Combinatorial matrix analysis and algebra

我的工作重点是开发技术和结构，探索数学的两个领域：代数和图论之间的相互作用。 数学最基本的任务之一就是求方程的解。19世纪初发现的一个特别奇怪的事实是，相对简单的5次或更高的多项式方程不允许有代数解。因此，发展了数值方法来寻找多项式方程的根。一种方法找出Frobenius伴随矩阵的特征值，例如在matlab软件中的ROUTS命令中使用。 我们最近对循环间伴随矩阵的惊人发现可能为Frobenius伴随矩阵提供有用和有效的替代，以及多项式的根的新的界。我计划使用较新的循环间伴随矩阵来提供新的界，并刻画在哪些条件下这些新的界比Frobenius或更新的Fiedler矩阵的新界更尖锐。 我计划研究非稀疏伴随矩阵和广义伴随矩阵的性质。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从于平衡的版本，提供条件良好的矩阵，同时保留其有利的结构。我还计划探索非稀疏模式，以确定哪些模式最适合使用高效的拆分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起源于诺贝尔经济学奖获得者P.Samuelson的工作，但在其他建模环境中的兴趣越来越大。最近，矩阵模式分析被应用于机器学习，也被用于检测生物和生态系统中周期性的可能性。我不打算直接专注于应用程序，而是将重点放在开发可能在未来的建模和算法应用程序中被证明有用的理论上。因此，我的工作是纯粹的数学。我的贡献之一将是开发精细的惯性任意模式的结构、技术和属性。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关于图寻址的一个经典问题开发一些代数技术。我还将继续发展与良好覆盖图的边理想有关的图的性质，特别是集中于图的顶点可分解性和脱落顶点集的性质。 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.