Stochastic comparability and dual q-functions

In this paper we discuss the relationship and characterization of stochastic comparability, duality, and Feller–Reuter–Riley transition functions which are closely linked with each other for continuous time Markov chains. A necessary and sufficient condition for two Feller minimal transition functions to be stochastically comparable is given in terms of their density q-matrices only. Moreover, a necessary and sufficient condition under which a transition function is a dual for some stochastically monotone q-function is given in terms of, again, its density q-matrix. Finally, for a class of q-matrices, the necessary and sufficient condition for a transition function to be a Feller–Reuter–Riley transition function is also given.

Applications of Feller-Reuter-Riley transition functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.

Condition assessment of a 63-year old reinforced concrete structure exposed to the North Sea

This paper describes the condition of a reinforced concrete balustrade consisting of some 1000 individual beam elements all exposed similarly to the hostile marine environment of the North Sea at Arbroath, Scotland since 1943. A comparison is made of the condition of the original construction with the condition of repairs carried out in 1968 and in 1993. It is shown that the 1943 construction shows very little corrosion-induced cracking and little rust staining even though it does not appear to be of high construction quality. Only a very low percentage of the balustrade beams have been replaced. In contrast the beam installed in 1968 and later in 1993 show very considerable and large concrete cracks directly attributable to the corrosion of the longitudinal reinforcement, even though the concrete is of a higher quality and density. A detailed condition survey and statistics of crack sizes are presented in the paper. It is found that the higher corrosion resistance of the 1943 concrete is generally consistent with the concrete electrical resistivity measurements but the degree of corrosion of the reinforcing bars is inconsistent with chloride penetration measurements. The results are compared with the very few observations available in the literature for ageing concrete structures in marine environments. The results cast doubt on the conventional wisdom that chloride content at the reinforcement correlates well with reinforcement corrosion.