Number Theory and Combinatorics

数论和组合学

• 批准号: 0457003
• 负责人: George Andrews
• 金额: --
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457003&HistoricalAwards=false
• 关键词: Number; Theory; Combinatorics

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.

乔治·安德鲁斯（George Andrews）的项目摘要此预言中涵盖的第一个主题是分区和概率。在这里，建议在分区理论中霍洛伊德（Holroyd），利格特（Liggett）和罗米克（Romik）开始的研究，希望得到（1）更好的渐近学，（2）更广泛的应用，以及（3）与Ra​​manujan的模拟theta功能的进一步联系。 第二个主题是整个功能和罗杰斯·拉马努扬的身份，研究了埋葬在拉马努扬丢失笔记本中的两个真正令人惊奇的公式所产生的进一步含义。 第三个主题是恩格尔转型，它是基于A.和J. Knopfmacher引起的一些最初的惊人扩展定理。 该算法在Q系列扩展起着重要作用的任何数学主题（包括统计力学）中都应有用。 第四个主题是冈田猜想，是多年开放的话题。 最近，在与Paule和Schneider的联合合作中，PI给出了该猜想的Q = 1案例的新证明。这个新的证据清楚地表明，应该能够将其扩展到一般Q。 第五个主题是Ramanujan和部分分数，是通过尝试更好地理解Ramanujan丢失笔记本中一些更重要的公式的尝试提出的。改善以前的工作已经取得了巨大进展。希望可以进一步开发这些方法来阐明模拟theta函数的一般理论。 第6节（最后一节）涉及PI研究与改善教师教育的努力的交集。 简而言之，很明显，构图理论（有序分区）可以在初级和中等教育中发挥更大的作用。 该项目用于分区和Q系中的问题，尤其是同时（1）推进数学分支的中心理论，并且（2）在数学和数学科学的其他分支中具有很大的应用潜力。关于分区和概率的部分特别体现了这一理念。 该主题的应用已经在蜂窝自动机中提出。 恩格转换研究的工作研究了一种算法，其中在统计物理学中存在潜在的应用。在整个功能，部分分数和冈田锥体上的工作应开发出具有超出当前重点的大量应用的方法。第6节，最后一节涉及研究与改善教师教育的努力的交集。 组成理论（有序分区）很可能在初级和中等教育中起着有用的辅助作用。