Calculating the matrix exponential of a constant matrix on time scales

We propose a method which simplifies the main result obtained in [A. Zafer, The exponential of a constant matrix on time scales, ANZIAM J. 48 (2006) 99–106] to calculate the matrix exponential of a constant matrix on a time scale.     

Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection–diffusion problem with characteristic layers

We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection–diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first‐order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second‐order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Stability of Timoshenko systems with past history

We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.     

Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion

The main aim of this paper is to investigate the exponential stability of the Euler method for a stochastic age-structured population system with diffusion. The definition of exponential mean square stability of numerical method is introduced. It is proved that the Euler scheme is exponentially stable in mean square sense. An example is given for illustration.     

Counting labeled claw-free cubic graphs by connectivity

Claw-free cubic graphs are counted with given connectedness and order. Tables are provided for claw-free cubic graphs with given connectedness. This builds on methods for counting general cubic graphs by connectivity previously developed by Chae, Palmer, and Robinson, and on the earlier enumeration of all claw-free cubic graphs by McKay, Palmer, Read, and Robinson.     

CZTS‐based materials and interfaces and their effects on the performance of thin film solar cells

Cu₂ZnSnS₄ (CZTS) and its related materials such as Cu₂ZnSnSe₄ (CZTSe) and Cu₂ZnSn(S,Se)₄ (CZTSSe) have attracted considerable attention as an absorber material for thin film solar cells due to the non‐toxicity, elemental abundance, and large production capacity of their constituents. Despite the similarities between CZTS‐based materials and Cu(In,Ga)Se₂(CIGS), the record efficiency of CZTS‐based solar cells remains significantly lower than that of CIGS solar cells. Considering that the difference between the two lies in the choice of the absorber material, the cause of the lower efficiency of CZTS‐based solar cells can be isolated to the issues associated with CZTS‐based materials and their related interfaces. Herein, these issues and the work done to understand and resolve them is reviewed. Unlike existing review papers, every unique region of CZTS‐based solar cells that contributes to its lower efficiency, namely: (1) the bulk of the absorber, (2) the grain boundaries of the absorber, (3) the absorber/buffer layer interface, and (4) the absorber/back contact interface are surveyed. This review also intends to identify the major unresolved issues and the potential improvement approaches of realizing sizable improvements in the solar cells' efficiency, thus providing a guide as to where research efforts should be focused. (© 2014 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)     

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

Asymptotic behavior for second order stochastic evolution equations with memory

In this paper, a general second order integro-differential evolution equation with memory driven by multiplicative noise is considered. We prove the existence of global mild solution and asymptotic stability of the zero solution using Lyapunov function techniques. Moreover, we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations.