Operator Analysis and Applications

算子分析及应用

• 批准号: 2054199
• 负责人: John McCarthy
• 金额: $ 45万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054199&HistoricalAwards=false
• 关键词: Operator; Analysis; Applications

Operator Theory was originally developed as a mathematical language to describe Quantum Mechanics. It has spread to many other areas in pure and applied mathematics, such as Control Theory. Control Theory is the design of systems like automatic pilots and self-driving cars that must react, but not over-react, to their environment. As the systems get more complex, and more features are incorporated, new problems arise. The PI will work on developing new mathematical tools to deal with these problems. This project also contributes to US workforce development through the training of graduate students.The PI will study problems in Operator theory, function theory, and the interaction between them. Non-commutative functions are functions whose input is two (or more) operators, and whose output is an operator. This is a relatively new field, but already has applications in control theory, algebraic geometry, and semi-definite programming. The PI will study non-commutative functions, which can often shed light on the commutative case, and have recently become important in Control Theory. The methods will be a combination of techniques from multivariable operator theory, functional analysis and several complex variables. In addition, the PI will work on using mathematical models to study the development of Alzheimer's disease, the analysis of acoustic signals, and risk measurement models for COVID-19.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.

操作者理论最初是作为数学语言开发的，可以描述量子力学。它已经传播到纯数学和应用数学的许多其他领域，例如控制理论。控制理论是自动飞行员和自动驾驶汽车等系统的设计，这些系统必须对其环境做出反应但不会反应过度的系统。随着系统变得越来越复杂，并且会纳入更多功能，因此出现了新的问题。 PI将致力于开发新的数学工具来解决这些问题。该项目还通过培训研究生为美国的劳动力发展做出了贡献。PI将研究操作员理论，功能理论及其之间的相互作用中的问题。非共同功能是功能，其输入是两个（或更多）运算符，并且其输出是操作员。这是一个相对较新的领域，但已经在控制理论，代数几何和半明确编程中具有应用。 PI将研究非交通功能，这些功能通常可以阐明交换案例，并且最近在控制理论中变得很重要。这些方法将是多变量运算符理论，功能分析和几个复杂变量的技术的组合。此外，PI将努力使用数学模型研究阿尔茨海默氏病的发展，声学信号的分析以及COVID-119的风险测量模型。这项奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来支持的，审查了审查的审查标准。

项目成果

Symbol functions for symmetric frameworks

对称框架的符号函数

• DOI: 10.1016/j.jmaa.2020.124895
• 发表时间: 2021
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Kastis E

The common range of co-analytic Toeplitz operators on the Drury-Arveson space

Drury-Arveson 空间上协同分析 Toeplitz 算子的公共范围

• DOI: 10.1007/s11854-022-0265-9
• 发表时间: 2023
• 期刊: Journal d'Analyse Mathématique
• 作者: Aleman, Alexandru;Hartz, Michael;McCarthy, John E.;Richter, Stefan
• 通讯作者: Richter, Stefan

White matter hyperintensity longitudinal morphometric analysis in association with Alzheimer disease

与阿尔茨海默病相关的白质高信号纵向形态测量分析

• DOI: 10.1002/alz.13377
• 发表时间: 2023
• 期刊: Alzheimer's & Dementia
• 作者: Strain, Jeremy Fuller;Phuah, Chia‐Ling;Adeyemo, Babatunde;Cheng, Kathleen;Womack, Kyle B.;McCarthy, John;Goyal, Manu;Chen, Yasheng;Sotiras, Aristeidis;An, Hongyu
• 通讯作者: An, Hongyu

Asymptotic Müntz–Szász theorems

渐近 MüntzâSzász 定理

• DOI: 10.4064/sm220713-19-2
• 发表时间: 2023
• 期刊: Studia Mathematica
• 影响因子: 0.8
• 作者: Agler, Jim;McCarthy, John E.
• 通讯作者: McCarthy, John E.

Complete norm-preserving extensions of holomorphic functions

全纯函数的完整保模扩展

• DOI: 10.1007/s11856-022-2415-2
• 发表时间: 2023
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: Agler, Jim;Kosiński, Łukasz;McCarthy, John E.
• 通讯作者: McCarthy, John E.