构造非均匀剖分上局部支集样条函数的积分方法

１　引　　言 给定递增数列｛ｘｉ｝,｛ｙｉ｝,｛ｚｋ｝．对于Ｒ２,由直线｛ｘ＝ｘｉ）和｛ｙ＝ｙｊ｝形成的剖分称为非均匀矩形剖分;再连接每个矩形［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］的正斜率对角线形成的剖分称为非均匀Ｉ－型三角剖分,记为　１;若连接每个矩形的两条对角线,则所得剖分为非均匀Ⅱ－型三角剖分,记为　２．对于Ｒ３,由平面｛ｘ＝ｘｉ｝,｛ｙ＝ｙｊ｝,｛ｚ＝ｚｋ｝形成的剖分称为非均匀长方体剖分;若再连接每个小长方体［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］×［ｚｋ,ｚｋ＋１］的过点（ｘｉ,ｙｊ,ｚ…     

The Bezout Number for Piecewise Algebraic Curves

The computation of the Bezout number, the maximum number of intersection points between two piecewise algebraic curves whose common points are finite, is considered. A piecewise algebraic curve is a curve determined by a bivariate spline function. It is found that the maximum number of intersections depends not only on the degrees and the differentiability of the spline functions, but also on the structure of the partition on which the spline functions are defined.     

The Bases of the Non-Uniform Cubic Spline Space $S_{3}^{1,2}(\Delta_{mn}^{(2)})$

In this paper, the dimension of the nonuniform bivariate spline space $S_{3}^{1,2}(\Delta_{mn}^{(2)})$ is discussed based on the theory of multivariate spline space. Moreover, by means of the Conformality of Smoothing Cofactor Method, the basis of $S_{3}^{1,2}(\Delta_{mn}^{(2)}) $composed of two sets of splines are worked out in the form of the values at ten domain points in each triangular cell, both of which possess distinct local supports. Furthermore, the explicit coefficients in terms of B-net are obtained for the two sets of splines respectively.     

On branched continued fractions rational interpolation over pyramid-typed grids

In this paper, three-term recurrence relations for branched continued fractions are determined. Based on the algorithm of partial inverse differences in tensor-product-like manner, the finite branched continued fractions can be applied to rational interpolation over pyramid-typed grids in R ³. By means of the three-term recurrence relations, a characterization theorem is valid. Then an error estimation is worked out. Based on the relationship between the partial inverse differences and partial reciprocal ones, and the partial reciprocal derivatives as well, the BCFs osculatory interpolation with its algorithm is stated which shows it feasibility of partial derivable functions in BCFs expansion at one point.     

Zirconium‐Mediated Multicomponent Reactions of 1,3‐Butadiynes with Ylidenemalononitriles to Form Functionalized 1,8‐Naphthyridine and Cyclopenta[b]pyridine Derivatives

Zirconocene‐mediated multicomponent reactions of 1,3‐butadiynes with ylidenemalononitriles in the absence or presence of CuCl have been developed. In the absence of CuCl, 1,3‐butadiyne couples with three molecules of ylidenemalononitriles to yield azazirconacycles bearing a hexahydro‐1,8‐naphthyridine skeleton with high stereoselectivity. In the presence of CuCl, cyclopenta[b]pyridine or cyclopenta[b]quinolin‐1‐one derivatives are obtained via transmetalation of Zr?C bond to Cu?C bond as the key reaction step. These domino‐type reactions proceed with high chemo‐, regio‐ and/or stereoselectivities, and allowing the formation of multiple C?N and C?C bonds in a single operation.     

非对角二元二次函数逼近的存在性和局部性质

§ 1　IntroductionThis paper is concerned with the properties of the simple off-diagonal bivariatequadratic Hermite-Padé approximation.Thisapproximation may be defined asfollows(see,for example,[1 ] ) .Let f(x,y) be a bivariate function,analytic in some neighbourhood of the origin(0 ,0 ) ,whose series expansion about the origin is known.Let a0 (x,y) ,a1 (x,y) ,a2 (x,y) bebivariate polynomials,a0 (x,y) = ki,j=0 a(0 )ij xiyj,a1 (x,y) = ni,j=0 a(1 )ij xiyj,a2 (x,y) = mi,j=0 a(2 )ij xiyj,such th…     

Numerical Integration Based on Bivariate Quartic Quasi-Interpolation Operators

In this paper,we propose a method to deal with numerical integral by using two kinds of C~2 quasi-interpolation operators on the bivariate spline space,and also dis- cuss the convergence properties and error estimates.Moreover,the proposed method is applied to the numerical evaluation of 2-D singular integrals.Numerical exper- iments will be carried out and the results will be compared with some previously published results.     

关于非均匀剖分下多元样条空间S_2~1(△_(mn)~(2))的拟插值算子

１引言随着计算机科学技术的发展,多元样条在力学和计算机辅助几何设计（ＣＡＧＤ）中的应用越来越引起人们极大兴趣．然而,由于一般剖分下样条空间的研究有相当的难度,迄今为止只对于一些特殊剖分的样条空间取得了一定的进展,如：矩形剖分,均匀的１－型,２－型三角剖分等．王仁宏和崔锦泰讨论了均匀２－型三角剖下的拟插值算子以及其逼近性质,鉴于在工程和实际应用中均匀剖分具有一定局限性,作者在文献（［１］,［３］）的基础上,对于非均匀２－型三角剖分,给出了一类拟插值算子,并研究了它的逼近性质．同时,利用其构造了一类…     

A nodal basis of Cμ-rational spline functions on triangulations

Let ρ be a triangulation of a polygonal domain D⊂R² with vertices V={v_i:l≤i≤N_v} and RS^k(D, ρ)={u∈C^k(D): ≠ T∈ρ, u/_T is a rational function}. The purpose of this paper is to study the existence and construction of C^μ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem H^μ(V,U)={find u∈U: D^αu(v_i)=D^αf(v_i),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RS^μ. this solution needs only the knowledge of partial derivatives of order≤μ at v_i. The explicit repesentations of all C^μ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.