Differential Inequalities Applied to The Stochastic Birth and Death Process

We describe an approach to a countably infinite system of ordinary differential equations belonging to the theory of the stochastic birth and death process. The main novelty in our method is the systematic use of a classical theorem on sub- and supersolutions for finite linear systems of the form y '( t ) = Ay ( t ). It leads in a simple way to the minimal solution and some of its properties. For convenience a proof of the theorem is given at the end.     

The Arboreal Approach to Pairs of Involutions in Rank Two

We use ideas of J.-P. Serre to obtain a geometric classification of the integral p-adic rank two representations of the infinite dihedral group.     

Free and Non-Free Subgroups of the Group UT(∞, ℤ)

We provide a method to find free groups of rank two in the group of infinite unitriangular matrices. Our groups are generated by two block-diagonal matrices, namely of the form A = diag(C, C, C…), B = diag(I _t , C, C…), where C is a matrix of finite dimension.

We give a necessary and sufficient condition for A and B defined above to generate a free group when C is a transvection. We formulate a sufficient condition to generate a free group, when C is a product of any number of commuting transvections.

We provide a classification of groups defined above, when C is of degree 3 or 4.     

一类板弯曲方程的辛本征函数展开方法

本文研究一边简支对边滑支边界条件的矩形板方程的无穷维Hamilton算子本征函数系,证明该无穷维Hamilton算子广义本征函数系在Cauchy主值意义下是完备的,为应用辛本征函数展开法求解该平面弹性问题提供理论基础.进而推导出原方程的通解,并对该平面弹性问题指出什么样的边界条件可按此方法求解.最后应用具体的算例说明所得结论的合理性.     

On a bulk arrival bulk service infinite service queue

This paper considers an infinite server queue in continuous time in which arrivals are in batches of variable size X and service is provided in groups of fixed size R. We obtain analytical results for the number of busy servers and waiting customers at arbitrary time points. For the number of busy servers, we obtain a recursive relation for the partial binomial moments both in transient and steady states. Special cases are also discussed     

A Regularity Result for CR Mappings Between Infinite Type Hypersurfaces

The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M′ be two real analytic curves in ? and f a holomorphic function defined on one side of M, extending continuously through M, and mapping M into M′. Then f has a holomorphic extension across M. In this paper, we extend this classical theorem to higher complex dimensions for a class of hypersurfaces of infinite type.     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Infinite domain potential problems by a new formulation of singular boundary method

This study proposes a new formulation of singular boundary method (SBM) and documents the first attempt to apply this new method to infinite domain potential problems. The essential issue in the SBM-based methods is to evaluate the origin intensity factor. This paper derives a new regularization technique to evaluate the origin intensity factor on the Neumann boundary condition without the need of sample solution and nodes as in the traditional SBM. We also modify the inverse interpolation technique in the traditional SBM to get rid of the perplexing sample nodes in the calculation of the origin intensity factor on the Dirichlet boundary condition. It is noted that this new SBM retains all merits of the traditional SBM being truly meshless, free of integration, mathematically simple, and easy-to-program without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). We examine the new SBM by the four benchmark infinite domain problems to verify its applicability, stability, and accuracy.