arXiv subjects

A. A. Kazakov

Publications and source records attributed to A. A. Kazakov.

3 recordsLinked to original sources

arXiv2026

Metric properties of electrical networks and the graph reconstruction problems

Using the generalized Temperley trick, we demonstrate the explicit embedding of circular electrical networks into totally non-negative Grassmannians. Building on this result, we show that the effective resistances between boundary nodes of circular electrical networks satisfy the Kalmanson property, and we provide the full characterization of planar electrical Kalmanson metrics. Additionally, we present a graph reconstruction algorithm with applications in phylogenetic network analysis as well as the numerical solution of the Calderon problem.

math.CO↗

arXiv2025

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

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.

math-ph↗

arXiv2023

The state-sum invariants for virtual knots

We construct the new non-trivial state--sum invariants for virtual knots and links by a generalization of the powerful Carter--Saito--Jelsovsky--Kamada--Langford theorem for classical knots. The main result of this work is based on cohomology quandle theory and colorings of virtual knot and link diagrams by quandle elements.

math.QA↗