Combinatorial matrix analysis and algebra

我的工作重点是开发技术和结构，以探索数学的两个领域：代数和图论的相互作用。数学最基本的任务之一是求方程的解。19世纪早期发现的一个特别奇怪的事实是，5次或更高次的相对简单的多项式方程不允许代数解。因此，已经发展了求多项式方程根的数值方法。一种方法是找到Frobenius伴矩阵的特征值，例如在MATLAB软件的根命令中使用。****我们最近令人惊讶的发现可能为Frobenius伴矩阵提供有用和有效的替代，以及多项式根上的新界。我计划使用较新的环间伴矩阵来提供新的界，并表征在哪些条件下这些新界比Frobenius的界或最近的Fiedler矩阵的界更清晰。****我计划探索非稀疏伴矩阵和广义伴矩阵的性质。有证据表明，这些也可能提高算法的效率；一个特别的目标是找到容易屈服于平衡的版本，提供条件良好的矩阵，同时保留其优势结构。我还计划探索非稀疏模式，以确定哪种模式最适合高效的分割技术和更清晰的界限。****我计划揭示矩阵模式的特征值属性，确定哪些矩阵模式（如符号模式）具有特定的有用属性。符号模式矩阵的研究在一定程度上起源于诺贝尔经济学奖得主萨缪尔森的工作，但在其他建模环境中也越来越受到关注。近年来，矩阵模式分析已被应用于机器学习，也用于检测生物和生态系统中周期性的可能性。我不打算直接关注应用程序，相反，我的重点将放在开发理论上，这些理论可能在未来的建模和算法应用中证明是有用的。因此，我的工作是纯数学。我的贡献之一将是开发构造、技术和精细的惯性任意模式的属性。作为一种独特的方法，我将使用零模式来深入了解符号模式的组合结构。****我还计划探索图的代数性质（例如网络结构）：我计划为Graham和Pollack关于图寻址的经典问题开发一些代数技术。我还将继续开发与覆盖良好的图的边缘理想相关的图的属性，特别是关注图的顶点分解性和脱落顶点集的属性。***我的项目涉及适合研究生和本科生HQP培训的部分

The focus of my work is developing techniques and structures for exploring the interplay of two areas of mathematics: algebra and graph theory.***One of the most fundamental tasks of mathematics is finding the solutions of equations. A particularly curious fact, discovered in the early 19th century, is that relatively simple polynomial equations, of degree 5 or higher, do not allow for algebraic solutions. Consequently, numerical methods have been developed to find roots of polynomial equations. One method finds the eigenvalues of a Frobenius companion matrix, used for example in the roots command in MATLAB software. ****Our recent and surprising discovery of intercyclic companion matrices may provide useful and efficient alternatives to the Frobenius companion matrix, as well as new bounds on the roots of a polynomial. I plan to use the newer intercyclic companion matrices to provide new bounds, and to characterize under which conditions these new bounds are sharper than those from Frobenius, or the more recent Fiedler matrices. ****I plan to explore properties of the non-sparse companion matrices and generalized companion matrices. There is evidence that these too may improve efficiency of algorithms; one particular goal is to find versions that readily succumb to balancing, to provide well-conditioned matrices, while retaining their advantageous structure. I also plan to explore non-sparse patterns to determine which are most amenable to efficient splitting techniques and sharper bounds.****I plan to uncover eigenvalue properties of matrix patterns, determining which matrix patterns (such as sign patterns) have specific useful properties. The study of sign pattern matrices has some origins in the work of Nobel Laureate P. Samuelson in economics, but has growing interest in other modeling contexts. Recently matrix pattern analysis has been applied to machine learning, and also to detect the possibility of periodicity in biological and ecological systems. I do not plan to focus directly on applications, but instead my focus will be on developing theory that may prove useful in future modeling and algorithmic applications. As such my work is pure mathematics. One of my contributions will be to develop constructions, techniques, and properties of refined inertially arbitrary patterns. As one unique approach, I will use zero patterns to gain insight into the combinatorial structure of the sign patterns. ****I also plan to explore algebraic properties of graphs (e.g. network structures): I plan to develop some algebraic techniques for a classic problem of Graham and Pollack on graph addressing. I will also continue to develop properties of graphs related to the edge ideal of a well-covered graph specifically focusing on the vertex-decomposability of a graph and properties of the set of shedding vertices. ***My projects involve parts that are suitable for training of HQP at the graduate and undergraduate level.**