Polynomial inclusions: open problems and potential applications

多项式包含：开放问题和潜在应用

• 批准号: 1410273
• 负责人: Liping Liu
• 金额: $ 19.1万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1410273&HistoricalAwards=false
• 关键词: Polynomial; inclusions; open; problems; potential

The goal of this project is to study a new geometric concept, polynomial inclusions, and their applications. It was first discovered by Newton that an interior point in a hollow ellipsoid feels no gravitational force, and that for a filled ellipsoid the gravity inside is a quadratic function of the position. Polynomial inclusions are a generalization of ellipsoids in terms of Newtonian potential. An interior point in a hollow polynomial inclusion feels a gravitational force that is a polynomial of the position. Polynomial inclusions have proven to be useful for many modeling and design problems. Specifically, the following applications pertaining to polynomial inclusions will be investigated: (i) predictive models for design of synthetic complex materials, (ii) optimal structures for fusion reactors and minimum field concentration, and (iii) inverse source problem for medical and geological imaging. This research project will be integrated with educational and outreach activities such as one-to-one mentoring, educational module development, training graduate students, and fostering interdisciplinary collaborations. The Newtonian potential induced by a polynomial inclusion of degree k is precisely a polynomial of degree k inside the body. The interest in polynomial inclusions arises from that partial differential equations admit closed-form simple solutions for polynomial inclusions as for ellipsoids. The following fundamental technical problems concerning polynomial inclusions will be addressed in the project: (i) proving the existence and uniqueness of polynomial inclusions, (ii) numerically computing polynomial inclusions, (iii) explicit parametrization of polynomial inclusions and (iv) employing polynomial inclusions to solve aforementioned engineering problems. The outcomes of this project may have an impact on the classic Hilbert's sixteenth problem as well as industrial problems ranging from structural engineering, fusion designs and to imaging. The project will also strengthen the connection between the Department of Mathematics and the Department of Mechanical Aerospace Engineering at Rutgers University by means of recruiting students of diverse backgrounds into the research team.

本计画的目标是研究一个新的几何概念，多项式内含及其应用。牛顿首先发现，空心椭球的内部点不受重力的影响，而对于填充的椭球，内部的重力是位置的二次函数。多项式内含物是椭球体在牛顿势方面的推广。中空多项式内含物中的一个内部点受到的引力是其位置的多项式。多项式包含已被证明对许多建模和设计问题是有用的。具体而言，将研究下列与多项式包体有关的应用：(i)设计合成复杂材料的预测模型，（ii）聚变反应堆的最佳结构和最小场浓度，以及（iii）医学和地质成像的逆源问题。该研究项目将与教育和推广活动相结合，如一对一指导、教育模块开发、研究生培训和促进跨学科合作。由k次多项式包涵引起的牛顿势在物体内部恰好是k次多项式。多项式内含物之所以引起人们的兴趣，是因为偏微分方程对于多项式内含物和椭球体一样，都有闭形式的简单解。本项目将解决多项式内含物的以下基本技术问题：(i)证明多项式内含物的存在性和唯一性；（ii）多项式内含物的数值计算；（iii）多项式内含物的显式参数化；（iv）利用多项式内含物解决上述工程问题。这个项目的结果可能会对经典的希尔伯特第十六问题以及从结构工程、聚变设计到成像等工业问题产生影响。该项目还将通过招募不同背景的学生加入研究团队，加强罗格斯大学数学系和机械航空航天工程系之间的联系。