构造性分拆理论研究

项目名称： 构造性分拆理论研究

项目编号： No.11526136

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 刘海

作者单位： 上海对外经贸大学

项目金额： 3万元

中文摘要： 本项目旨在研究构造性方法在分拆理论中的应用。该理论由当代组合数学权威、美国科学院院士George Andrews教授引领，有着很强的数学物理背景，是当代组合学研究的前沿课题之一。同时，在涉及partial theta函数和mock theta函数分拆等式的构造性证明方面我们已经取得重要进展，为继续研究奠定了基础。 项目将利用组合数学的方法和技巧来构造分拆双射和对合，给出一些著名分拆等式和q-级数等式的组合证明，如：Andrews’ partial theta等式的更一般推广等式，关于两个partial theta函数和与差等式，涉及mock theta函数的等式和子分拆等式，并解决一些公开问题。 本项目的研究成果将有助于加深对分拆等式和q-级数等式的理解，为分拆理论在q-级数、组合数学、数论和物理学等领域的应用提供更多的理论支持。

中文关键词： theta 函数；分拆；组合证明；模形式；同余

英文摘要： Our objective is to study applications of constructive method in the theory of partitions. Professor George Andrews, who is the master of algebraic combinatorics and an elected member of American Academy of Arts and Sciences, directs the developments of the theory of partitions. The theory of partitions has a strong background on mathematical physics, and is one of the most active fields in combinatorics. We have made important achievements in constructive proofs of partitions identities involving partial theta functions and mock theta functions, which lays a strong basis for further study. We will construct partition bijections and involutions using the methods and techniques of combinatorics, in order to give combinatorial proofs of some important partitions identities and q-series identities, such as a more general identity of Andrews' partial theta identity, identities concerning the sum or difference of two partial theta functions, identities involving mock theta functions and subpartitions identities, and then solve some open problems. Research findings of the proposal will help to strengthen the understanding of partitions and q-identities, and provide more theoretical supports for applications of the theory of partitions in q-series, combinatorics, number theory and physics.

英文关键词： theta function；partition；combinatorial proof；modular form；congruence