This paper is devoted to computational problems related to Markov chains (MC) on a finite state space. We present formulas and bounds for characteristics of MCs using directed forest expansions given by the Matrix Tree Theorem. These results are applied to analysis of direct methods for solving systems of linear equations, aggregation algorithms for nearly completely decomposable MCs and the Markov chain Monte Carlo procedures.

The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we give formulae for the dyadic diaphony of digital $(0,s)$-sequences over ${\mathbb{Z}}_2$, $s=1,2$. These formulae show that for fixed $s\in \{1,2\}$, the dyadic diaphony has the same values for any digital $(0,s)$-sequence. For $s=1$, it follows that the dyadic diaphony and the diaphony of special digital $(0,1)$-sequences are up to a constant the same. We give the exact asymptotic order of the dyadic diaphony of digital...