Inverse problem in cylindrical electrical networks

2011/04/26 by Thomas Lam, Lam, Thomas, Pavlo Pylyavskyy +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.1104.4998

openalex publication_date 2011/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Verdière, 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.

Citations

• Circular planar graphs and resistor networks

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