The electrical response matrix of a regular 2n-gon

Details The electrical response matrix of a regular 2n-gon The electrical response matrix of a regular 2n-gon

2007-03-28

arxiv.org/abs/math/0703826v2

Proceedings of the American Mathematical Society 137(6):2015--2025, 2009

Mathematics

Classical Analysis and ODEs

10 pages, 4 figures; v2 adds more background information

Scientific paper

Consider a unit-resistive plate in the shape of a regular polygon with 2n sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2n-gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests.

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