Number Theory and Combinatorics

乔治·安德鲁斯的项目摘要本论文涉及的第一个主题是划分和概率。本文建议将Holroyd，Liggett和Romik开始的研究建立在分拆理论的基础上，希望得到(1)更好的渐近性，(2)更广泛的应用，(3)与Ramanujan的模拟theta函数进一步联系。第二个主题，完整的函数和罗杰斯-拉马努扬的身份，考察了埋藏在拉马努詹的丢失的笔记本中的两个真正令人惊叹的公式所产生的进一步影响。第三个主题，恩格尔变换，建立在A.和J.Knopfmacher的一些原始的令人惊叹的展开定理的基础上。该算法在Q级数展开起重要作用的任何数学学科(包括统计力学)中都应该是有用的。第四个话题，冈田猜想，是一个多年来一直悬而未决的话题。最近，PI与Paule和Schneider合作，对这个猜想的q=1情形给出了新的证明。这个新的证明清楚地表明，人们应该能够将它推广到一般的Q。第五个主题，Ramanujan和部分分数，是先前为了更好地理解Ramanujan的Lost Notebook中的一些更深奥的公式而提出的。在改进以前的工作方面已经取得了很大的进展，希望这些方法可以进一步发展，以阐明模拟theta函数的一般理论。第六节，也就是最后一节，讨论了国际教师联合会的研究与他改进教师教育的努力之间的交集。简而言之，很明显，作文理论(有序分区)可以在中小学教育中发挥更重要的作用。这个项目致力于研究划分和Q级数中的问题，特别是那些同时(1)促进了这一数学分支的中心理论和(2)在数学和数学科学的其他分支中具有巨大应用潜力的问题。关于划分和概率的章节特别集中体现了这一哲学。这一主题的应用已经在元胞自动机中得到了应用。关于恩格尔变换的工作研究了一种算法，其中有可能应用于统计物理中的问题。关于整函数、部分分数和冈田定理的工作应该开发出具有实质性应用的方法，而不是目前的焦点。第六节，也就是最后一节，涉及研究与改进教师教育的努力的交集。作文理论(有序划分)在中小学教育中可以起到很好的辅助作用。

Project Summary for George Andrews The first topic covered in this propsoal is Partitions and Probability. Here it is proposed to ground the study begun by Holroyd, Liggett, and Romik in the theory of partitions with the hope of getting (1) better asymptotics, (2) broader applications, and (3) further connections with Ramanujan's mock theta functions. The second topic, Entire Functions and the Rogers-Ramanujan Identities, examines the further implications that arise from two truly amazing formulas that lay buried in Ramanujan's Lost Notebook. The third topic, Engel Transformations, builds upon some of the original amazing expansion theorems due to A. and J. Knopfmacher. This algorithm should be useful in any mathematical subject (including statistical mechanics) where q-series expansions play a substantial role. The fourth topic, The Okada Conjecture, is one that has been open for many years. Recently in joint work with Paule and Schneider, the PI gave a new proof of the q = 1 case of this conjecture. This new proof clearly suggests that one should be able to extend it to general q. The fifth topic, Ramanujan and Partial Fractions, was suggested by prior attempts to better understand some of the more recondite formulas in Ramanujan's Lost Notebook. Great progress has been made in improving previous work; it is hoped that these methods may be further developed to elucidate a general theory of mock theta functions. Section 6, the final section, concerns the intersection of the PI's research with his efforts to improve teacher education. Put succinctly, it is clear that the theory of compositions (ordered partitions) could play a much more substantial role in primary and secondary education. This project is devoted to problems in partitions and q-series, especially ones that simultaneously (1) advance the central theory of this branch of mathematics and (2) have substantial potential for application in other branches of mathematics and mathematical sciences. The section on partitions and probability especially epitomizes this philosophy. Applications of this topic have already been made to cellular automata. The work on Engel Transformations studies an algorithm wherein there are potential applications to problems in statistical physics. Work on entire functions, partial fractions and the Okada conejcture should develop methods with substantial applications beyond the current focus. Section 6, the final section, concerns the intersection of research with efforts to improve teacher education. The theory of compositions (ordered partitions) could well play a useful auxiliary role in primary and secondary education.