Related papers: On the discrete bicycle transformation

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean…

We consider the Berglund-H\"ubsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the…

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…

The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate…

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to M\"obius…

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{\'e} equation. We use the generalised monodromy map for this equation to give…

The emergence of non-configurational symmetry is studied in a minimal example. The system under scrutiny consists of a dimeric hexagonal complex with configurational $C_3$ symmetry, formulated as a tight-binding model. An accidental…

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is…

We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with…

Mechanical unfolding and refolding of ubiquitin are studied by Monte Carlo simulations of a Go model with binary variables. The exponential dependence of the time constants on the force is verified, and folding and unfolding lengths are…

Multidimensional Consistency becomes more and more important in the theory of discrete integrable systems. Recently, we gave a classification of all 3D consistent 6-tuples of equations with the tetrahedron property, where several novel…

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation…

Based on a novel discretization procedure which has recently been proposed and applied in the construction of a canonical discrete analogue of confocal coordinate systems, an explicit method of constructing discrete analogues of ellipsoids…

We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the evolution towards the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the…

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and…

We study dynamics of a probe particle, which performs biased diffusive motion in a one-dimensional adsorbed monolayer of mobile hard-core particles undergoing continuous exchanges with a vapor phase. In terms of a mean-field-type approach,…

Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such…

The mathematical structure of higher-dimensional physical phase spaces of the nondiagonal Bianchi IX model is analyzed in the neighborhood of the cosmological singularity by using dynamical system methods. Critical points of the Hamiltonian…

We construct explicit cohomogeneity two metrics of G_2 holonomy, which are foliated by twistor spaces. The twistor spaces are S^2 bundles over four-dimensional Bianchi IX Einstein metrics with self-dual (or anti-self-dual) Weyl tensor.…