样条函数在离散数学中的应用

In this project, the applications of spline method in discrete mathematics will be systematically studied. Multivariate splines is an essentiallyimportant tool in computational mathematics and approximation theory. nevertheless, a series of problems emerged in descrete 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 descrete mathematics and combinatorial number theory. In this project, with multivariate spline functions, a novel method for ralated descrete mathematics problems will be studied. For instance, in combinatorics,the log-concavity for some combinatorial sequences and combinatorial enumeration will be investigated by spline theory. Cube slicing in descrete 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 enumeration which usually considered as counting discrete objects. This method is the bridge betweent the related problems in spline theory and descrete mathematics.Therefore, it''ll provide a novel analysis method for study descrete objects.Some problems in parallel programs compiler will also be taken into consideration. Furthermore, the relating software design is taken in as part of the project.

本项目拟系统的开展离散数学中样条方法的研究。多元样条原本是计算数学与函数逼近论里的重要工具。然而近来，人们发现借助已发展成熟的多元样条理论亦可对离散数学的一系列问题进行研究。本项目在此基础上，利用多元样条的方法研究离散几何和组合数学中的相关问题。拟发展一种系统的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，组合序列的对数凹性质等相关问题进行研究。离散几何方面主要对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。该方法成为样条函数与离散数学两个领域相关结果之间的桥梁，从而为离散对象的研究提供了一种新的分析方法。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研究各相关部分的软件。

本项目基本按照研究计划执行，围绕离散数学中的样条函数方法的理论和应用开展研究工作，主要研究重点为：离散几何学中的样条方法理论研究，计数组合学中的多元样条方法研究，多元样条在正交多项式渐近分析中的应用。组合数学方面主要针对组合数论和组合序列的对数凹性质等相关问题进行研究。离散几何方面主要对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。以样条函数为桥梁，沟通了计算数学与离散数学两个领域的相关结果，为离散对象的研究提供了一种新的分析方法。并且对计算多面体混合体积的快速算法设计等技术问题展开了研究。

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