Related papers: Integrable systems on hexagonal lattices and consi…
Using the fixed point alternative theorem we establish the orthogonal stability of quadratic functional equation of Pexider type $f(x+y)+g(x-y)=h(x)+k(y)$, where $f, g, h, k$ are mappings from a symmetric orthogonality space to a Banach…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…
Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in $\mathbb Z^d$. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of…
Existence, stability and dynamics of soliton complexes, centered at the site of a single transverse link connecting two parallel 2D (two-dimensional) lattices, are investigated. The system with the on-site cubic self-focusing nonlinearity…
In this paper, we present new multiplicity fixed point theorems for operators acting on Cartesian products of two normed linear spaces. We show that Leggett-Williams type conditions in each component of the system guarantee the existence of…
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
Quadratic systems of equations appear in several applications. The results in this paper are motivated by quadratic systems of equations that describe equilibrium behavior of physical infrastructure networks like the power and gas grids.…
We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a…
We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems…
In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case is when the two small masses tend to the same Lagrangian point $L_4$ (or $L_5$). In \cite{Xia}, Z. Xia…
We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the…
The integrability in quadratures of normality equation for spatially homogeneous dynamical systems in two-dimensional space is shown. The classical symmetries of this equation are calculated and the corresponding self-similar solutions are…
We consider generalizations of the familiar fifteen-piece sliding puzzle on the 4 by 4 square grid. On larger grids with more pieces and more holes, asymptotically how fast can we move the puzzle into the solved state? We also give a…
The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on…
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…
This paper proposes a novel and simple algorithm of facet enumeration for convex polytopes. The complexity of the algorithm is discussed. The algorithm is implemented in Matlab. Some simple polytopes with known H-representations and…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
An analog of the lattice KdV equation of Nijhoff et al. is constructed on a hexagonal lattice. The resulting system of difference equations exhibits soliton solutions with interesting local structure: there is a nontrivial phase shift on…
We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is…