Combinatorial and spectral analysis of matrix patterns

拟议的研究探讨了两个不同的数学领域之间的相互作用，即线性代数和图论。 正在探索的一个特殊类型的问题是，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到对结果的显著限制，而与数组中单个数字的大小无关。这些符号模式可以使用图论建模，并且该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可以用这些模型发现哪些限制，以及这些模型如何帮助证明缺乏限制。 对符号模式的研究是由诺贝尔经济学奖获得者P.萨缪尔森引发的。经济学中的各种问题都涉及到建模的情况下，不可能得到确切的数字，但更多的定性信息是可用的。这样的问题也出现在生态学、种群生物学、流行病建模、化学和社会学中。模式分析在逆问题中也很有用。这些问题可能出现在工程问题中，例如结构分析。 逆问题通常涉及试图用一些规定的行为物理地重建系统。通常，系统会受到限制。模式分析可能有助于确定，如果给定的约束条件，逆问题是可行的。 我将专注于模式的数学，以发展理论，可能是有用的建模应用程序的未来发展。因此，这个项目是纯数学。我将研究各种抽象模式，以确定图论信息提供指示器的方式，这些指示器指示模式使用方式中的固有限制。我将开发用于确定基于模式的限制可以知道什么的方法。

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.