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Modern Aspects of Multivariable Operator Theory and Matrix Analysis

多变量算子理论和矩阵分析的现代方面

基本信息

批准号：
2000037
负责人：
Hugo Woerdeman
金额：
$ 24.9万
依托单位：
Drexel University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000037&HistoricalAwards=false
关键词：
Modern Aspects Multivariable Operator Theory

项目摘要

Operator theory and matrix analysis are both fundamental areas of mathematics. In large part, they were originally developed to provide a theoretical basis for quantum mechanics and other physical phenomena. An increasing number of application areas have emerged as a result of breakthroughs in operator theory and matrix analysis, which testify to their underlying importance to science and engineering. The areas of application closest to the research in this project are control systems engineering, electrical engineering, signal processing, image processing, and quantum computation. The principal investigator will build on his past work to further develop the interplay between the subdisciplines of free function theory, operator completions, statistical signal processing, matrix inequalities, and optimization. To accomplish this goal the principal investigator will continue existing collaborations as well as develop new ones, and engage actively with both graduate students and undergraduate students in the emerging research. The principal investigator will continue to maintain an intellectual environment fostering student involvement and development, providing the students with the skills, experience, and confidence to successfully pursue a career in the mathematical sciences. Thus, the project will yield both new, impactful mathematical results, as well as highly trained mathematicians prepared to join the scientific and educational workforce crucial to this nation. Many questions in system and control theory, filter design, signal and image processing come down to function theoretic questions. The case of several variables is a highly active research area where the techniques of multivariable operator theory are highly effective. The specific themes of the current project include (i) Matrix completions, (ii) Moment problems, (iii) Free function theory, (iv) Realizations, (v) Determinantal representations, (vi) Numerical range and radius, and their generalizations, (vii) Hypergeometric functions, and (viii) Inverse eigenvalue problems. This combination of areas will lead to new avenues of research that are of interest to different research groups. All projects also have a computational component, allowing for the implementation of the results and the potential to be used by researchers in all areas of science and engineering. The principal investigator will continue running an Analysis Seminar at Drexel University featuring local and international researchers, as well as Drexel students. The principal investigator and his students will disseminate the results via conference presentations and publications in a variety of leading mathematical journals and preprint servers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

算子理论和矩阵分析都是数学的基础领域。在很大程度上，它们最初的开发是为了为量子力学和其他物理现象提供理论基础。由于算子理论和矩阵分析的突破，出现了越来越多的应用领域，这证明了它们对科学和工程的潜在重要性。与该项目研究最接近的应用领域是控制系统工程、电气工程、信号处理、图像处理和量子计算。首席研究员将在他过去的工作基础上进一步发展自由函数理论、算子完成、统计信号处理、矩阵不等式和优化等子学科之间的相互作用。为了实现这一目标，首席研究员将继续现有的合作并开发新的合作，并积极与研究生和本科生参与新兴研究。首席研究员将继续维持一个促进学生参与和发展的智力环境，为学生提供成功从事数学科学职业的技能、经验和信心。因此，该项目将产生新的、有影响力的数学成果，并培养训练有素的数学家，准备加入对这个国家至关重要的科学和教育队伍。系统和控制理论、滤波器设计、信号和图像处理中的许多问题都可以归结为函数论问题。多变量的情况是一个非常活跃的研究领域，其中多变量算子理论技术非常有效。当前项目的具体主题包括（i）矩阵完成，（ii）矩问题，（iii）自由函数理论，（iv）实现，（v）行列式表示，（vi）数值范围和半径及其概括，（vii）超几何函数，以及（viii）逆特征值问题。这种领域的结合将带来不同研究小组感兴趣的新研究途径。所有项目还具有计算组件，允许结果的实施以及科学和工程所有领域的研究人员使用的潜力。首席研究员将继续在德雷克塞尔大学举办分析研讨会，由当地和国际研究人员以及德雷克塞尔学生参加。首席研究员和他的学生将通过会议演讲和在各种领先数学期刊和预印本服务器上的出版物来传播结果。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Completing an Operator Matrix and the Free Joint Numerical Radius

完成算子矩阵和自由联合数值半径

DOI：
10.1007/s11785-022-01273-0
发表时间：
2022
期刊：
Complex Analysis and Operator Theory
影响因子：
0.8
作者：
Rosa, Kennett L.;Woerdeman, Hugo J.
通讯作者：
Woerdeman, Hugo J.

Minimal Realizations and Determinantal Representations in the Indefinite Setting

不定环境中的最小实现和行列式表示

DOI：
10.1007/s00020-022-02697-1
发表时间：
2022
期刊：
Integral Equations and Operator Theory
影响因子：
0.8
作者：
Jackson, Joshua D.;Woerdeman, Hugo J.
通讯作者：
Woerdeman, Hugo J.

Upper bounds for positive semidefinite propagation time

正半定传播时间的上限

DOI：
10.1016/j.disc.2022.112967
发表时间：
2022
期刊：
Discrete Mathematics
影响因子：
0.8
作者：
Hogben, Leslie;Hunnell, Mark;Liu, Kevin;Schuerger, Houston;Small, Ben;Zhang, Yaqi
通讯作者：
Zhang, Yaqi

The autoregressive filter problem for multivariable degree one symmetric polynomials

多元一阶对称多项式的自回归滤波问题