Banach Spaces with a Focus on Sobolev-Style Spaces, Frame Theory, and Quantum Graphs

Banach 空间，重点关注 Sobolev 式空间、框架理论和量子图

基本信息

批准号：
1900985
负责人：
Javier Chavez-Dominguez
金额：
$ 15.43万
依托单位：
University of Oklahoma Norman Campus
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：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900985&HistoricalAwards=false
关键词：
Banach Spaces Focus Sobolev Style

项目摘要

Banach spaces are a useful and powerful abstract framework to understand real-world data such as images, sound, or experimental results, and they do so in at least two different levels. First, for a particular instance of data, a Banach space provides a way to rigorously quantify various characteristics of that data. At a second level of abstraction, studying the structure of the whole space consisting of all possible instances of data has been crucial to the solution of certain problems, such as the prediction of the future behavior of the system being modeled. For this project, the spaces under consideration come from both signal processing (which deals with the problem of storing information about an object by considering it as a sum of simpler ones) and Quantum Information Science (which studies a mathematical framework for communications where one can encode information not as a string of 0s and 1s as today's computers do, but rather in the state of a quantum-mechanical system). The PI seeks to advance the knowledge of some spaces coming from the two aforementioned practical settings -- as well as related analytical considerations -- via fundamental research, including research conducted by undergraduate students under the PI's supervision. In addition, through his outreach and mentoring of postdocs, the PI will contribute to growing and diversifying the group of students and researchers in STEM fields.The project is divided into three parts. The first is inspired by an uncertainty principle in time-frequency analysis, which is related to Sobolev-style inequalities and spaces associated to finite graphs endowed with an extra "magnetic" structure. The name comes from the fact that the presence of a magnetic potential in some quantum-mechanical models of bonds between atoms is modeled not just with a graph, but also with an additional assignment of a complex number of modulus one to each edge of the graph. The second part of the project seeks to generalize the theory of frames, i.e. overcomplete bases, from the Hilbert space setting to the general Banach space one, where we no longer enjoy the advantages of having a large group of symmetries. Significant work has already been done in this direction, but mostly in the infinite-dimensional setting, and the PI will continue developing the nascent theory of frames on finite-dimensional Banach spaces. The third part is focused on quantum graphs, which are linear spaces of complex-valued matrices that come from Quantum Information Theory and can be considered as generalizations of classical combinatorial graphs. The PI will investigate quantum versions of a variety of classical results in graph theory, particularly those related to the aforementioned Sobolev-style inequalities.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.

Banach空间是一个有用且有力的抽象框架，可理解现实世界中的数据，例如图像，声音或实验结果，并且至少在两个不同的层面上这样做。首先，对于数据的特定实例，Banach空间提供了一种严格量化该数据的各种特征的方法。在第二级抽象上，研究由所有可能的数据实例组成的整个空间的结构对于解决某些问题的解决方案至关重要，例如预测被建模的系统的未来行为。对于此项目，所考虑的空间来自两个信号处理（这涉及通过将对象视为简单的信息来存储有关对象的信息的问题）和量子信息科学（其中研究了通信的数学框架，其中一个人可以将信息编码为当今计算机的0s和1s字符串，而是在当今的计算机上，而是在量子机械系统的状态下）。 PI试图通过基础研究来促进来自上述两个实际环境以及相关分析考虑的一些空间的知识，包括在PI的监督下由本科生进行的研究。此外，通过他对博士后的推广和指导，PI将有助于成长和多样化STEM领域的学生和研究人员。该项目分为三个部分。第一个是受时频分析中不确定性原理的启发，该原理与Sobolev风格的不等式和与有限图相关的空间有关，并具有额外的“磁性”结构。该名称来自以下事实：在某些原子之间的键合键模型中存在磁电势不仅是用图建模的，而且还可以在图形的每个边缘进行附加分配复杂数量的模量。该项目的第二部分旨在概括框架理论，即从希尔伯特（Hilbert）空间环境到一般的巴纳克（Banach Space One），我们不再享受拥有一大批对称性的优势。已经朝这个方向完成了重要的工作，但主要是在无限维度的环境中进行的，PI将继续在有限维的BANACH空间上发展框架的新生理论。第三部分集中在量子图上，量子图是来自量子信息理论的复杂值矩阵的线性空间，可以被视为经典组合图的概括。 PI将研究图理论中各种经典结果的量子版本，尤其是与上述Sobolev式不平等相关的量子版本。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛的影响审查标准来通过评估来进行评估的。