Recent results concerning the existence of a common nontrivial invariant subspace and reflexivity for families of commuting linear bounded Hilbert space operators will be presented; starting with the families of linear transformations on finite dimensional space, through families of isometries, jointly quasinormal operators and spherical isometries, finishing with N-tuples of contractions with dominating spectra. This paper is based On the notes for the series of lectures given in the Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China in November and December of 1995.

In virtue of the notion of likelihood ratio the limit properties of the sequences of dependent nonnegative integer-valued random variables are studied, and a class of small deviation theorems are obtained. In the proof an approach of applying the tool of generating function together with the method of splitting intervals to the study of the strong laws is proposed.

We introduce the Wiener space and then consider wavelets which are not necessarily compactly supported but have a decay condition at infinity. Under the Wiener condition, several scaling functions and their dual functions have the same rate of decay at infinity. Furthermore, multiwavelets and their bi-orthogonal multiwavelets have the same rate of decay at infinity and the same number of vanishing moments.

We prove theorems on the existence of fixed points and the structure of fixed point sets for asymptotic 1-set contraction mappings T on certain subsets of Banach spaces by assuming some condition on T. We also prove some fixed point theorems for a sum of asymptotic 1-set contraction and compact (strongly continuous) mappings in real Banach spaces (reflexive real Banach spaces).

In this paper a general theorem on $|\overline{N}, p_n; \delta |_k$ summability factors, which generalizes a theorem of Bor [3] on $|\overline{N}, p_n|_k$ summability factors, is proved.

Let $R$ be a prime ring with extended centroid $C$ and $\rho$ a nonzero right ideal of $R$. In this paper we investigate the derivations $\delta$, $d$ on $R$ such that $[\delta (x), d(x)] \in C$ for all $x \in \rho$. As an application, we prove that any centralizing additive mapping $f$ on $\rho$ must be of the form $f(x)=\lambda x+\mu (x)$ for all $x\in\rho$, where $\lambda \in C$ and $\mu : \rho\to C$, except when $[\rho, \rho ]\rho =0$.

This paper studies algorithmic aspects of majority domination, which is a variation of domination in graph theory. A majority dominating function of a graph $G=(V,E)$ is a function $g$ from $V$ to $\{-1,1\}$ such that $\sum_{u\in N[v]}\,g(u)\geq 1$ for at least half of the vertices $v\in V$. The majority domination problem is to find a majority dominating function $g$ of a given graph $G=(V,E)$ such that $\sum_{v\in V}\,g(v)$ is minimized. The concept of majority domination was introduced by Hedetniemi and studied by Broere {\em et al}., who gave exact values for the majority domination numbers of complete graphs, complete bipartite graphs, paths, and unions of two complete graphs. They also proved that the majority domination problem is NP-complete for general graphs; and asked if the problem NP-complete for trees. The main result of this paper is to give polynomial-time algorithms for the majority domination problem in trees, cographs, and $k$-trees with fixed $k$.