Problems in Computational Number Theory

计算数论中的问题

• 批准号: RGPIN-2014-04154
• 负责人: Hare, Kevin
• 金额: $ 1.02万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587634
• 关键词: Problems; Computational; Number; Theory

The primary focus of my research is in number theory. Typically I deal with problems with a computational or analytic flavour. I have additional interests in discrete geometry, ergodic theory, dynamical systems and fractals. Below I have listed some of the types of problems that I am interested in. This is in no way a comprehensive list. An algebraic integer is the root of a monic polynomial with integer coefficients. There are many special classes of algebraic integers that have been the focus of much research. Examples include cyclotomic numbers, Pisot numbers and Salem numbers. Properties concerning their Mahler measure, beta-expansions, spectra and Bernoulli convolutions are still not fully understood for many of these classes. A significant part of my research has historically been concerned with these properties. This is expected to continue. These investigations have a wide variety of applications. Many problems in this area are accessible to undergraduate or starting graduate students, and will often be the focus of HQP (Highly Qualified Personal) research. As a second example let q be a positive integer greater than 1. We denote by s_q(n) the sum of the q-ary digits of n. For example, if q = 10, then s_10(1729) = 1 + 7 + 2 + 9 = 19. In recent years, much effort has been made to get a better understanding of the distribution properties of s_q regarding certain subsequences of the positive integers. Particular attention has been given to the ratio s_q(p(n))/s_q(n) where p(n) is a polynomial with integer coefficents. Questions concerning which values this ratio can achieve are still not fully understood, and is something I plan to futher investigate. One of the main open conjectures about this is the average value of this ratio. Many problems in this area are accessible to undergraduate or starting graduate students, and will often be the focus of HQP research. As a last example, consider a set of N x N matrices. If we take long products of matrices from this set, one natural question to ask is: how quickly does this grow? More specifically, how quickly can the supremum of the matrix norm of long products of matrices grow? A second question might be: What does a "big" product look like? For example, is a "big" product dominated by a particular matrix, or pattern of matrices occurring multiple times? These sorts of questions have been looked at in the study of the joint spectral radius of a set of matrices. This is a very active area of research, with many unanswered and important questions. Much of this research will be conducted with Nikita Sidorov. A significant part of this proposal is for the training of highly qualified personal. I have had good success with training students, at the undergrad, graduate and postdoctoral level. I have numerous ideas for projects that are just waiting for the right student.

我研究的主要重点是数论。 通常我处理的问题与计算或分析的味道。 我对离散几何、遍历理论、动力系统和分形有额外的兴趣。 下面我列出了一些我感兴趣的问题类型。 这绝不是一个全面的清单。 一个代数整数是具有整数系数的一元多项式的根。 有许多特殊类别的代数整数一直是许多研究的焦点。 例子包括分圆数，皮索数和塞勒姆数。 关于它们的马勒测度、β-展开、谱和伯努利卷积的性质对于许多这些类仍然没有完全理解。 我的研究的一个重要部分历来与这些属性有关。 预计这种情况将继续下去。 这些研究有着广泛的应用。 这一领域的许多问题是本科生或研究生开始接触，往往是HQP（高素质的个人）研究的重点。 作为第二个例子，让q是大于1的正整数。 我们用s_q（n）表示n的q进制数字之和。 例如，如果q = 10，则s_10（1729）= 1 + 7 + 2 + 9 = 19。 近年来，人们对s_q关于正整数的某些连续性的分布性质作了大量的研究。特别注意了s_q（p（n））/s_q（n）的比值，其中p（n）是一个整数系数的多项式。 关于这个比率可以达到什么值的问题仍然没有完全理解，这是我计划进一步研究的问题。 关于这一点的一个主要的公开讨论是这个比率的平均值。 这一领域的许多问题是本科生或刚开始研究的学生可以解决的，并且通常是HQP研究的重点。 作为最后一个例子，考虑一组N × N矩阵。 如果我们从这个集合中取矩阵的长积，一个自然的问题是：它增长得有多快？ 更具体地说，矩阵的长积的矩阵范数的上确界增长的速度有多快？ 第二个问题可能是：一个“大”产品是什么样子的？ 例如，一个“大”产品是由一个特定的矩阵或矩阵模式多次出现主导的？ 这类问题在研究一组矩阵的联合谱半径时已经被研究过了。 这是一个非常活跃的研究领域，有许多尚未回答的重要问题。 这项研究的大部分将与尼基塔·西多罗夫一起进行。 这项建议的一个重要部分是培训高素质的人员。 我在培养学生方面取得了很大的成功，包括本科生、研究生和博士后。 我有很多项目的想法，只是在等待合适的学生。