REU Site: Inverse Problems for Electrical Networks

REU的电力网络反问题项目将在2015-2017年的每个夏天为华盛顿大学输送8名本科生，研究电力网络的反问题和图上的相关问题。学生还将研究组合学中的问题和数论中的专题。对于电网络来说，最简单的反问题是，当网络的边界电流对边界电压的响应是已知的，但没有内部知识时，确定电网络中每个导体的电导。想象一下，有一个盒子，里面有一个未知的电路，从盒子里伸出来的引线可以测量电流和电压。这个问题的解决办法可以解释为提供一种电子版的x射线。反问题的一个简化版本是带电场的地雷探测问题。该计划的目的是引导学生进行独立的研究项目，这些项目可以在八周的时间内完成。在课程开始前，学生们将获得阅读材料，并在第一周向他们介绍研究问题。学生将学习如何在支持、协作的氛围中制定和解决研究问题。重点将放在沟通结果上。在校生将担任助教并指导在校生。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技术难题。近年来，学生们发现了一些奇异的电子网络，这些网络无法通过边界数据恢复，但几乎是唯一确定的。换句话说，与给定的边界测量相兼容的网络只有有限的可能性。这些被称为n对1网络。这个项目的学生发明了这些网络，并发现了它们的特性。新的例子是通过将某些“四星”图粘合在一个看起来像仙人掌或鞋带图的模式中来构建的。现在这样的图有许多不同的族，我们将对它们进行研究。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一些有趣的性质，例如，一个峰集中所有的峰都是它对应的峰多项式的零。这些和其他主题将是本课程的基础。

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.