【作者】 衣娜；

【导师】 刘焕文；

【作者基本信息】 广西民族大学 ， 计算数学， 2008， 硕士

【摘要】 平面上正规三角剖分下的二元样条函数空间Snr(?)在众多领域都有着广泛的应用,因而其维数问题受到了数学和计算机领域学者们的持续关注.但人们很早就发现其维数不仅依赖于三角剖分的拓扑性质,而且强烈地依赖于剖分的几何形状,使得维数的确定非常困难.对于一般正规三角剖分,维数问题得到解决的情形是n≥3r +2以及n = 4, r = 1.本学位论文主要研究非收缩三角剖分与广义非收缩三角剖分下,样条函数空间Snr(?)的维数问题.首先通过讨论星型域上二元样条空间的最小决定集的分布情况得到了一类特殊三角剖分–非收缩三角剖分下的二元样条函数空间Snr(?)的维数;然后引入了两个构造三角剖分的新算子,定义了一类广义非收缩三角剖分,它是对Farin(2006)提出的非收缩三角剖分的一个推广,其中包含了一些众所周知的难以在其上讨论维数的剖分,如Morgan-Scott剖分, Robbins剖分等.并讨论了广义非收缩三角剖分下二元三次一阶光滑样条函数空间的维数问题.更多还原

【Abstract】 The bivariate spline spaces S nr(?) on regular triangulations have very pop-ular use in many fields. Therefore, its dimension problem has been continuouslyconcerned by the mathematics and computer scholars. However, it is found thatthe dimensions of S nr(?) depend not only on the topological properties of trian-gulations, but also on the geometric shape. This makes it di?cult to determinethe dimensions. For the general regular triangulations, when n≥3r + 2 andn = 4, r = 1, the problem of dimensions has been solved.In this paper, we focus on the problem of dimensions of spline functionspace S nr(?) under the unconstricted triangulations and generalized unconstrictedtriangulations. The contents are arranged as follows. First, the dimensions ofbivariate spline function spaces S nr(?) under a class of special triangulations–unconstricted triangulations are achieved by studying the distribution of minimaldetermining set on star-type regional. Then, by introducing two new operatorsin triangulation construction, a kind of generalized unconstricted triangulation isdefined, which is an expansion of the unconstricted triangulations introduced byFarin in 2006. It contains some well-known triangulations, such as the Morgan-Scott and the Robbins triangulations, on which the dimension problems are dif-ficult to be solved. And then, the dimensions of bivariate C1 cubic spline spacesover generalized unconstricted triangulations is determined.更多还原