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.

该项目的目标是研究新的几何概念、多项式包含及其应用。牛顿首先发现，空心椭球体中的内部点感觉不到重力，而对于填充椭球体，内部点的重力是位置的二次函数。多项式包含是椭球在牛顿势方面的推广。 空心多项式包含中的内部点感受到的重力是该位置的多项式。多项式包含已被证明对于许多建模和设计问题很有用。具体来说，将研究与多项式包含物有关的以下应用：（i）合成复杂材料设计的预测模型，（ii）聚变反应堆的最佳结构和最小场集中，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外展活动相结合，例如一对一指导、教育模块开发、培训研究生和促进跨学科合作。由 k 次多项式包含引起的牛顿势恰好是体内 k 次多项式。对多项式包含的兴趣源于偏微分方程承认多项式包含与椭圆体一样的封闭形式简单解。该项目将解决多项式包含的以下基本技术问题：（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以及（iv）利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。