graph $H$ with a positive weight function $w'$ such that $w(v)=w'(v)$ for all vertices $v$ in $G$ and whose $w'$-median is $G$. This is a generalization of a previous result for the case in which all weights are 1. The second result is that for any $n$-tournament $T$ without transmitters, there exists an integer $m\leq 2n-1$ and an $m$-tournament $T'$ whose kings are exactly the vertices of $T$. This improves upon a previous result for $m\leq 2n$.