混合体积、对数凹序列和B样条函数

本文主要通过样条函数方法研究与之相关的离散几何学和组合学问题.在离散几何学方面主要考虑超立方体切面(cube slicing)体积和混合体(mixed volume)的样条表示,利用B样条函数的几何解释,将超立方体切面问题转化为与之等价的样条函数问题,分别给出Laplace和P′olya关于超立方体切面定理的样条证明,将样条函数与混合体积联系起来,给出一类混合体积的样条解释.利用这种解释可以得到一类具有对数凹性质的组合序列,从而部分地回答了Schmidt和Simion所提出的关于混合体积的公开问题.在组合数学方面主要考虑多种组合多项式与样条函数的关联以及组合序列对数凹性质的样条方法研究.本文借助丰富的样条函数理论,不但验证了离散几何学和组合数学中很多现有的结果,而且得到了一系列离散数学对象的新性质,建立了离散数学问题与具有连续性特质的样条函数之间的内在联系.

Mixed volumes,log-concave sequences and B-splines

In this paper, a series of problems emerged in discrete geometry and combinatorics related to spline functions are systematically studied. For example, in discrete geometry, the spline representations of cube slicing and mixed volumes of polytopes are considered. With the geometric interpretations of B-splines, the volume of cube slicing can be considered as an equivalent problems in spline theory. Based on the connection, a simple proof for Laplace and Polya＇s formulas in cub slicing is given by spline theory. A class of mixed volumes are given by the relations between splines and the mixed volumes. A class of log-concave sequence are derived by the connection. Therefore, the open problem proposed by Schmidt and Simion is partially solved. In combinatorics, the combinatorial polynomials and log-concavity for some combinatorial sequences are investigated by spline theory. With the well developed spline theory, not only the existing results in discrete geometry and combinatorics have been verified, but also a novel method for solving related discrete mathematics problems has been studied. Splines as functions of a continuous nature provide an analysis method in combinatorial enumerations which are usually considered as counting discrete objects. This method provides a novel analysis method for studying discrete objects.