The occurrence of felt earthquakes due to gas production in Groningen has initiated numerous studies and model attempts to understand and quantify induced seismicity in this region. The whole bandwidth of available models spans the range from fully deterministic models to purely empirical and stochastic models. In this article, we summarise the most important model approaches, describing their main achievements and limitations. In addition, we discuss remaining open questions and potential future directions of development.

In evolutionary theory, a key issue in selection theory is the expected time for a given amount of allele frequency change to occur. Crow and Kimura, by assuming weak selection, presented explicit results for several important cases of the directional selection and of the stochastic process. Those results played an important role in the theory of population genetics. In this paper, first we show that the weak selection assumption can be removed from most of the results of Crow and Kimura, and then we generalize those results to the most general selection model. Next, we estimate the errors of the deterministic formulae produced by proving that the deterministic formulae are limits of the corresponding stochastic formulae when the size of the population tends to infinity. Finally, we present a result which removes the restriction of Kimura’s corresponding results for a favourite recessive selection model, and we also observe that the conclusion made by Kimura about the favourite dominant selection might not be correct.

In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the virtual (actual) system time. The proposed method is analytically quite simple and easy to implement.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

Nous étudions ici la lubrification d'un palier dans une zone très localisée où l'épaisseur du film est particulièrement faible. Le problème ainsi posé est un problème de lubrification élastohydrodynamique rugueuse où il y a de fortes variations relatives d'épaisseur de film. Nous comparons les résultats obtenus dans cette situation par les équations de Navier-Stokes d'une part et par l'équation de Reynolds d'autre part. Puis nous regardons les influences mutuelles des différentes fréquences spatiales de la rugosité et du film lubrifiant.Les travaux en lubrification sont nombreux. Nous rappelons le classement bien connu des situations de lubrification, explicité sur une courbe de Stribeck, qui ordonne selon les épaisseurs croissantes de film, la lubrification limite, la lubrification mixte et la lubrification en film complet. Ceci nous conduit à considérer notre situation micro-élastohydrodynamique comme intermédiaire entre les lubrifications limite et mixte. Nous proposons ensuite une revue bibliographique montrant les deux approches, statistiques et déterministes, utilisées généralement pour intégrer la rugosité dans les équations de la lubrification élastohydrodynamique. Nous justifions ainsi le choix d'une approche déterministe pour notre étude.Les hypothèses et les méthodes utilisées par notre modèle étant ainsi précisées, nous présentons alors les résultats concernant les deux objectifs cités précédemment.

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm } of cyclic languages, where Lm = akm | k ∈ N. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language L m with isolated cut point on one-way probabilistic finite automata is $p_1^{\alpha_1}+ p_2^{\alpha_2} +\cdots +p_s^{\alpha_s}$ , with $p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_s^{\alpha_s}$ being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept L m with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms.

The aim of this paper is to investigate the distribution of the extinction times, T, of the stochastic logistic process from both the numerical and the theoretical standpoint. The problem is approached first by deriving formulae for the moments of T; it is then shown that in most cases T is, very nearly, a gamma variate. Some simulated results are given and these agree well with the theory. Furthermore, a consideration of the process conditioned on non-extinction is shown to be an effective way of obtaining a large t (time) description of the unconditioned process. Finally, a more general form of the model in which the death-rate as well as the birth-rate is ‘density-dependent' is considered, and by comparison with the usual form of the model the effect on T of this additional factor is assessed.