24 citations · 108 across the 21 of their papers we have counts for
22 papers
The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
Turgay Akyar, Artem Beliakov, Konstantin Delchev +6
Place one grain at every nonsink vertex of the wired square, and let be the order of this operation in the sandpile group. Thus is the least positive …
Many-point tropical relaxation and the Monge--Ampère equation
Nikita Kalinin, Ernesto Lupercio, Higinio Serrano +1
In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Ampère Dirichlet problem on bounded convex plan…
Symmetry Emergence in Self-Organized Criticality
Ernesto Lupercio, Mikhail Shkolnikov
We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of th…
Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes
Nikita Kalinin, Mikhail Shkolnikov
In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction p…
Algebraic Limits of Sandpiles
Mikhail Shkolnikov
The paper contributes to building algebraic foundations of self-organized criticality answering a previously unsolved question about the limiting structure of the extended sandpile…
Wave fronts and caustics in the tropical plane
Grigory Mikhalkin, Mikhail Shkolnikov
The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine -space is determined by the integer tangent vectors. Tropical isomorphisms are…
Relaxation in one-dimensional tropical sandpile
Mikhail Shkolnikov
A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a re…
Shrinking dynamic on multidimensional tropical series
Nikita Kalinin, Mikhail Shkolnikov
Let be a compact convex domain. An -tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on that vanishes on $\partialΩ…
Non-commutative amoebas
Grigory Mikhalkin, Mikhail Shkolnikov
The group of isometries of the hyperbolic 3-space is one of the simplest non-commutative complex Lie groups. Its quotient by the maximal compact subgroup naturally maps it back to…
Sandpile monomorphisms and limits
Moritz Lang, Mikhail Shkolnikov
We introduce a tiling problem between bounded open convex polyforms with directed and uniquely colored edges. If there exists a tiling of the polyform…
Quantum groups as hidden symmetries of quantum impurities
Enderalp Yakaboylu, Mikhail Shkolnikov, Mikhail Lemeshko
We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as -deformed Lie algebras. In particular, we show that if the s…
Harmonic dynamics of the Abelian sandpile
Moritz Lang, Mikhail Shkolnikov
The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group i…
Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry
Nikita Kalinin, Aldo Guzmán-Sáenz, Yulieth Prieto +3
Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each ot…
Sandpile solitons via smoothing of superharmonic functions
Nikita Kalinin, Mikhail Shkolnikov
Let be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that unde…
Tropical formulae for summation over a part of SL(2, Z)
Nikita Kalinin, Mikhail Shkolnikov
Let , let stand for such that . Define \begin{equation} \label{…
Introduction to tropical series and wave dynamic on them
Nikita Kalinin, Mikhail Shkolnikov
The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dy…
The number and summation by
Nikita Kalinin, Mikhail Shkolnikov
We obtained a new formula for .
Tropical curves in sandpiles
Nikita Kalinin, Mikhail Shkolnikov
We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation of the maximal stable state is obtained by adding extra…
Sandpiles on the heptagonal tiling
Nikita Kalinin, Mikhail Shkolnikov
We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations…
Tropical curves in sandpile models
Nikita Kalinin, Mikhail Shkolnikov
A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one g…
Bipartite knots
S. Duzhin, M. Shkolnikov
We give a solution to a part of Problem 1.60 in Kirby's list of open problems in topology thus answering in the positive the 1987 conjecture by J.Przytycki concerning the existence…
A formula for the HOMFLY polynomial of rational links
Sergei Duzhin, Mikhail Shkolnikov
We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the g…