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Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience. [ABSTRACT FROM AUTHOR]

This paper deals with (globally) random substitutions on a finite set of prototiles. Using renormalization tools applied to objects from operator algebras, we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic R d action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schrödinger operators obtained from aperiodic tilings in the construction. [ABSTRACT FROM AUTHOR]

We present numerical simulations of electromagnetic millimeter-wave propagation in a two-dimensional lattice of dielectric rods arranged in a tenfold Penrose tiling. We find (i) isotropic photonic band gap as expected for quasicrystals and (ii) localized states. We demonstrate that the high frequency edge of the second band gap is characterized by a very small refractive index (fast light). We study the transmission of electromagnetic waves in the frequency range corresponding to fast light and demonstrate that it is related to tunneling through localized states. We use the fast light phenomenon to design a focusing device—a planoconcave lens. [ABSTRACT FROM AUTHOR]

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) > 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, T c (f) < T c (w) . [ABSTRACT FROM AUTHOR]

A three-dimensional pentagon-shaped stereo-tiling concept has been realized through use of penta-graphene carbon structures, although mathematically, regular planar pentagon shapes cannot be used to completely tile the Euclidean plane. Two applications of the findings of this study have been considered from the mathematical and engineering viewpoints. First, the proposed discovery facilitates math-rule-based generation of beautiful designs comprising star shapes formed using regular pentagons. The underlying mathematical logic and hexagon-division rules have been deduced to obtain the proposed pentagonal stereo-tiling pattern comprising equal-length bonds, and two aesthetic designs have been derived using the deduced logic. A fence-like structure exclusively comprising pentagram stars has been designed and given the name "Star Walls." Emphasis has also been laid on application of the proposed stereo-tiling concept in industrial design operations, such as emboss manufacturing. To this end, finite element analysis of embossed steel sheets has been performed to verify the feasibility of the said industrial application. [ABSTRACT FROM AUTHOR]

We present in this paper some non-conventional applications of affine Weyl groups Waff of rank 2, the symmetry group of the tiling/lattice. We first develop and present the tools for applications requiring tilings of a real Euclidean plane R2. We then elucidate the equivalence of these tilings with 2D projections of knots. The resulting mathematical structure provides a framework within which is encompassed recent work utilizing knot theory for modeling the structure and function of genetic molecules, specifically the action of particular enzymes in altering the topology of DNA in site-specific recombination. [ABSTRACT FROM AUTHOR]

The free energies of six-vertex models on a general domain D with various boundary conditions are investigated with the use of the n-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit (height,width)→(∞,∞). It is derived that the free energies of the model on the domain D are classified through the densities of left/down arrows on the boundary. Specifically, the free energy is identical to that obtained by Lieb [Phys. Rev. Lett. 18, 1046 (1967); 19, 108 (1967); Phys. Rev. 162, 162 (1967)] and Sutherland [Phys. Rev. Lett 19, 103 (1967)] with the cyclic boundary condition when the densities are both equal to 12. This fact explains several results already obtained through the transfer matrix calculation. The relation to the domino tiling (or dimer, or matching) problems is also noted. [ABSTRACT FROM AUTHOR]

It is shown that two coupled oscillators perturbed by periodic kicks generate a thin stochastic web in the four-dimensional phase space, which differs from the Arnold web. Under some resonance-type condition the web possesses a quasicrystal-type symmetry. In three-dimensional coordinate space, the web’s symmetry corresponds to the icosahedral one and, due to that, the original four-dimensional map can be considered as a dynamical generator of the quasicrystal-type tiling of three-dimensional space. [ABSTRACT FROM AUTHOR]

Although time-frequency analysis is effective for characterizing dispersive wave signals, the time-frequency tilings of most conventional analysis methods do not take into account dispersion phenomena. An adaptive time-frequency analysis method is introduced whose time-frequency tiling is determined with respect to the wave dispersion characteristics. In the dispersion-based time-frequency tiling, each time-frequency atom is adaptively rotated in the time-frequency plane, depending on the local wave dispersion. Although this idea can be useful in various problems, its application to the analysis of dispersive wave signals has not been made. In this work, the adaptive time-frequency method was applied to the analysis of dispersive elastic waves measured in waveguide experiments and a theoretical investigation on its time-frequency resolution was presented. The time-frequency resolution of the proposed transform was then compared with that of the standard short-time Fourier transform to show its effectiveness in dealing with dispersive wave signals. In addition, to facilitate the adaptive time-frequency analysis of experimentally measured signals whose dispersion relations are not known, an iterative scheme for determining the relationships was developed. The validity of the present approach in dealing with dispersive waves was verified experimentally. © 2005 Acoustical Society of America. [ABSTRACT FROM AUTHOR]

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that they have no absolutely continuous part, that they have a uniformly discrete pure point part on the axis Re3, and that they are otherwise supported on a set of concentric cylinder surfaces around this axis. For SCD tilings with additional properties, more detailed results are given. [ABSTRACT FROM AUTHOR]

Let M be an arbitrary Riemannian homogeneous space, and let Ω be a space of tilings of M, with finite local complexity (relative to some symmetry group Γ) and closed in the natural topology. Then Ω is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Γ. This result extends previous results of Anderson and Putnam, of Ormes, Radin, and Sadun, of Bellissard, Benedetti, and Gambaudo, and of Gähler. In particular, the construction in this paper is a natural generalization of Gähler’s. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact Abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean space, which is motivated by the study of quasicrystallographic random tilings. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Introduces a notion of equivalence on tilings which is formulated in terms of their local structure. Example of an almost-groupoid; Notion of topological equivalence; Inverse semigroups of groupoids; Universal groupoid of an inverse semigroup; Mutual local derivability; Construction of local morphisms from local derivation rules.

Observes the acoustic vibrational modes of piezocomposites in perfect and random Penrose tilings. Comparison between perfect Penrose tiling (PPT) and random Penrose tiling (RPT); Kinds of connected PZT-methacrylate piezocomposites; Demonstration of the resonance spectra of PPT and RPT.

The effective conductivity of a two-phase two-dimensional composite with diamond-shaped tiling is considered. This analysis, based on a projection of the boundary conditions on linear combinations of solutions to the electrostatic equation that are orthonormal on the boundary, generalizes results by Keller [J. Math. Phys. 28, 2516 (1987)] and others. Numerical results are given for several conductivity ratios of the two phases and for varying obtuse angles of the tiles. Special emphasis is given to very large and very small conductivity differences, and very elongated tiles. [ABSTRACT FROM AUTHOR]

A result on nonperiodic tilings is generalized and related to the problem of the origin of crystalline symmetry. [ABSTRACT FROM AUTHOR]