Combinatorial and spectral analysis of matrix patterns

矩阵模式的组合和谱分析

• 批准号: 203336-2011
• 负责人: VanderMeulen, Kevin
• 金额: $ 0.73万
• 依托单位: Redeemer University College
• 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=569932
• 关键词: Combinatorial; spectral; analysis; matrix; patterns

The proposed research explores the interplay between two different areas of mathematics, namely linear algebra and graph theory. One particular type of problem being explored is how mathematics on arrays of numbers is affected by qualitative information, such as the pattern of positive, negative and zero entries in the array. Sometimes this information alone is sufficient to observe significant restrictions on results, independent of the magnitude of the individual numbers in the array. These sign patterns can be modeled using graph theory and this theory can provide insights into the restrictions for given patterns. Part of this research entails determining what restrictions can be discovered with these models, and likewise, how these models can help demonstrate lack of restrictions. The study of sign patterns was sparked by Nobel Laureate P. Samuelson in economics. Various problems in economics involve modeling situations when it is impossible to get exact numbers, but the more qualitative information is available. Such problems occur also in ecology, population biology, modeling epidemics, chemistry and sociology. Pattern analysis can also be helpful in inverse problems. These problems can arise in engineering problems, such as in structural analysis. An inverse problem generally involves trying to physically reconstruct a system with some prescribed behaviours. Usually there are constraints on the system. Pattern analysis could be useful in determining if, given the constraints, the inverse problem is feasible. I will focus on the mathematics of patterns in order to develop theory that may be useful in future development of modeling applications. As such this project is pure mathematics. I will be studying various abstract patterns to determine ways in which the graph theoretical information provides indicators as to restrictions inherent in the way the pattern can be used. I will be developing the methods used to determine what can be known about restrictions based on patterns.

拟议的研究探讨了两个不同领域的数学领域之间的相互作用，即线性代数和图理论。 探索的一种特殊类型的问题是数字数组上的数学如何受定性信息的影响，例如数组中正，负和零条目的模式。有时，仅此信息就足以观察到对结果的显着限制，而与阵列中单个数字的大小无关。可以使用图理论对这些符号模式进行建模，该理论可以提供有关给定模式限制的见解。这项研究的一部分需要确定这些模型可以发现哪些限制，并且同样，这些模型如何帮助证明缺乏限制。 签名模式的研究是由诺贝尔奖获得者P. Samuelson在经济学中引发的。经济学上的各种问题涉及在无法获得确切数字的情况下进行建模情况，但是可以使用更多的定性信息。这种问题也发生在生态学，人群生物学，建模流行病，化学和社会学。模式分析也可能在反问题中有所帮助。这些问题可能在工程问题中出现，例如结构分析。 一个逆问题通常涉及试图通过某些规定的行为进行物理重建系统。通常，系统上有限制。模式分析可能在确定鉴于约束的情况下是否可行，可能是有用的。 我将重点介绍模式的数学，以发展可能对建模应用的未来开发有用的理论。因此，这个项目是纯粹的数学。我将研究各种抽象模式，以确定图形理论信息为使用模式使用方式固有的限制提供指标。我将开发用于确定基于模式的限制的方法。