Inverse problem in cylindrical electrical networks

TL;DR

This paper investigates the inverse Dirichlet-to-Neumann problem for cylindrical electrical networks, introducing an electrical R-matrix transformation and proposing a conjectural solution extending previous work on circular networks.

Contribution

It defines a birational electrical R-matrix transformation and formulates a conjectural solution for the inverse problem on cylindrical networks, supported by results on purely cylindrical cases.

Findings

Conjectural solution extends prior circular network results.

Electrical R-matrix transformation is introduced and studied.

Results apply to certain purely cylindrical networks.

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, and of de Verdi\`ere, Gitler, and Vertigan 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.