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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Electrical response matrix of a regular $ 2n$-gon


Authors: Nathaniel D. Blair-Stahn and David B. Wilson
Journal: Proc. Amer. Math. Soc. 137 (2009), 2015-2025
MSC (2000): Primary 31A25; Secondary 30C20, 82B20, 05C05
Published electronically: January 21, 2009
MathSciNet review: 2480283
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Abstract: 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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Additional Information

Nathaniel D. Blair-Stahn
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195

David B. Wilson
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09734-2
PII: S 0002-9939(09)09734-2
Received by editor(s): April 3, 2007
Received by editor(s) in revised form: May 1, 2008
Published electronically: January 21, 2009
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.