Numerical solution of the three‐dimensional parabolic equation with an integral condition

Developement of numerical methods for obtaining approximate solutions to the three dimensional diffusion equation with an integral condition will be carried out. The numerical techniques discussed are based on the fully explicit (1,7) finite difference technique and the fully implicit (7,1) finite difference method and the (7,7) Crank‐Nicolson type finite difference formula. The new developed methods are tested on a problem. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithms. The results of numerical testing show that the numerical methods based on the finite difference techniques discussed in the present article produce good results. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 193–202, 2002; DOI 10.1002/num.1040     

一类积分不等式的推广

对一些基本的积分不等式进行了推广，给出了含有n个无关变元的更广泛的非线性积分不等式．利用所得的不等式讨论了某些非线性积分方程解的有界性．     

BOUNDEDNESS OF MAXIMAL SINGULAR INTEGRALS

The authors study the singular integrals under the Hormander condition and the measure not satisfying the doubling condition. At first, if the corresponding singular integral is bounded from L2 to itself, it is proved that the maximal singular integral is bounded from L∞to RBMO except that it is infinite μ-a.e. on Rd. A sufficient condition and a necessary condition such that the maximal singular integral is bounded from L2 to itself are also obtained. There is a small gap between the two conditions.     

Continuity of Multilinear Operators in Weighted Herz Spaces with Extreme Exponents

Continuity is obtained of some multilinear operators related to certain integral operators for the weighted Herz spaces with extreme exponents. The operators include the Littlewood–Paley and Marcinkiewicz operators.     

Probability Interpretation of the Integral of Fractional Order

We establish a relation between stable distributions in probability theory and the fractional integral. Moreover, it turns out that the parameter of the stable distribution coincides with the exponent of the fractional integral. It follows from an analysis of the obtained results that equations with the fractional time derivative describe the evolution of some physical system whose time degree of freedom becomes stochastic, i.e., presents a sum of random time intervals subject to a stable probability distribution. We discuss relations between the fractal Cantor set (Cantor strips) and the fractional integral. We show that the possibility to use this relation as an approximation of the fractional integral is rather limited.     

首次穿过边界概率及其在金融中的应用

本文把数学和管理科学有机结合,为数学应用提出问题,得出新结果,推广了J.Michel Harrison(1985)[1]第43页的命题27,并给出了在金融中的应用.     

Tribute to Endre Csáki and Pál Révész

Summary This article provides a glimpse of some of the highlights of the joint work of Endre Csáki and Pál Révész since 1979. The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.     

On Non-Continuous Dirichlet Processes

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C ¹ transformation.     

非完整约束奇异广义力学系统的Poincaré-Cartan积分

对受高阶微商非完整约束并用奇异Lagrange量描述的广义力学系 统，基于广义Apell-Четаев约束条件，并考虑到系统的内在约束，导出了该非完整 约 束奇异广义力学系统的广义Poincaré-Cartan积分不变量. 并证明了该不变量与非完整约束 奇异广义力学系统的广义正则方程等价.     

Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.AMS subject classification: primary 90B15, secondary 60K25, 68M20