Methods in nonlinear analysis, applications to differential and variational problems, and scientific computations

非线性分析方法、微分和变分问题的应用以及科学计算

• 批准号: 41779-2008
• 负责人: Krawcewicz, Wieslaw
• 金额: $ 1.22万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=421223
• 关键词: Methods; nonlinear; analysis; applications; differential

In my research I am developing new mathematical methods to investigate the impact of symmetries on dynamical systems. One can easily observe that almost every design (e.g. buildings, networks, configurations of devices, etc.) has certain symmetries, which are expressions of our tendencies for elegance. Very little is known, how symmetries affect the performance or reliability of such design (e.g. the earthquake resistance of buildings, electric systems, chemical processes). Even in simple models there are difficult problems leading to complicated analysis involving advanced algebraic and analytic method. My research area, the Equivariant Analysis, is focused on creating new tools (for example the equivariant degree) allowing effective and complete study of symmetric dynamical processes. My interests are two folded: theoretical and applied. In the theoretical part, based on the most recent achievements in various mathematical fields (such as algebraic and equivariant topology, transformation groups, algebra, differential geometry, differential equations, functional analysis, etc.), I am trying to understand the complex nature of abstract nonlinear symmetric equations, so it becomes possible to extract essential information about the solutions and their symmetric properties. This information leads to a development of new computational methods (involving algebraic, topological and analytical routines). In the applied part, we establish standard settings and create comprehensive routines, which allow usage of computers (e.g. Maple package was created for several groups of symmetries) in order to provide the user with quick and precise way to analyze dynamical systems, making our theoretical achievements accessible to applied mathematicians and even engineers. Our methods were used, for example, to investigate symmetric configurations of (electrical, chemical or biological) oscillators, in population models, or electrical transmission lines.

在我的研究中，我正在开发新的数学方法来研究对称性对动力系统的影响。我们可以很容易地观察到，几乎每一个设计（如建筑、网络、设备配置等）都有一定的对称性，这是我们对优雅倾向的表达。很少有人知道，对称如何影响这种设计的性能或可靠性（例如，建筑物，电力系统，化学过程的抗震能力）。即使在简单的模型中，也存在一些难题，需要运用先进的代数和解析方法进行复杂的分析。我的研究领域是等变分析，专注于创建新的工具（例如等变度），允许对对称动态过程进行有效和完整的研究。我的兴趣分两方面：理论和应用。在理论部分，基于各个数学领域（如代数和等变拓扑、变换群、代数、微分几何、微分方程、泛函分析等）的最新成果，我试图理解抽象非线性对称方程的复杂性，从而可以提取有关解及其对称性质的基本信息。这些信息导致了新的计算方法（包括代数、拓扑和分析例程）的发展。在应用部分，我们建立了标准设置，并创建了全面的例程，这些例程允许使用计算机（例如，为几个对称组创建了Maple包），以便为用户提供快速准确的分析动力系统的方法，使我们的理论成果能够被应用数学家甚至工程师使用。例如，我们的方法被用于研究（电、化学或生物）振荡器在人口模型或输电线路中的对称配置。