Inverse problems related to electrical networks and the geometry of non-negative Grassmannians

TL;DR

This paper introduces a novel solution to the discrete Calderon problem by embedding electrical networks into non-negative Grassmannians, linking electrical network theory with combinatorics and matrix positivity.

Contribution

It presents an explicit embedding of electrical networks into non-negative Grassmannians and connects the problem to combinatorial and matrix positivity theories.

Findings

New solution to the discrete Calderon problem.

Established connection between electrical networks and non-negative Grassmannians.

Linked electrical impedance tomography with combinatorial properties.

Abstract

We provide a new solution to the classical black box problem (the discrete Calderon problem) in the theory of circular electrical networks. Our approach is based on the explicit embedding of electrical networks into non-negative Grassmannians and generalized chamber ansatz for it. Also, we reveal the relation of this problem with the combinatorial properties of spanning groves and the theory of totally non-negative matrices. Key words: electrical networks, discrete Calderon problem, discrete electrical impedance tomography, non-negative Grassmannians, twist for positroid variety, Temperley trick, totally non-negative matrices, effective resistances.