Matrix Analysis and Applications

矩阵分析与应用

• 批准号: RGPIN-2014-06449
• 负责人: Lancaster, Peter
• 金额: $ 0.8万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589011
• 关键词: Matrix; Analysis; Applications

Many physical systems (aircraft, string instruments, oceans, skyscrapers, etc.) experience oscillatory behaviour in some form. This behaviour may be beneficial, harmless, or catastrophic depending on the applied loads and the environment. However, mathematical models can be used to analyse vibrations and they have certain mathematical features in common. In particular, in the context of small vibrations linear mathematical/computational models achieve extraordinary success in the prediction and modification of oscillatory systems. Theories have evolved over a period of about three-hundred years, but new mathematical and computational techniques have developed explosively in the last fifty years. Problems in this area provide a major stimulus for my research. However, (as in the past) I also venture into generalizations of these ideas concerning properties of more general matrix and operator-valued functions, and into other areas of application such as mathematical biology, geophysics, and the design of algorithms. In particular, and together with collaborators where appropriate, I plan to continue with research in the direction of so-called "inverse problems", in which performance characteristics are prescribed (generally in the form of spectral data), and then systems are designed to reproduce these characteristics.

许多物理系统（飞机、弦乐器、海洋、摩天大楼等）都以某种形式经历振荡行为。这种行为可能是有益的、无害的，也可能是灾难性的，这取决于所施加的负载和环境。然而，数学模型可以用来分析振动，它们具有某些共同的数学特征。特别是在小振动的情况下，线性数学/计算模型在预测和修正振荡系统方面取得了非凡的成功。理论已经发展了大约三百年，但是新的数学和计算技术在过去的五十年里得到了爆炸性的发展。这一领域的问题为我的研究提供了重要的动力。然而，（和过去一样）我也将这些关于更一般的矩阵和算子值函数的性质的思想进行了推广，并将其应用于数学生物学、地球物理学和算法设计等其他领域。