二元三次样条空间S31,2(Δmn(2)) 的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2 型三角剖分上二元三次样条空间S₃^1,2(Δ_mn⁽²⁾)的若干样条拟插值算子. 这些变差缩减算子由样条函数B_ij¹支集上5 个网格点或中心和样条函数B_ij²支集上5 个网格点处函数值定义. 这些样条拟插值算子具有较好的逼近性,甚至算子V_mn（f） 能保持近最优的三次多项式性. 然后利用连续模,分析样条拟插值算子V_mn（f）一致逼近于充分光滑的实函数. 最后推导误差估计.     

样条的机械化求解

本文在逐次分解法的基础上,给出一种样条机械化求解方法．该方法对多项式样条,有理样条乃至更一般样条的研究都是十分有效的．它适用于三角剖分,矩形剖分乃至更一般的代数曲线剖分     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件,首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分,并确定了此剖分下二次样条空间的维数．     

PIECEWISE SEMIALGEBRAIC SETS

Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of Rn satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of C^μ piecewise semialgebraic sets are also discussed.     

What is the Inverse of a Basis?

A formula is given for the inverse of the linear map, from coordinate space to a linear space, induced by a basis for that linear space, which is then connected to various basic Applied Linear Algebra constructs.This revised version was published online in October 2005 with corrections to the Cover Date.     

Discrete elastica

This paper develops a discrete approach to the design of planar curves that minimize cost functions dependent upon their shape. The curves designed by using this approach are piecewise linear with equal length segments and obey various types of endpoint constraints.     

Stable Estimation in Dimension Reduction

We introduce stable estimation procedures for several aspects of a sufficient dimension-reduction matrix. We first propose a stable method for estimating structural dimension, which only selects the correct directions in the central subspace with no false positive selection. We then provide a Grassmann manifold sparse estimate for the central subspace. By using subsampling, we develop an ensemble method to obtain a stable nonsparse estimate for the central subspace. This ensemble idea is also used to stabilize the choice of the number of slices in sliced inverse methods. Theoretical results are established, and the efficacy of the proposed stable methods is demonstrated by simulation studies and the analysis of Hitters’ salary data. Supplementary materials for this article are available online.