A Kind of Generalization of the Curve Type Node Configuration in R^S（S 〉 2）

A kind of generalization of the Curve Type Node Configuration is given in this paper,and it is called the generalized node configuration CTNCB in RS(S＞2).The related multivariate polynomial interpolation problem is discussed.It is proved that the CTNCB is an appropriate node configuration for the polynomial space PSn (S＞2).And the expressions of the multivariate Vandermonde determinants that are related to the Odd Curve Type Node Configuration in R2 are also obtained.     

THE ELLIPTIC TYPE NODE CONFIGURATION AND INTERPOLATION IN R~2

1.IlltroductionInthispaper,weusetheusualmultivariatenotationac=yi'w'B,iii=if ... j.(if,'',j.EZ )[l'a]andletPnbethe(bivariate)polynomialspaceofallreal(bivariate)polynomialsofdegreeatmostn.NowweintroducetheconceptoftheCurveTypeNodeConfiguration(CTNC):De…     

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C (SLTNCC) in multivariate polynomial interpolation as the result of different kinds of transformations of lines (such as parallel translations, rotations). Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.     

双曲线型结点组上的插值

本文讨论了R~2上的弧线型结点组之一:双曲线型结点组上插值的适定性问题。     

竖线型结点组上的插值及向高维情形的推广

本文讨论了Ｒ＾２中竖线型结点组插值的适定性，得到了相应的插值多项式，并将这些结果推广到Ｒ＾ｓ（ｓ〉２）的情形。     

一类非线性椭圆边值问题解的存在性

目前 ,对 s——拉普拉斯算子△s的研究是较为活跃的数学课题 .原因在于算子 -△s与许多物理现象有关 .比如 :反射扩散问题 ,石油提取问题等等 .基于此因 ,在文 [3]的基础上 ,我们将继续研究以下非线性边值问题在 Ls(Ω) ,( 1 2 nn+1 )中解的存在条件 .-△su +g( x,u) =f几乎处处在Ω中-〈 ,| u|s- 2 u〉 =0几乎处处在Γ上其中 f∈Ls( Ω)给定 ,Ω Rn( n 1 ) ,△su=div( | u|s- 2 u) ,g∶Ω× R→ R满足 Caratheodory条件 .本文把文 [3]关于非线性边值问题 @在 Lp( Ω) ( 2 p<+∞ )空间中解的存在性的研究推广到 Ls( Ω) ( 1 2 nn+1 )空间中 .     

多项式空间的对偶及其在多元插值中的应用

本文通过把域Ｋ上ｎ元多项式环看成域Ｋ上的无限维向量空间Ａ，把ｎ维仿射空间Ｋ＾ｎ中的每一点看成Ａ上的线性泛函，从而Ｋ＾ｎ为对偶空间Ａ＾＊的子集，利用对偶空间的理论得到了一些有趣的理论结果，弄清了Ｋ＾ｎ上点有限拓扑的结构，给出了判定给定结点组是否是给定多项式空间的适定结点组的判定准则，最后还给出了构造理想对偶基的一种算法。     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Local Existence of Strong Solutions to the Generalized MHD Equations

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms $\Lambda^{2\alpha}u$ for the velocity field and $\Lambda^{2\alpha}b$ for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in $H^s(\mathbb{R}^n)\,(s>\max \left\{\frac{n}{2}+1-2\alpha, 0\right\})$ for any $\alpha\geq0$.     

ON AN EXTENSION OF ABEL-GONTSCHAROFF＇S EXPANSION FORMULA

We present a constructive generalization of Abel-Gontscharoff＇s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for constructing the basis functions of the interpolants are given.     

一种对称损失函数下刻度参数的最优同变估计

本文研究刻度参数分布族(1/σ)f(x/σ)中刻度参数在损失函数L(σ,δ)=(σ-δ)~2/σδ下的最小风险同变估计及其最小最大性。     

多元波动率模型的一些新进展

对多个资产收益率的协方差矩阵建立动态模型是一个非常重要的问题。本文就近些年来该方面研究的一些主要进展进行了综述,特别地介绍了几种基于数据降维技术发展起来的能够适用于高维情形的多元GARCH模型,另外,对于多元波动率的模型诊断与比较方法以及条件协方差矩阵的预测等方面的研究成果也作了分析。     

Asymptotic Conditions for Degree Diminution Along Prescribed Lines

One of the problems in bivariate polynomial interpolation is the choice of a space of polynomials suitable for interpolating a given set of data. Depending on the number of data, a usual space is that of polynomials in 2 variables of total degree not greater than k. However, these spaces are not enough to cover many interpolation problems. Here, we are interested in spaces of polynomials of total degree not greater than k whose degree diminishes along some prescribed directions. These spaces arise naturally in some interpolation problems and we describe them in terms of polynomials satisfying some asymptotic interpolation conditions. This provides a general frame to the interpolation problems studied in some of our recent papers.     

某类二阶非线性系统初值问题的奇摄动*

本文研究某类二阶非线性向量微分方程初值问题ε'x"=f(t,x,x',ε),x(0,ε)=a,x'(0,ε)=β的奇摄动,其中r>0为任意常数,ε>0为小参数,x,f,α,β∈Rn.在适当的假设下,利用多参数展开法和对角化技巧,证得摄动问题解的存在和导出解的高阶的一致有效渐近展开式.     

The homogenization of the three dimensional skew polynomial algebras of type I

This paper studies a special case of graded central extensions of three dimen-sional Artin-Schelter regular algebras, see [9, §3]. The algebras are homoge-nizations of two classes of three dimensional skew polynomial algebras. We refer to these algebras as Type I and Type II algebras. We describe the non-commutative projective geometry and compute the finite dimensional simple modules for the homogenization of Type I algebras in the case that α is not a primitive root of unity. In this case, all finite dimensional simple modules are quotients of line modules that are homogenizations of Verma modules.     

Phase II monitoring of changes in mean from high‐dimensional data

The generalized T² chart (GT‐chart), which is composed of the T² statistic based on a small number of principal components and the remaining components, is a popular alternative to the traditional Hotelling's T² control chart. However, the application of the GT‐chart to high‐dimensional data, which are now ubiquitous, encounters difficulties from high dimensionality similar to other multivariate procedures. The sample principal components and their eigenvalues do not consistently estimate the population values, and the GT‐chart relying on them is also inconsistent in estimating the control limits. In this paper, we investigate the effects of high dimensionality on the GT‐chart and then propose a corrected GT‐chart using the recent results of random matrix theory for the spiked covariance model. We numerically show that the corrected GT‐chart exhibits superior performance compared to the existing methods, including the GT‐chart and Hotelling's T² control chart, under various high‐dimensional cases. Finally, we apply the proposed corrected GT‐chart to monitor chemical processes introduced in the literature.     

Directions for computing truncated multivariate Taylor series

Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of variables by fixing a direction in . After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or ``interpolated'. The sequence of univariate calculations is more efficient than multivariate methods, although previous work indicates a space cost for this savings and significant cost for the reconstruction. We completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction. By appropriate choice of directions, the reconstruction reduces to a sequence of Lagrange polynomial interpolation problems in for which a divided difference algorithm computes the coefficients of a Newton form. Another algorithm collects like terms from the Newton form and returns the desired multivariate coefficients.

     

C~n中F(p,q,s)型空间上的Bergman型算子

设p0,s≥0,q+s-1,q+n-1.讨论了C~n中单位球上F(p,q,s)到本身或A(p,q,s)空间到L(p,q,s)空间上的Bergman型算子的有界性条件.     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

Larson理想中算子的插值

设a为nest代数，R^∞为A的Larson理想，R(A)为A的根，[R(A)]^s为R（A）的强拓扑闭。在文本中，我们给出[R(A)]^s的纯代数构造；且引进了一个新算子集合I，并证明了：若A的不变子空间格为几乎原子的，则R^∞＝[R(A)]^s=I。利用上述结果，我们研究了当A的不变子空间格为几乎原子时的Larson理想中的算子插值问题。我们得到算子方程AX＝Y在R^∞中有解A的充分必要条件。