Poisson’s equation for discrete-time single-birth processes

We consider Poisson’s equation for discrete-time single-birth processes, and we derive its solutions by solving a linear system of infinitely many equations. We apply the solution of Poisson’s equation to obtain the asymptotic variance. The results are further applied to birth–death processes and the scalar-valued GI/M/1-type Markov chains.     

Entire large solutions of semilinear equations involving the fractional Laplacian

We investigate the existence and nonexistence of positive entire large solutions of a class of semilinear equations involving the fractional Laplacian. Sharp lower and upper bounds of the minimal solution, if it exists, are also given.     

Stability analysis for stochastic jump systems with time-varying delay

This paper deals with the mean-square asymptotic stability of stochastic Markovian jump systems with time-varying delay. Based on a new stochastic inequality and convex analysis property, some novel stability conditions are presented. In the derivation, the information of the time-varying delay is retained and the estimation of it by the worst-case enlargement is not involved. Some special cases of the systems under consideration are also investigated. Illustrative examples are given to show the effectiveness of the proposed approach.     

A new level‐dependent coarse grid correction scheme for indefinite Helmholtz problems

In this paper, we construct and analyze a level‐dependent coarse grid correction scheme for indefinite Helmholtz problems. This adapted multigrid (MG) method is capable of solving the Helmholtz equation on the finest grid using a series of MG cycles with a grid‐dependent complex shift, leading to a stable correction scheme on all levels. It is rigorously shown that the adaptation of the complex shift throughout the MG cycle maintains the functionality of the two‐grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state‐of‐the‐art MG‐preconditioned Krylov methods, for example, complex shifted Laplacian preconditioned flexible GMRES. Copyright © 2013 John Wiley & Sons, Ltd.     

On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times

The structural properties of the moments of the time to ruin are studied in dependent Sparre Andersen models. The moments of the time to ruin may be viewed as generalized versions of the Gerber–Shiu function. It is shown that structural properties of the Gerber–Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in the model of Willmot and Woo (2012), which has Coxian interclaim times and arbitrary time-dependent claim sizes. Structural quantities needed to determine the moments of the time to ruin are specified under this model. Numerical examples illustrating the methodology are presented.     

Magnetic,Mössbauer and optical spectroscopic properties of the AFe3O(PO4)3 (A = Ca,Sr, Pb) series of powder compounds

AFe₃O(PO₄)₃ (A = Ca, Sr and Pb) powder compounds were studied by means of X-ray diffraction (XRD), electron-probe microanalysis (EPMA) coupled with wavelength dispersion spectroscopy (WDS), Raman and diffuse reflectance spectroscopies, specific heat and magnetic properties measurements. Magnetization, magnetic susceptibility and specific heat measurements carried out on AFe₃O(PO₄)₃ (A = Sr, Ca and Pb) powders firmly establish a series of three ferromagnetic (FM)-like second order phase transitions spanned over the 32–8 K temperature range. Room temperature Mössbauer spectroscopy and associated DFT calculations confirm the existence of three crystallographically non equivalent Fe³⁺ sites in the three compounds. Mössbauer spectra recorded as a function of temperature in the PbFe₃O(PO₄)₃ compound also establishes the occurrence of two purely magnetic and reversible phase transitions at 32 and 10 K. Diffuse reflectance measurements reveal two broad absorption bands at 1047 and 837 nm, in both PbFe₃O(PO₄)₃ and SrFe₃O(PO₄)₃ powders, with peak cross sections ∼10⁻²⁰ cm² typical of spin-forbidden and forced electric dipole intraconfigurational transitions.     

“Painlevé 34” equation: Equivalence test

We give the complete solution of the Equivalence Problem for “Painlevé 34” equation.     

三阶柯西差分方程在几类群上的解

在几类群上讨论了三阶柯西差分方程解的存在性问题,将二阶柯西差分方程的已有结论进一步推广到三阶的情况,并给出在不同群上的一般解.     

LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LINARD DIFFERENTIAL EQUATION

We investigate the generalized polynomial Linard differential equations. Using the averaging theory of first and second order, we obtain the maximum number of limit cycles of the system.