Birth-death processes with mass annihilation and state-dependent immigration

A birth-death process is subject to mass annihilation at rate β with subsequent mass immigration occurring into state j at rateα j . This structure enables the process to jump from one sector of state space to another one (via state 0) with transition rate independent of population size. First, we highlight the difficulties encountered when using standard techniques to construct both time-dependent and equilibrium probabilities. Then we show how to overcome such analytic difficulties by means of a tool developed in Chen and Renshaw (1990, 1993b); this approach is applicable to many processes whose underlying generator on E\{0} has known probability structure. Here we demonstrate the technique through application to the linear birth-death generator on which is superimposed an annihilation/immigration process.

Birth-death processes with an instantaneous reflection barrier

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth-death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

An application of quasi-Newton methods for the numerical solution of interface problems

Quasi-Newton methods are applied to solve interface problems which arise from domain decomposition methods. These interface problems are usually sparse systems of linear or nonlinear equations. We are interested in applying these methods to systems of linear equations where we are not able or willing to calculate the Jacobian matrices as well as to systems of nonlinear equations resulting from nonlinear elliptic problems in the context of domain decomposition. Suitability for parallel implementation of these algorithms on coarse-grained parallel computers is discussed.

Neutron quasi-elastic scattering in disordered solids: a Monte Carlo study of metal-hydrogen systems

The dynamic structure factor of neutron quasi-elastic scattering has been calculated by Monte Carlo methods for atoms diffusing on a disordered lattice. The disorder includes not only variation in the distances between neighbouring atomic sites but also variation in the hopping rate associated with each site. The presence of the disorder, particularly the hopping rate disorder, causes changes in the time-dependent intermediate scattering function which translate into a significant increase in the intensity in the wings of the quasi-elastic spectrum as compared with the Lorentzian form. The effect is particularly marked at high values of the momentum transfer and at site occupancies of the order of unity. The MC calculations demonstrate how the degree of disorder may be derived from experimental measurements of the quasi-elastic scattering. The model structure factors are compared with the experimental quasi-elastic spectrum of an amorphous metal-hydrogen alloy.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller–Reuter–Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

Has cultural studies sounded the death-knell of classical sociology prematurely?: the view from Romani studies

The paper will argue that although Bryan S.Turner's recent defence of classical sociology was seen as apostacy by some, it points to real problems in the idealism and a-historicism of contemporary cultural studies. The paper will examine the importance of the classical sociological problematic in getting the field of Romani Studies started, and the continuing relevance of a sociological approach rooted in history and political economy. [From the Author]