Combinatorial and spectral analysis of matrix patterns

这项拟议的研究探索了两个不同的数学领域，即线性代数和图论之间的相互作用。正在探索的一种特定类型的问题是数字数组上的数学如何受到定性信息的影响，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对结果的重大限制，而与数组中单个数字的大小无关。这些符号模式可以用图论来建模，这个理论可以提供对给定模式的限制的见解。这项研究的一部分需要确定这些模型可以发现哪些限制，同样地，这些模型如何帮助证明缺乏限制。 对手势模式的研究是由诺贝尔经济学奖得主P·萨缪尔森引发的。经济学中的各种问题都涉及对情况进行建模，虽然不可能获得准确的数字，但可以获得的定性信息更多。这样的问题也出现在生态学、人口生物学、流行病模型、化学和社会学中。模式分析在反问题中也很有帮助。这些问题可能出现在工程问题中，例如在结构分析中。逆问题通常涉及试图以某些规定的行为在物理上重建系统。通常情况下，系统会受到一些限制。模式分析可能有助于确定在给定约束的情况下，反问题是否可行。 我将把重点放在模式的数学上，以便开发在未来的建模应用程序开发中可能有用的理论。因此，这个项目是纯粹的数学。我将研究各种抽象模式，以确定图论信息如何为模式的使用方式中固有的限制提供指标。我将开发用于确定关于基于模式的限制的可知信息的方法。

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.