In this paper, we first survery some Pompeiu-type theorems in $\Bbb R^n$. As consequences of some of these results, we obtain some Morera-type theorems in one and several complex variables. We also use the heat kernel of the sub-Laplacian to obtain a Pompeiu-type theorem for $L^{p,q}$ functions in the Heisenberg group.

Let $T$ be a $C_0$ contraction. Using the Jordan model, we prove that $T$ is quasisimilar to an irreducible operator if and only if $T$ is not quadratic and $T-\lambda I$ is not finite-rank for any complex number $\lambda$.

We show that a class of abstract differential operators generates fractionally integrated cosine functions, and give an application to partial differential operators on many function spaces.

This paper investigates the group testing problem in graphs as follows. Given a graph $G=(V,E)$, determine the minimum number $t(G)$ such that $t(G)$ tests are sufficient to identify an unknown edge $e$ with each test specifies a subset $X\subseteq V$ and answers whether the unknown edge $e$ is in $G[X]$ or not. Damaschke proved that $\lceil\log_2 e(G)\rceil \le t(G) \le \lceil\log_2 e(G)\rceil +1$ for any graph $G,$ where $e(G)$ is the number of edges of $G$. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. This paper verifies the conjecture for bipartite graphs $G$ with $e(G)\le 2^4$ or $2^{k-1} \lt e(G) \le 2^{k-1} + 2^{k-3}+2^{k-6}+19 \cdot 2^{\frac{k-7}{2}}-1$ for $k \ge 5$.

In this paper, two unidirectional binary $n$-cubes, namely, $Q_1(n)$ and $Q_2(n)$, proposed as high-speed networking schemes by Chou and Du, are studied. We show that the smallest possible length for any maximum fault-tolerant container from $a$ to $b$ is at most $n+2$ whether $a$ and $b$ are in $Q_1(n)$ or in $Q_2(n)$. Furthermore,we prove that the wide-diameters of $Q_1(n)$ and $Q_2(n)$ are equal to $n+2$. At last, we show that a conjecture proposed by Jwo and Tuan is true.

Let $T(\cdot)$ be a bounded $C_0$-semigroup on the Banach space $X$, with generator $A$. It is shown that the denseness of range $A$ is necessary and sufficient for the semigroup's mean stability with respect to suitable weights. Analogous results are valid for power bounded operators, tensor product semigroups, and cosine operator functions.

The aim of this paper is to investigate links of the recently introduced B-bounded semigroups [5] with other generalizations of semigroups, like C-regularized semigroups and C-existence families.

It is known that the determination of the profile for arbitrary graphs is NP-complete. The {\em composition } of two graphs $G$ and $H$ is the graph with vertex set $V(G)\times V(H)$ and $(u_1,v_1)$ is adjacent to $(u_2,v_2)$ if either $u_1$ is adjacent to $u_2$ in $G$ or $u_1=u_2$ and $v_1$ is adjacent to $v_2$ in $H$. The exact values of the profile of the composition of a path with other graphs, a cycle with other graphs, a complete graph with other graphs and a complete bipartite graph with other graphs are established.

The usual models of the chemostat assume that the competition is purely exploitative, the competition is only through the consumption of the nutrient. However, it is known that microorganisms can produce toxins against its competitors. In this work, we consider a model of competition in the chemostat between plasmid-bearing, plasmid-free organisms for a single nutrient where plasmid-bearing organism can produce a toxin (allelopathic agent) against the plasmid-free organism at some cost to its reproductive abilities. We give a characterization of the outcome of this competition in terms of the relevant parameters in hyperbolic cases. The global asymptotic behavior of the solutions is proved by using the perturbation of a globally stable steady state for a sufficiently small plasmid loss rate.