Related papers: Electric network and Hirota type $4$-simplex maps
We study polygon equations and their connections to simplex equations, which generalize the pentagon and Yang--Baxter equations, respectively. First, we show that certain "commutative" pairs of solutions of (dual) polygon equations give…
In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with…
This mini-course was given in the First Yaroslavl Summer School on Discrete and Computational Geometry in August 2012, organized by International Delaunay Laboratory "Discrete and Computational Geometry" of Demidov Yaroslavl State…
Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the…
We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the Lie algebras $e(3)$ and $so(4)$. Using this map we establish a connection between the deformed Kowalevski top on $e(3)$ proposed by Sokolov and…
We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - \Delta u= \varepsilon e^{\kappa u} +\beta\delta_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $\kappa>0$ and…
For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of…
This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used…
Network Models with couplings between link pairs are the simplest models for a class of networks with Higher Order interactions. In this paper we give an analytic, general solution to this family of Random Graph Models extending previous…
We apply a 3-dimensional approach to describe a new parametrization of the L-operators for the 2-dimensional Bazhanov-Stroganov (BS) integrable spin model related to the chiral Potts model. This parametrization is based on the solution of…
We derive a formally simple approximate analytical solution to the Poisson-Boltzmann equation for the spherical system via a geometric mapping. Its regime of applicability in the parameter space of the spherical radius and the surface…
We introduce and study a class of Thurston maps from the 2-sphere to itself which we call nearly Euclidean Thurston (NET) maps. These are simple generalizations of Euclidean Thurston maps.
As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of corresponding three-dimensional lattice models. We present the modified version of these equations which give the…
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of…
Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of…
The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to…
We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$.…
We find a simple brane configuration in the IKKT matrix model which resembles the standard model at low energies, with a second Higgs doublet and right-handed neutrinos. The electroweak sector is realized geometrically in terms of two…
We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of…
The large size of multiscale, distribution and transmission, power grids hinder fast system-wide estimation and real-time control and optimization of operations. This paper studies graph reduction methods of power grids that are favorable…