On incidence between strata of the Hilbert scheme of points on $$\mathbb{P}^{2}$$

The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand, Ph.D Thesis, Universite’de Nice, 2002 studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand’s result. This allows us to show that the technical condition is not necessary. Michel Van den Bergh is a director of research at the FWO.     

Strong Convergence of an Implicit Iteration Process for a Finite Family of Asymptotically Ф-pseudocontractive Mappings

Strong convergence theorems for approximation of common fixed points of asymptotically Ф-quasi-pseudocontractive mappings and asymptotically C-strictly- pseudocontractive mappings are proved in real Banach spaces by using a new composite implicit iteration scheme with errors. The results presented in this paper extend and improve the main results of Sun, Gu and Osilike published on J. Math. Anal. Appl.     

Numerical solution to the optimal feedback control of continuous casting process

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton–Jacobi–Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.     

Mathematical modeling and study of the process of solidification in metal casting

The process of melting and solidification in metal casting is considered. The process is modeled by a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the problem and its finite-difference approximation are given. A numerical algorithm is presented for solving the direct problem. The results are described and analyzed in detail. Primary attention is given to the evolution of the solidification front and to how it is affected by the parameters of the problem. Some of the results are illustrated by plots.     

Polynomial time approximation schemes and parameterized complexity

In this paper, we study the relationship between the approximability and the parameterized complexity of NP optimization problems. We introduce a notion of polynomial fixed-parameter tractability and prove that, under a very general constraint, an NP optimization problem has a fully polynomial time approximation scheme if and only if the problem is polynomial fixed-parameter tractable. By enforcing a constraint of planarity on the W-hierarchy studied in parameterized complexity theory, we obtain a class of NP optimization problems, the planar W-hierarchy, and prove that all problems in this class have efficient polynomial time approximation schemes (EPTAS). The planar W-hierarchy seems to contain most of the known EPTAS problems, and is significantly different from the class introduced by Khanna and Motwani in their efforts in characterizing optimization problems with polynomial time approximation schemes.     

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.     

一类偏积分微分方程二阶差分全离散格式

本文给出了数值求解一类偏积分微分方程的二阶差分全离散格式．时间方向采用了二阶向后差分格式,积分项的离散利用了Lubich的二阶卷积求积公式,给出了稳定性的证明、误差估计及收敛性的结果,并给出了数值例子．     

Pathwise Approximation of Random Ordinary Differential Equations

Standard error estimates for one-step numerical schemes for nonautonomous ordinary differential equations usually assume appropriate smoothness in both time and state variables and thus are not suitable for the pathwise approximation of random ordinary differential equations which are typically at most continuous or Hölder continuous in the time variable. Here it is shown that the usual higher order of convergence can be retained if one first averages the time dependence over each discretization subinterval.This revised version was published online in October 2005 with corrections to the Cover Date.     

A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noise. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA are given.