REU Site: Inverse Problems for Electrical Networks

REU 站点：电力网络的反问题

• 批准号: 1460937
• 负责人: James Morrow
• 金额: $ 24.55万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-04-01 至 2018-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1460937&HistoricalAwards=false
• 关键词: REU; Site; Inverse; Problems; Electrical

The REU program on Inverse Problems for Electrical Networks will bring eight undergraduates to the University of Washington in each of the summers of 2015-2017 to work on inverse problems for electrical networks and related problems on graphs. The students will also investigate problems in combinatorics and special topics in number theory. The simplest inverse problem for an electrical network is to determine the conductance of each of the conductors in an electrical network when the boundary current response of the network to boundary voltages is known, but no interior knowledge is available. Imagine a box in which there is an unknown electric circuit with leads protruding from the box at which current and voltage can be measured. The solution of the problem can be interpreted as providing an electrical version of an x-ray. A simplified version of the inverse problem is the problem of mine detection with an electrical field. The intent of the program is to guide students into independent research projects which can be completed in the eight week period. The students will be given reading material before the program starts and introduce them to research problems in the first week. Students will learn how to formulate and attack research problems in a supportive, collaborative atmosphere. An emphasis will be placed on communicating results. Prior students will work as TAs and mentor current students. Alumni of the program will be frequent visitors and provide additional help. There are difficult technical questions that students have asked and answered. In recent years students have discovered exotic electrical networks that cannot be recovered by boundary data but are almost uniquely determined. In other words there are only a finite number of possibilities for networks compatible with the given boundary measurements. These are called networks n-to-1 networks. Students in the program invented these networks and discovered their properties. New examples are constructed by gluing certain "four-star" graphs in a pattern that looks something like a cactus or shoelace graph. There are now many different families of such graphs, which will be investigated. Another topic is properties of permutations, especially peak points of permutations. Students have recently found interesting properties of peak polynomials, for instance that all peaks in a peak set are zeros of its corresponding peak polynomial. These and other topics will be the building blocks of this program.

在电气网络逆问题的REU计划将带来八个本科生在2015年至2017年的每个夏天的华盛顿大学工作的电气网络和相关问题的反问题的图形。学生还将研究组合数学和数论中的特殊主题的问题。 最简单的反问题是当网络对边界电压的边界电流响应是已知的，但没有内部知识可用时，确定电网络中每个导体的电导。想象一个盒子，里面有一个未知的电路，从盒子里伸出的引线可以测量电流和电压。 该问题的解决方案可以解释为提供了X射线的电子版本。 逆问题的简化版本是利用电场进行地雷探测的问题。 该计划的目的是引导学生进入独立的研究项目，可以在八周内完成。 在课程开始前，学生将获得阅读材料，并在第一周向他们介绍研究问题。 学生将学习如何制定和攻击的支持，协作的气氛中的研究问题。 重点将放在交流结果上。 以前的学生将担任助教和导师目前的学生。 该计划的校友将是常客，并提供额外的帮助。学生们提出并回答了一些困难的技术问题。近年来，学生们发现了奇异的电网络，不能恢复的边界数据，但几乎是唯一确定的。换句话说，只有有限数量的可能性，网络兼容给定的边界测量。这些网络称为n对1网络。 该项目的学生发明了这些网络，并发现了它们的性质。 新的例子是通过将某些“四星”图粘在一个看起来像仙人掌或鞋带图的图案中来构造的。 现在有许多不同的家庭这样的图，这将是调查。 另一个主题是排列的性质，特别是排列的峰值点。学生们最近发现了峰多项式的有趣性质，例如，峰集中的所有峰都是其相应峰多项式的零。这些和其他主题将是这个计划的基石。