In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor Method,the basis ofS_(3)^(1,2)(Δ_(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.
A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.
Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.
The effect of Pr substitution for Dy on the magnetic and magnetostrictive properties, anisotropy, spin reorientation and Mssbauer effect of a series of Tb0.3Dy0.7-xPrx(Fe0.9Al0.1)1.95 (x=0,0.1,0.20,0.25,0.30,0.35) alloys at room temperature have been investigated. It was found that a small amount of Pr substitution is beneficial to a decrease in the magnetocrystalline anisotropy for the Tb0.3Dy0.7-xPrx(Fe0.9Al0.1)1.95 alloys. The magnetostriction decreases drastically with increasing x and the magnetostrictive effect disappears for x>0.2. However, the magnetostriction exhibits a slightly bigger value at x=0.1 than the free alloys and is saturated more easily with the magnetic field H. The saturation magnetization and Curie temperature decrease monotonously, but the spontaneous magnetostriction increases linearly with increasing x, whereas the spin reorientation temperature increases first, then decreases rapidly and reaches the maximum at x=0.1. The analysis of Mssbauer spectra indicated that the easy magnetization direction in the {110} plane deviates slightly from the main axis of symmetry with the Pr concentration x, namely spin reorientation. Compared with Al substitution for Fe, the effect of Pr substitution for Dy on spin reorientation is relatively small. The hyperfine field increases with Pr concentration increasing, and the isomer shifts and the quadrupole splitting (QS) show weak concentration dependence.
ZHU ChunGang & WANG RenHong School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a Cl-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.
