离散数学中的样条方法研究

项目名称： 离散数学中的样条方法研究

项目编号： No.11301060

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 许艳

作者单位： 东北财经大学

项目金额： 21万元

中文摘要： 本项目拟系统开展离散数学中样条方法的研究。产生于逼近论的样条函数原本是计算数学的重要工具，然而近来人们发现借助已发展成熟的样条函数理论也可以对离散数学的一系列问题进行研究。本项目在此基础上，利用多元样条方法研究离散几何和组合数学中的相关问题。拟发展一种系统研究此类离散数学问题的新方法。组合数学方面主要针对组合数论、整数剖分、组合序列的对数凹性质等相关问题进行研究。离散几何方面主要针对凸多面体的混合体积计算问题以及超立方体的切面问题进行研究。与传统的组合方法相比，具有连续性特征的样条函数为组合计数等离散问题的研究提供了新的解析方法。该方法成为沟通样条函数与离散数学两个领域相关结果之间的桥梁，为离散对象的研究提供一种新的分析方法，同时也为样条理论的发展提供新的组合学工具。本项目还将研究与多面体混合体积计算相关的并行程序编译问题，如嵌套循环的内存分配、进程间的数据交换以及各相关部分的软件。

中文关键词： 多元样条；混合体积；组合计数；离散几何；渐近分析

英文摘要： In this project, the applications of spline method in discrete mathematics will be systematically studied. Multivariate splines is an essentially important tool in computational mathematics and approximation theory. Nevertheless, a series of problems emerged in discrete mathematics are solved with the help of the well developed multivariate spline theory, recently. Based on this point, we are going to deal with the problems arising in discrete mathematics and combinatorial number theory. In this project, with multivariate spline functions, a novel method for related discrete mathematics problems will be studied. For instance, in combinatorics, the log-concavity for some combinatorial sequences and combinatorial enumerations will be investigated by spline theory. Cube slicing in discrete mathematics and mixed volumes of polytopes are going to be considered as well. Compared with the traditional combinatorial methods, splines as functions of a continuous nature provide an analysis method in combinatorial enumerations which usually considered as counting discrete objects. This method is the bridge between the related problems in spline theory and discrete mathematics. Therefore, it'll provide a novel analysis method for study discrete objects. Some problems in parallel programs compiler will also be taken into cons

英文关键词： Multivariate splines；Mixed volume；Combinatorial enumeration；Discrete geometry；Asymptotic analysis