Inverse problem in cylindrical electrical networks

Thomas Lam, Pavlo Pylyavskyy

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9 Scopus citations

Abstract

In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical R-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow [Linear Algebra Appl., 283 (1998), pp. 115-150] and of de Verdiere, Gitler, and Vertigan [Comment. Math. Helv., 71 (1996), pp. 144-167] for circular planar electrical networks. We show that our conjectural solution holds for certain "purely cylindrical" networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson.

Original languageEnglish (US)
Pages (from-to)767-788
Number of pages22
JournalSIAM Journal on Applied Mathematics
Volume72
Issue number3
DOIs
StatePublished - 2012

Keywords

  • Cylinder
  • Electrical networks
  • Inverse problem
  • Kenyon-Wilson groves
  • R-matrix
  • Total positivity

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