Digits, distributions, and rigidity in dynamical systems

动力系统中的数字、分布和刚度

• 批准号: RGPIN-2016-04643
• 负责人: Berger, Arno
• 金额: $ 1.09万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615206
• 关键词: Digits; distributions; rigidity; dynamical; systems

Dynamical systems tools and techniques are indispensable to modern science and technology as they enable accurate modelling of processes ranging from space flight to cell growth to economic shocks. The wide scope of dynamical systems and the vast amounts of data generated by them have made the study of their quantitative and computational aspects an important research area. To this area the proposed program will contribute in several ways. The research topics can roughly be grouped as they relate, respectively, to digits, distributions, and rigidity. Digits. The program's main long-term goal is to establish a comprehensive theory of digits, their properties and distributions, for dynamical systems. A prominent phenomenon in this context is the surprising ubiquity of one particular logarithmic distribution often referred to as Benford's Law. Regarding the latter, many mathematical questions remain which the program will address, building not least on results obtained under the applicant's previous Discovery Grant. Recent advances in the theory of dynamical systems will be utilized to precisely quantify the randomness inherent to the digits generated by familiar one- and multi-dimensional systems. Distributions. Another important goal is to carry out an in-depth analysis of probabilities induced by digit statistics, notably on the circle, the discretization and (constrained) approximation properties of which are only partly understood at present. This analysis will include the study of novel quantitative concepts such as distortion (with respect to scale, base, or sum) and spread of probability measures for which no theoretical framework exists at present, but also refinements of classical results in uniform distribution theory. Rigidity. A third common theme of the program is rigidity. Informally put, in case of rigidity far fewer objects with certain properties exist than might reasonably be expected. Many deep rigidity results have been established in dynamics. To this existing zoo, the program will introduce two new species, namely digit rigidity and equivalence rigidity. Both forms of rigidity provide unexpected insights into the behavior of linear systems, arguably the simplest class of dynamical systems widely used throughout the sciences. The core parts of the proposed program are highly interrelated. Together, they provide a deepened understanding of important quantitative aspects of dynamical systems. The results established under this program will be of interest to other areas of pure and applied mathematics, notably number theory, optimal transport, and statistics. They will lead to new or more reliable statistical tests and data plausibility checks (e.g., with regard to potential fraud in accounting, election, or scientific data) as well as to improved numerical schemes that will benefit all scientists who use dynamical systems as models for real-world processes.

动态系统工具和技术对于现代科学和技术来说是必不可少的，因为它们可以准确地建模从太空飞行，细胞增长到经济冲击。动态系统的广泛范围以及它们产生的大量数据使其定量和计算方面成为重要的研究领域。在这个领域，拟议的计划将以几种方式做出贡献。研究主题分别与数字，分布和刚性分别进行分组。 数字。该计划的主要长期目标是为动态系统建立一个全面的数字理论，其属性和分布理论。在这种情况下，一个突出的现象是一种特殊的对数分布的惊人无处不在，通常称为本福德定律。关于后者，该计划将解决许多数学问题，尤其是基于申请人以前的发现赠款获得的结果。动态系统理论的最新进展将被用来精确量化熟悉的一维系统生成的数字固有的随机性。 分布。另一个重要的目标是对数字统计引起的概率进行深入分析，尤其是在圈子上，离散化和（约束）近似属性的概率，其目前仅部分了解。该分析将包括对诸如变形（相对于规模，基础或总和）等新型定量概念的研究以及目前尚无理论框架的概率度量的扩散，但也以统一分布理论的经典结果对经典结果进行了改进。 刚性。该程序的第三个共同主题是刚性。在刚性的情况下，非正式地说，具有某些特性的物体要少于合理的预期。在动态中已经建立了许多深度刚度结果。对于现有的动物园，该计划将引入两个新物种，即数字刚度和等效性刚度。两种形式的刚度都提供了对线性系统行为的意外见解，这可以说是在整个科学中广泛使用的最简单的动力系统类别。 拟议程序的核心部分高度相互关联。他们一起，对动态系统的重要定量方面进行了深入的了解。根据该计划建立的结果将引起其他纯数学和应用数学领域，尤其是数字理论，最佳运输和统计数据。它们将导致新的或更可靠的统计检验和数据可比性检查（例如，关于会计，选举或科学数据的潜在欺诈），以及改进的数值方案，这些方案将使所有使用动态系统作为现实世界过程的模型的科学家受益。