Related papers: Hexagonal circle patterns and integrable systems: …
It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…
An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $\Psi$ living in a Lie group $G$, which satisfies some…
Let $S$ be a closed, orientable surface of genus $g\geq 2$. We consider Delaunay circle patterns on $S$ equipped with a complex projective structure. We prove that the space of complex projective structures on $S$ equipped with a Delaunay…
The sets of the integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.
Three circles define each of the Brocard points of a triangle. If one adds the three circles through a pair of vertices and the orthocentre one has nine circles. It is described how each of the nine centres of these circles lies at the…
The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…
We give a formula for the number of symmetric tilings of hexagons on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent non-opposite sides. We show that for certain families of such regions,…
We present three equivalence classes of rational non-invertible multidimensional compatible maps. These maps turns out to be idempotent and by construction they admit birational partial inverses (companion maps) which are Yang-Baxter maps.…
We propose a nonperturbative framework to study general correlation functions of single-trace operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory at large $N$. The basic strategy is to decompose them into fundamental building…
A lucasene is a hexagon chain that is similar to a fibonaccene, an $L$-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after…
We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected…
We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge…
The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b_1 + b_2 + b_3 + b_4)^2 = 2(b_1^2 + b_2^2 + b_3^2 + b_4^2). We show…
Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the…
We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable…
We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black-white lattice. For each one of these equations, two different three-leg forms are constructed, leading to…
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the…
The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of…
We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum…