Combinatorics and Number Theory

Li 9622938近年来，数论在其他领域的应用引起了越来越多的兴趣。这项调查将利用数论和组合学之间丰富的相互作用。所要研究的问题要么起源于组合学，要么直接应用于组合学，但问题本身是数论的。粗略地说，这次调查中的问题可以分为两个部分。第一部分讨论了Ramanujan图的特征和作为特征值的分布及其与Hecke算子特征值的关系。PI将继续对有理函数域上的Krousterman猜想进行研究。这一猜想意味着某些Krousterman和是函数域上Hecke算子的特征值。该方法包括对函数域上曲线上的L函数及其与L函数之间的联系的研究。第二部分是利用p-ady域的加法特征标和乘法特征标定义的部分特征标和得到了较好的结果。如果建立了这些界限，这些界限将立即在信息论中得到应用，以设计出更好的低偏相关周期序列族。该方法结合了研究者估计p-adady域特征标的全特征标和和有限域特征标的部分特征标和的技巧。国际数学家学会计划在一九九六至九七年度开办“数论及应用”研究生课程，内容包括她过去在这方面的工作，以及这项建议的成果。这项研究属于数论和组合学的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是垫子血液学中最古老的分支之一，几个世纪以来一直出于纯粹的审美原因而被追寻。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。组合学也是现代通信中不可或缺的工具。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。大型网络的设计，如电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

Li 9622938 The applications of Number Theory to other fields have been attracting increasing interest in recent years. This investigation will exploit the rich interplay between Number Theory and Combinatorics. The problems to be investigated either have their origin in Combinatorics or have immediate application to Combinatorics, yet the problems themselves are Number- Theoretic. Roughly speaking, the problems in this investigation can be divided into two parts. The first part concerns the distribution of the character sums occurring as eigenvalues of Ramanujan graphs and their connection with eigenvalues of Hecke operators. The PI will continue her investigation of the Kloosterman conjecture over rational function fields. This conjecture implies that certain Kloosterman sums are eigenvalues of Hecke operators over function fields. The proposed approach includes the study of L-functions attached to curves over function fields and their connection to L-functions attached to representations. The second part is to obtain good results for partial character sums defined using additive and multiplicative characters of p-adic fields. If established, these bounds will have immediate applications in information theory to design better families of periodic sequences with low partial correlation. The proposed approach is a combination of the investigator's technique in estimating full character sums involving characters of p-adic fields and estimating partial character sums involving characters of finite fields. The PI plans to give a graduate course on "Number Theory with Applications" in 1996-97, which will include her past work in this area as well as the results from this proposal. This research falls into the general mathematical fields of Number Theory and Combinatorics. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mat hematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Combinatorics is also an indispensable tool in modern communications. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.