## 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 language | English (US) |
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Pages (from-to) | 767-788 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 2012 |

## Keywords

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