On the arithmetic product of combinatorial species

We introduce two new binary operations on combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series in the context of species. It allows us to introduce the notion of multiplicative species, a lifting to the combinatorial level of the classical notion of multiplicative arithmetic function. Interesting combinatorial constructions are introduced; cloned assemblies of structures, hyper-cloned trees, enriched rectangles, etc. Recent research of Cameron, Gewurz and Merola, about the product action in the context of oligomorphic groups, motivated the introduction of the modified arithmetic product. By using the modified arithmetic product we obtain new enumerative results. We also generalize and simplify some results of Canfield, and Pittel, related to the enumerations of tuples of partitions with the restrictions met.     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

In this paper, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation. Then, the large deviation estimates of small perturbation to equations of this type are obtained. We finally apply them to SDEs with the kernel of fractional Brownian motion with Hurst parameter H∈(0,1).     

广义Stirling数偶的理论及应用

The object of this expository paper is to sum up several results concerning generalized Stirling number (GSN) pairs investigated earlier by the author. Also expounded in some detail are two kinds of extended GSN pairs with applications (illustrative examples).     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

The RH boundary value problem of the k-monogenic functions

In this paper we study the Riemann and Hilbert problems of k-monogenic functions. By using Euler operator, we transform the boundary value problem of k-monogenic functions into the boundary value problems of monogenic functions. Then by the Almansi-type theorem of k-monogenic functions, we get the solutions of these problems.     

On a graded mesh method for a class of weakly singular Volterra integral equations

In this paper a class of weakly singular Volterra integral equations with an infinite set of solutions is investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation. The aim of this paper is to improve the convergence rates by a graded mesh method. The convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings.     

A random Euler scheme for Carathéodory differential equations

We study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte–Carlo method for integration problems.