On some generalized Sister Celine's polynomials

Certain generalizations of Sister Celine's polynomials are given which include most of the known polynomials as their special cases. Besides, generating functions and integral representations of these generalized polynomials are derived and a relation between generalized Laguerre polynomials and generalized Bateman's polynomials is established.     

Error estimation of Hermite spectral method for nonlinear partial differential equations

Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.

     

Hermite插值多项式在不同基下的显式表示

本文利用对偶基的概念,导出了 Ｈｅｒｍ ｉｔｅ 插值多项式在不同基下的显式表示,这给人们对 Ｈｅｒｍ ｉｔ插值多项式在不同基下从一种表示转换到另一种表示带来极大的方便     

基于面积约束的统计直方图曲线拟合

我们提出用分段三次Hermite插值曲线拟合统计直方图的新方法.先根据统计直方图的特点选取Hermite插值曲线在插值点处的导数值和可调整的插值点,然后根据面积约束确定调整值,从而得到拟合曲线.所得拟合曲线与统计直方图有面积相等的约束,并且拟合曲线是C1连续的光滑曲线.所给方法简单、实用.     

固定网格下的特征线法求解溃坝问题

溃坝问题是典型的非线性双曲方程的Riemann问题,其数值求解的难点在于对间断面的捕捉以及避免间断面处在数值计算过程中产生数值色散,因而为求解此问题所产生的各种数值计算方法的优劣也体现在这两个方面。本文针对溃坝问题提出一种新的计算方法。该方法基于对偶变量推导的浅水波方程,根据方程的特点,从方程的特征值和黎曼不变量出发,采用高精度的激波捕捉方法计算黎曼不变量的位置随时间的变化,然后映射至不随时间变化的固定网格。根据黎曼不变量的位置,采用保形分段三次Hermite插值将物理量映射至网格节点。计算结果显示,该方法不仅操作简单,计算量小,而且结果准确。     

Excitation of Forbidden Electronic Transitions in Atoms by Hermite–Gaussian Modes

Photoexcitation of trapped ions by Hermite–Gaussian (HG) modes from guided beam structures is proposed and investigated theoretically. In particular, simple analytical expressions for the matrix elements of induced atomic transitions are derived that depend both on the parameters of HG beams and on the geometry of an experiment. By using these general expressions, the ${}^2{S}_{1/2}\to {}^2{F}_{7/2}$ electric octupole (E3) transition is investigated in an Yb⁺ ion, localized in the low–intensity center of the HG₁₀ and HG₀₁ beams. It is shown how the corresponding Rabi frequency can be enhanced by properly choosing the polarization of incident light and the orientation of an external magnetic field, which defines the quantization axis of a target ion. The calculations, performed for experimentally feasible beam parameters, indicate that the achieved Rabi frequencies can be comparable or even higher than those observed for the conventional Laguerre–Gaussian (LG) modes. Since HG-like modes can be relatively straightforwardly generated with high purity and stability from integrated photonics, these results suggest that they may form a novel tool for investigating highly-forbidden atomic transitions.