相关论文: Dispersion relations of periodic quantum graphs as…

There are totally 11 kinds of Archimedean tiling for the plane. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tiling, namely the triangular…

This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete tacnode kernel). Here this kernel is used…

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of…

Two-dimensional colloidal suspensions subject to laser interference patterns with decagonal symmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodically along one direction [J. Mikhael et al.,…

We construct via usual graph theory a class of associative dialgebras as well as a class of coassociative L-coalgebras. Tiling of the (n^2,1)-De bruijn graphs are also obtained and constructions of cubical trialgebras, (notion defined by…

The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other…

We use methods from algebra and discrete geometry to study the irreducibility of the dispersion polynomial of a discrete periodic operator associated to a periodic graph after changing the period lattice. We provide numerous applications of…

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…

We study the electronic transport in quasiperiodic separable tight-binding models in one, two, and three dimensions. First, we investigate a one-dimensional quasiperiodic chain, in which the atoms are coupled by weak and strong bonds…

In the present paper we discuss four ways of looking at rhombile tilings: stacking 3-dimensional cubes, elements of groups, and configurations of lines and points.

Here I present several theorems about trapezoids tilings. The first one is related to trapezoids with rational base relation, the other ones are related to those with base relation from quadratic number field.

We compute the number of rhombus tilings of a hexagon with sides n, n, N, n, n, N, where two triangles on the symmetry axis touching in one vertex are removed. The case of the common vertex being the center of the hexagon solves a problem…

For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the…

In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…

We discuss dispersion representations for the triangle diagram $F(p_1^2,p_2^2,q^2)$, the single dispersion representation in $q^2$ and the double dispersion representation in $p_1^2$ and $p_2^2$, with special emphasis on the appearance of…

\noindent The algebraic characterization of classes of locally isomorphic aperiodic tilings, being examples of quantum spaces, is conducted for a certain type of tilings in a manner proposed by A. Connes. These $2$-dimensional tilings are…

We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap…

The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…

We investigate a quantum generalization of the Mycielski construction for quantum graphs. In particular, we analyze the quantum symmetries of quantum Mycielskians and their relation to the symmetries of the underlying quantum graphs. We…

We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…