CAREER: Scaling and self-similarity in nonlinear science-education and research

职业：非线性科学教育和研究中的标度和自相似性

• 批准号: 0748482
• 负责人: Govind Menon
• 金额: $ 56.07万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0748482&HistoricalAwards=false
• 关键词: CAREER; Scaling; self; similarity; nonlinear

Date: November 21, 2007Proposal: DMS- 0748482PI: Menon, Govind Institution: Brown UniversityTitle: CAREER: Scaling and self-similarity in nonlinear science-education and researchABSTRACTEmpirical scaling laws are often the only signature of order in complex nonlinear problems. Such laws arise in diverse areas such as the study of biological networks, earthquakes and fully turbulent flows. The investigator's research vision is to bring some of these laws within the scope of rigorous mathematical analysis. To this end, seemingly unrelated problems are treated within a unified framework that combines mathematical methods from dynamical systems, partial differential equations and probability theory. Explicit connections between these areas are used to provide a fundamental understanding of the microscopic origin of `universal' scaling laws in (a) models of domain coarsening in materials science and physical chemistry, (b) dynamic scaling of transport coefficients in polymer chemistry, (c) models of turbulence. Analogies with classical probability are also used to guide new investigations of universality in dynamical systems.These research goals are closely tied to concrete improvements in education and training. Scaling and self-similarity are used as a unifying theme in a program for curricular innovation for undergraduates that stresses the extraction of simple quantitative answers from complex models. This includes the development of new freshman and senior seminars and an REU program.This mathematical research is motivated by problems of fundamental technological significance in fluid mechanics, materials science and physical chemistry. Studies of domain growth in materials science improve understanding of the stability of binary alloys and coarsening of nanoscale islands on thin-film substrates. Improved numerical methods for the simulation of polymers advance the analysis of biochemical flows, such as the manipulation of individual strands of DNA in confined channels. A deeper understanding of universality in dynamical systems enhances knowledge of the transition to chaos in several fluid flows, and is the first step in the control of such flows. The plans for undergraduate education place early emphasis on the basic utility of scaling analysis in emerging areas for the application of mathematics such as biology, physical chemistry and geosciences. They also include the development of innovative expositions at the undergraduate level of successful mathematical research in these fields. This contributes to a growing demand for mathematical sophistication in these areas, and is critical for sustained impact. The educational plans stress small classes, undergraduate research, and outreach to under-represented groups, in the firm belief that careful, individual mentoring of students is the foundation for a diverse and talented scientific workforce.

日期：年月日2007年11月21日提案：DMS-0748482 PI：Menon，Govind机构：布朗大学标题：职业生涯：非线性科学教育和研究中的标度和自相似性摘要经验标度律往往是复杂非线性问题中唯一的阶次特征。这些定律出现在不同的领域，如生物网络、地震和完全湍流的研究。研究者的研究愿景是将其中一些定律纳入严格的数学分析范围。为此，看似无关的问题被处理在一个统一的框架内，结合了动力系统，偏微分方程和概率论的数学方法。这些领域之间的显式连接被用来提供一个基本的理解微观起源的“通用”标度律在（a）在材料科学和物理化学领域粗化模型，（B）动态标度的传输系数在聚合物化学，（c）湍流模型。与经典概率的类比也被用来指导对动力系统普适性的新研究。这些研究目标与教育和培训的具体改进密切相关。尺度和自相似性被用作一个统一的主题，在课程创新的本科生，强调简单的定量答案提取复杂的模型。这包括新的大一和大四研讨会和REU计划的发展。这种数学研究的动机是在流体力学，材料科学和物理化学的基本技术意义的问题。在材料科学领域，畴生长的研究提高了对二元合金的稳定性和薄膜基底上纳米级岛的粗化的理解。用于模拟聚合物的改进的数值方法推进了生化流的分析，例如在受限通道中操纵DNA的单个链。更深入地了解动力系统的普遍性，增强了对几种流体流向混沌过渡的认识，也是控制这种流的第一步。本科教育计划早期强调在生物学、物理化学和地球科学等新兴数学应用领域中尺度分析的基本效用。它们还包括在这些领域成功的数学研究的本科层次的创新博览会的发展。这有助于这些领域对数学复杂性的需求不断增长，并对持续影响至关重要。教育计划强调小班，本科研究和推广到代表性不足的群体，坚信对学生的认真，个别辅导是多元化和有才华的科学工作者的基础。