Lozenge tilings of hexagons with arbitrary dents

Eisenkölbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

Spherical f-Tilings by Two Noncongruent Classes of Isosceles Triangles-Ⅱ

In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by Em,m ≥ 2,m ∈ N,Fk,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.     

Are layered two-dimensional quasicrystals periodic in the third direction?

Two-dimensional quasicrystals have generally been believed to be quasiperiodic in theXY plane and periodic in the Z direction. This is not necessarily the case. A layered material with equidistantly spaced layers and a random tiling two-dimensional quasicrystal in each layer is shown to exhibit delta-function diffraction spots even when the phason strain fields in different layers are completely uncorrelated. Surprisingly, such a Z-aperiodic quasicrystal shows true-peaks, while a more ordered Z-periodic quasicrystal shows less sharp, power-law-decaying peaks.     

Statistical mechanics of two dimensional tilings

Reduced dimensionality in two dimensions is a topic of current interest. We use model systems to investigate the statistical mechanics of ideal networks. The tilings have possible applications such as the 2D locations of pore sites in nanoporous arrays (quantum dots), in the 2D hexagonal structure of graphene, and as adsorbates on quasicrystalline crystal surfaces. We calculate the statistical mechanics of these networks, such as the partition function, free energy, entropy, and enthalpy. The plots of these functions versus the number of links in the finite networks result in power law regression. We also determine the degree distribution, which is a combination of power law and rational function behavior. In the large-scale limit, the degree of these 2D networks approaches 3, 4, and 6, in agreement with the degree of the regular tilings. In comparison, a Penrose tiling has a degree also equal to about 4.     

THE SCENERY FLOWFOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING

The authors define the scenery flow of the torus. The flow space is the union of all flat 2-dimensional tori of area $1$ with a marked direction (or equivalently, the union of all tori with a quadratic differential of norm 1). This is a $5$-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant $3$-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of: \item{$\bullet$} stacking and rescaling periodic tilings of the plane; \item{$\bullet$} symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the $S$-adic system generated by two substitutions; \item{$\bullet$} zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; \item{$\bullet$} the natural extension of two-dimensional continued fractions; \item{$\bullet$} induction on exchanges of three intervals; \item{$\bullet$} rescaling on pairs of transverse measure foliations on the torus, or the Teichm\"uller flow on the twice-punctured torus.     

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid G _n of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QAD _n−1 of order n−1 with the path P _n ⁽²⁾ on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular that the characteristic polynomials of the above graphs satisfy the equality P(G _n )=P(P _n ⁽²⁾)[P(QAD _n−1)]². On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described “linear squarishness” property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases. Research supported in part by NSF grant DMS-0500616.     

The global minimum of energy is not always a sum of local minima—A note on frustration

A classical lattice gas model with translation-invariant, finite-range competing interactions, for which there does not exist an equivalent translation-invariant, finite-range nonfrustrated potential, is constructed. The construction uses the structure of nonperiodic ground-state configurations of the model. In fact, the model does not have any periodic ground-state configurations. However, its ground-state—a translation-invariant probability measure supported by ground-state configurations—is unique.     

Integer Cech cohomology of a class of n‐dimensional substitutions

A class of non‐periodic tilings in n‐dimensions is considered. They are based on one‐dimensional substitution tilings that force the border, a property preserved in the construction for higher dimensions. This fact allows to compute the integer?ech cohomology of the tiling spaces in an efficient way. Several examples are analyzed, some of them with PV numbers as inflation factors, and they have finitely or infinitely generated torsion‐free cohomologies. Copyright © 2010 John Wiley & Sons, Ltd.     

The combinatorics of N.G. de Bruijn

In memoriam: N.G. de Bruijn.