Matrix Inequalities in Engineering and Science

矩阵不等式在工程和科学（DMS 9704534） 李志光和罗伊·马蒂亚斯 矩阵不等式理论在许多领域都有应用， 电子工程，统计学，经济学，心理学，图论， 微分方程，运筹学和优化，算子理论， 量子物理、数值分析和控制理论。但不少 这些领域的研究人员无法使用现有的结果和证据， 矩阵理论中的技术来有效地发展他们的理论，因为他们的 专业知识不在矩阵理论中，并且缺乏沟通 他们和矩阵理论的研究人员。主要调查人员已经 在矩阵不等式方面有相当丰富的经验，并从 抄袭、改编或借用许多技术来研究这个问题。在这 项目，他们计划发展和应用矩阵不等式理论， 应用程序中出现的不同问题。 特别是，强调 将通过概念而不是计算来证明矩阵不等式 技术，并把它们放在更一般和更容易获得的设置中， 非专业人士也可以很容易地使用这些结果。 值得一提的是，这个项目与一些 有联邦战略利益的地区 例如，在高性能 计算，一个人需要能够有效和准确地进行计算。 在许多情况下，计算对舍入误差非常敏感 由机器本身引入。快速收敛方法的设计 矩阵不等式理论极大地简化了这些情况。 作为 在第一段中提到的，矩阵不等式是重要的工具， 控制理论，这是一个制度，是至关重要的设计制造 工厂，飞机，汽车等。矩阵不等式在 其他学科，包括量子物理和半定规划， 提出，经济学，统计学和心理学中出现的问题将 也要考虑。

Matrix Inequalities in Engineering and Sciences (DMS 9704534) Chi-Kwong Li and Roy Mathias The theory of matrix inequalities has applications in many fields including electrical engineering, statistics, economics, psychology, graph theory, differential equations, operations research and optimization, operator theory, quantum physics, numerical analysis and control theory. However, many researchers in these areas are not able to use the existing results and proof techniques in matrix theory to develop their theory effectively because their expertise is not in matrix theory and there is a lack of communication between them and researchers on matrix theory. The principal investigators have considerable experience in matrix inequalities and have developed from scratch, adapted, or borrowed many techniques to study the subject. In this project, they plan to develop and apply the theory of matrix inequalities to different problems that arise from applications. In particular, the emphasis will be on proving matrix inequalities by conceptual rather than computational techniques, and putting them in more general and accessible settings, so that the results can be readily used by non-specialists. It is worth mentioning that this project has strong connections with a number of areas of Federal strategic interest. For example, in high-performance computing, one needs to be able to do computation efficiently and accurately. In many situations, computations are very sensitive to round-off error introduced by the machine itself. The design of fast convergent methods in these cases is greatly facilitated by the theory of matrix inequalities. As mentioned in the first paragraph, matrix inequalities are important tools in control theory, a regime that is crucial to the design of manufacturing plants, aircraft, automobiles, etc. Applications of matrix inequalities in other subjects, including quantum physics and semi-definite programming are proposed , and problems arising in economics, statistics and psychology will also be considered.