Probabilistic solution of random homogeneous linear second-order difference equations

This paper deals with the computation of the first probability density function of the solution of random homogeneous linear second-order difference equations by the Random Variable Transformation method. This approach allows us to generalize the classical solution obtained in the deterministic scenario. Several illustrative examples are provided.     

一个Erdis问题在二次曲线下的推广

著名数学家Paul Erd■s曾提出过这样一个有趣的问题:"设U为平面上一些单位圆所成之集,如果平面上任一条直线总和U中至少一个圆相交或相切,那么是否可以肯定:对任给的自然数m,总能找到一条直线,它至少和U中的m个圆相交或相切?"张景中,杨路等人解决了这个问题.本文探讨了该问题在二次曲线下的推广,得到了和原Erd■s问题相似的结论.     

Pattern memory analysis of second-order neural networks with time-delay

In this paper, the pattern memory of the second-order neural networks with time-delay are investigated based on the stability theory. Several sufficient conditions are obtained such that the equilibrium point is locally exponential stable when the equilibrium point located in the saturation region. These conditions can be directly derived from the synaptic connection weights and the external input of the second-order neural networks, and be verified easily. In addition, four examples are given to show the effectiveness of the results.     

On Liouville type theorems for fully nonlinear elliptic equations with gradient term

In this paper, we prove a Hadamard property and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with a gradient term, both in the whole space and in an exterior domain.     

Interpretation of vicinal proton–proton coupling constants of heteroaromatic molecules in terms of topological electron density descriptors

The indirect vicinal proton–proton coupling constants for pyrrole, furan, thiophene and 15 related heteroaromatic compounds were calculated using the Khon–Sham approximation. An analysis of the four Ramsey contributions to the coupling constants was carried out showing that the Fermi contact term is always positive and dominant, although the remaining contributions have a nonnegligible net negative contribution. The trends observed for the proton–proton coupling constants were rationalized in terms of the properties of the electron density. It was found that electron delocalization between the corresponding hydrogen atoms plays a major role on the observed behavior with the charges of the carbon atoms bonded to them and the accompanying geometric variations being also of importance in the coupling mechanism. Copyright © 2010 John Wiley & Sons, Ltd.     

Oscillation of a Second-order Impulsive Neutral Differential Equation

Oscillation theorems for a second-order impulsive neutral differential equation are established, which extend the main results developed by Li et alLi et al, Oscillation of second order self-coajugate differential equation with impuls[es. J Comput Appl Math 197（2006）： 78-88] to the considered equation. Two examples are also inserted to illustrate our main results.     

Term rank preservers of Boolean matrices

We obtain some characterizations of linear operators that preserve the term rank of Boolean matrices. That is, a linear operator over Boolean matrices preserves the term rank if and only if it preserves the term ranks 1 and k(≠1) if and only if it preserves the term ranks 2 and l(≠2). Other characterizations of term rank preservers are given.     

On the stability and dissipation of wall boundary conditions for compressible flows

Characteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison theorems for diffusion processes

We prove comparison theorems for diffusion processes onR ^d. From these theorems we derive lower and upper bounds for the transition probabilities of a diffusion process. In contrast to the known estimates for fundamental solutions of parabolic equations our bounds do not depend on the moduli of continuity of the coefficients of the differential operator.     

Recent developments in the stereological analysis of particles

Recent developments in the stereological analysis of particles are reviewed. The trend has been towards methods which are applicable without specific assumptions about particle shape. Geometric samples of a local 3-d character are used. Stereological estimators of particle intensity, particle size distribution and particle interaction are presented and discussed.An earlier version of this article was presented at the Symposium on the Analysis of Statistical Information held in the Institute of Statistical Mathematics, Tokyo during December 5–8, 1989.