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，多项式包含及其应用。牛顿首先发现，在一个中空的椭球体中，内部的点感受不到重力，而对于一个充满的椭球体，内部的重力是位置的二次函数。多项式包含是椭球在牛顿势下的推广。 空心多项式包含体中的内点感受到的引力是位置的多项式。多项式包含已被证明是有用的许多建模和设计问题。具体而言，以下应用程序有关的多项式夹杂物将进行调查：（i）预测模型的合成复杂材料的设计，（ii）最佳结构的聚变反应堆和最小场浓度，（iii）逆源问题的医学和地质成像。 该研究项目将与教育和推广活动相结合，如一对一的指导，教育模块开发，培训研究生和促进跨学科合作。由一个k次多项式包含所引起的牛顿势，在物体内部就是一个k次多项式。对多项式包含的兴趣来自于偏微分方程允许多项式包含的封闭形式简单解。该项目将解决以下有关多项式包含的基本技术问题：（i）证明多项式包含的存在性和唯一性，（ii）数值计算多项式包含，（iii）多项式包含的显式参数化和（iv）使用多项式包含来解决上述工程问题。该项目的成果可能会对经典的希尔伯特第十六问题以及从结构工程，融合设计到成像的工业问题产生影响。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。