Related papers: Discrete Z^a and Painleve equations
An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…
There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its…
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the…
We extend the classical Schwarz-Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between…
We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced…
Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according…
A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the…
We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models…
A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables $Z_A$ and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed.…
We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrodinger equation that is used to write the solution for quantum graphs. Formulation of the problem and…
Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e…
Over the past few years it has been discovered that an "observable" can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which…
This paper first discusses irreducibility of a Painlev\'e equation $P$. We explain how the Painlev\'e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an…
A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.
A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…
The symmetry group of a (discrete) Painlev\'e equation provides crucial information on the properties of the equation. In this paper we argue against the commonly-held belief that the symmetry group of a given equation is solely determined…
We formulate the notion of parity for the periodic XXZ spin chain within the Quantum Inverse Scattering Method. We also propose an expression for the eigenvalues of the charge conjugation operator. We use these discrete symmetries to help…
We show that the discrete Painlev\'e-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula…
We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we…
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szeg\"o recurrence, Schur…