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This paper is a survey of results presented in the recent book Gyllenberg and Silvestrov [GS08].1 This book is devoted to studies of quasi-stationary phenomena for nonlinearly perturbed stochastic processes and systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and for risk processes are presented. The book also contains an extended bibliography of works in the area. [ABSTRACT FROM AUTHOR]

The notion of predictive likelihood stems from the fact that in the prediction problem there are two unknown quantities to deal with: the future observation and the model parameter. Since, according to the likelihood principle, all the evidence is contained in the joint likelihood function, a predictive likelihood for the future observation is obtained by eliminating the nuisance quantity, namely the unknown model parameter. This paper focuses on the profile predictive likelihood and on some modified versions obtained by mimicking the solutions proposed to improve the profile (parametric) likelihood. These predictive likelihoods are evaluated by studying how well they generate prediction intervals. In particular, we find that, at least in some specific applications, these solution usually improve on those ones based on the plug-in procedure. However, the associated predictive densities and prediction limits do not correspond to the optimal frequentist solutions already described in the literature. [ABSTRACT FROM AUTHOR]

In view of the role of reaction equations in physical problems, the authors derive the explicit solution of a fractional reaction equation of general character, that unifies and extends earlier results. Further, an alternative shorter method based on a result developed by the authors is given to derive the solution of a fractional diffusion equation. Fox functions and Mittag-Leffler functions are used for closed-form representations of the solutions of the respective differential equations. [ABSTRACT FROM AUTHOR]

Existence of oblique polar lines for the meromorphic extension of the current valued function ʃ|f|2λ|g|2μ□ is given under the following hypotheses: f and g are holomorphic function germs in ℂn+1 such that g is non-singular, the germ Σ := {df 〉 dg = 0} is one dimensional, and g is proper and finite on S := {df = 0}. The main tools we use are interaction of strata for f (see [4]), monodromy of the local system Hn-1 (u) on S for a given eigenvalue exp(−2iΠu) of the monodromy of f, and the monodromy of the cover gS. Two non-trivial examples are completely worked out. [ABSTRACT FROM AUTHOR]

This paper discusses the crack tip asymptotic fields of frictionless and frictional cohesive cracks in quasi-brittle materials. This has been made possible after reformatting the cohesive-law into a special but universal polynomial. For accurate simulation of crack growth in quasi-brittle materials using the extended/generalized finite element method (XFEM), the leading term of the true displacement asymptotic field is used as the enrichment function at the tip of a cohesive crack. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment. A statically admissible stress recovery (SAR) technique is used to recover the stresses at the crack tip. Finally, a pure mode I cohesive crack problem is analysed to demonstrate the characteristics of global responses and local fields obtained numerically by the XFEM. [ABSTRACT FROM AUTHOR]