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The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method
Computational Methods and Function Theory ( IF 2.1 ) Pub Date : 2021-08-31 , DOI: 10.1007/s40315-021-00409-1
Hiroyuki Miyoshi 1 , Darren Crowdy 1 , Rhodri Nelson 2
Affiliation  

The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.



中文翻译:

素函数、Fay 三割恒等式和 van der Pauw 方法

van der Pauw 方法是应用科学中众所周知的实验技术,用于测量给定样品的电导率或霍尔系数等物理量。它的流行归因于它的灵活性:相同的方法适用于任何形状的平面样本,只要它们简单连接即可。在数学上,该方法基于交叉比率恒等式。应用科学家最近做了很多工作,试图将范德堡方法扩展到有孔的样品(“有孔样品”)。在本文中,我们展示了两个新的函数理论成分与该应用领域的相关性:与多连接平面域的肖特基二重函数相关的素函数和涉及该素函数的 Fay 三割恒等式。我们这里关注的是单孔(双连接,或属一)情况。使用这些新的理论成分,我们能够证明应用科学文献中提出的几个数学猜想。

更新日期:2021-09-01
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