Operator Theory and Stable Polynomials

算子理论和稳定多项式

基本信息

批准号：
1900816
负责人：
Greg Knese
金额：
$ 19.1万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900816&HistoricalAwards=false
关键词：
Operator Theory Stable Polynomials

项目摘要

Operator theory is a broad and mature area of pure mathematics with close ties to the mathematics of quantum mechanics and control systems engineering. Indeed, von Neumann is often credited with formulating the foundations of operator theory as a language for quantum mechanics, while Nobert Wiener initiated an approach to engineering prediction problems (such as jet tracking) based on ideas from operator theory and harmonic analysis. In more recent decades, operator theory has been used in H-infinity control theory which has applications in automatic pilot design. On the other hand, a stable polynomial is not a field per se but a basic concept in mathematics that has become profoundly useful in the study of a diverse range of problems in mathematics: combinatorics (enumeration problems), graph theory (the study of networks), and operator theory. The purpose of this project is to use operator theory to study stable polynomials and vice versa. This project focuses on three problems: (1) the generalized Lax conjecture, (2) a quantitative understanding of linear preservers of stability, and (3) extensions of the theory of stable polynomials to entire functions. Semi-definite programming is a powerful technique in optimization and the generalized Lax conjecture is a bold assertion about the sets on which this theory can successfully be implemented. The conjecture claims that these feasibility sets have a geometric description in terms of hyperbolic polynomials (a slight generalization of the concept of a stable polynomial). Although evidently important in optimization, this would further establish links between stable polynomials and operator theory as the question is closely related to the concept of representing stable polynomials via operator theoretic determinantal representations. Problem (2) is about taking the highly successful theorems of Borcea-Branden that characterized the linear maps on stable polynomials that preserve stability and making them quantitative. How exactly do stability preservers modify zero sets? One approach to this problem is through analyzing how stability preservers affect various sums-of-squares formulas for stable polynomials. Problem (3) is about a natural extension of stable polynomials to various classes of entire functions (the Polya class, Hermite-Biehler class) and would require developing multivariable theories of Hilbert spaces of entire functions. This latter endeavor is important in its own right as the successes in one variable Hilbert spaces of entire functions have been profound (i.e. the Poltoratski approach to problems of uncertainty type).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.

操作员理论是纯数学的广阔而成熟的领域，与量子力学和控制系统工程的数学有着密切的联系。的确，冯·诺伊曼（Von Neumann）通常以将操作者理论的基础作为量子力学的语言而被认为是基于操作者理论和谐波分析的思想来提出工程预测问题（例如喷气跟踪）的方法。在最近几十年中，运算符理论已用于H-Infinity Control理论，该理论在自动飞行员设计中有应用。另一方面，稳定的多项式本身不是一个领域，而是数学中的基本概念，在研究数学中的各种问题：组合学问题（枚举问题），图理论（网络研究）和操作者理论方面已变得非常有用。该项目的目的是使用操作者理论研究稳定的多项式，反之亦然。该项目的重点是三个问题：（1）广义的洛杉矶国际念头，（2）对稳定性线性保留者的定量理解，以及（3）稳定多项式理论对整个功能的扩展。半明确编程是一种优化的强大技术，广义的LAX猜想是关于可以成功实施该理论的集合的大胆断言。该猜想声称这些可行性集在双曲多项式方面具有几何描述（对稳定多项式的概念的略有概括）。尽管在优化中显然很重要，但这将进一步建立稳定的多项式和操作者理论之间的联系，因为该问题与通过操作理论决定性确定表示代表稳定的多项式表示的概念密切相关。问题（2）是关于采用硼ce骨布兰登的非常成功的定理，这些定理表征了稳定多项式上的线性图，以保持稳定性并使它们进行定量。稳定性保留器如何修改零集？解决此问题的一种方法是分析稳定性保留器如何影响稳定多项式的各种方案公式。问题（3）是关于稳定多项式的自然扩展到各种类别的整个功能（Polya类，Hermite-Biehler类），并且需要开发整个功能的Hilbert Space的多变量理论。后者的努力本身就是重要的，因为一个整个功能的一个可变的希尔伯特空间中的成功是深刻的（即Poltoratski解决不确定性类型问题的方法）。这项奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的智力和广泛影响来评估CRETERIA的评估，这是值得的。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An [O iii] search for extended emission around AGN with H i mapping: a distant cloud ionized by Mkn 1

[O-iii] 使用 H-i 映射搜索 AGN 周围的扩展发射：由 Mkn 1 电离的遥远云

DOI：
10.1093/mnras/staa1510
发表时间：
2020
期刊：
Monthly Notices of the Royal Astronomical Society
影响因子：
4.8
作者：
Knese, Erin Darnell;Keel, William C;Knese, Greg;Bennert, Vardha N;Moiseev, Alexei;Grokhovskaya, Aleksandra;Dodonov, Sergei N
通讯作者：
Dodonov, Sergei N

Cyclicity Preserving Operators on Spaces of Analytic Functions in $${\mathbb {C}}^n$$

$${mathbb {C}}^n$$ 解析函数空间上的循环保留算子