This paper is based on a course delivered by the author at NCTS, National Chiao Tung University, Taiwan in Febuary 1999. We survey results related to structural aspects of graph homomorphism. Our aim is to demonstrate that this forms today a compact collection of results and methods which perhaps deserve its name : structural combinatorics. Due to space limitations we concentrate on a sample of areas only: representation of algebraic structures by combinatorial ones (graphs), the poset of colour classes and corresponding algorithmic questions which lead to homomorphism dualities, blending algebraic and complexity approaches.

In this paper, we give a unied approach to various forms of high temperature expansions. The variants of expansion include Mayer\'s expansion, cluster expansion and lace expansion. We also give a brief summary of applications of lace expansion to scaling limits of self-avoiding random walks, lattice trees and percolation above their critical dimensions.

The purpose of this note is to show that solutions of rst and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete.

This note addresses the general problem of the dynamical behavior for successive Gauss-Seidel transformations (shortly called Gauss-Seidelisations) of a given mapping over the $n$-cube. Complete results are given for $n=2$ and $n=3$, and then a natural conjecture is proved to be false for greater $n$. Thus this interesting problem remains still open for $n\geq 4$.

Existence criteria for oscillatory solutions of dierence equations have been obatined by many authors. It is therefore of interest to obtain additional conditions which are needed to yield stability of these solutions. In this paper, such conditions are derived for two functional dierence equations.

Consider the partly linear regression model $Y_i=X_i^\tau\beta+g(T_i)+\varepsilon_i$ $(i=1,\ldots,n)$, where $(X_i, T_i)$ are i.i.d. random design points, $\beta$ is a $p$-dimensional unknown parameter, $g(\cdot)$ is an unknown function on $[0, 1],$ $\varepsilon_i$ are i.i.d. random errors with mean 0 and variance $\sigma^2$. This paper is concerned with the {\bf LIL} of the estimators of $\beta$ and $\sigma^2.$

An analogue of the classical Heisenberg inequality is given for an innite-dimensional space. The proof relies on a commutation relationship and integration by parts formula for Gaussian measure. We also discuss when the equality holds.

In this paper the notion of stochastic conditional entropy is introduced, and the asymptotic relation between this notion and the relative entropy density is studied. A strong limit theorem which holds for arbitrary discrete information sources is obtained. In the proof an analytic technique to study the strong limit properties of discrete information sources is proposed.