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.

该项目的目的是研究一种新的几何概念，多项式包含及其应用。牛顿首先发现，空心椭圆形中的一个内点没有引力力，而对于填充的椭圆形，内部的重力是该位置的二次功能。多项式夹杂物在牛顿电位方面是椭圆形的概括。 空心多项式包容中的一个内点感觉是该位置多项式的重力。事实证明，多项式夹杂物对许多建模和设计问题很有用。具体而言，将研究与多项式夹杂物有关的以下应用：（i）合成复合物材料设计的预测模型，（ii）融合反应器的最佳结构和最小的场浓度，以及（iii）医疗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，例如一对一的指导，教育模块发展，培训研究生以及促进跨学科合作。由多项式k的多项式k诱导的牛顿电势恰好是体内度k的多项式。对多项式夹杂物的兴趣来自该部分微分方程，即对多项式夹杂物的封闭式简单解决方案，如椭圆形。在项目中将解决以下有关多项式包含的基本技术问题：（i）证明多项式夹杂物的存在和独特性，（ii）数值计算多项式夹杂物，（iii）使用多项式包容性工程的多项式夹杂物的明确参数化和（iiv）。该项目的结果可能会影响经典希尔伯特（Hilbert）的第十六个问题以及从结构工程，融合设计和成像的工业问题。该项目还将通过将不同背景的学生招募到研究团队的数学系与机械航空工程系之间的联系。