Related papers: Absence of absolutely continuous diffraction spect…
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…
Two-dimensional (2D) semi-Dirac materials feature a unique anisotropic band structure characterized by quadratic dispersion along one spatial direction and linear dispersion along the other, effectively hybridizing ordinary and Dirac…
We investigate tiling properties of spectra of measures, i.e., sets $\Lambda$ in $\br$ such that $\{e^{2\pi i \lambda x}: \lambda\in\Lambda\}$ forms an orthogonal basis in $L^2(\mu)$, where $\mu$ is some finite Borel measure on $\br$. Such…
We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We…
We present mathematical theory for understanding the transmission spectra of heterogeneous materials formed by generalised Fibonacci tilings. Our results, firstly, characterise super band gaps, which are spectral gaps that exist for any…
We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling…
A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral…
The diffraction spectra of lattice gas models on Z^d with finite-range ferromagnetic two-body interaction above T_c or with certain rates of decay of the potential are considered. We show that these diffraction spectra almost surely exist,…
We give a generalization of Lagarias' formula for diffraction by ideal crystals, and we apply it to the lattice case, in preparation for addressing the problem of quasicrystals and complex dimensions posed by Lapidus and van Frankenhuijsen…
Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
The diffraction spectrum of the dart-rhombus random tiling of the plane is derived in rigorous terms. Using the theory of dimer models, it is shown that it consists of Bragg peaks and an absolutely continuous diffuse background, but no…
Particle models with finitely many types of particles are considered, both on $\mathbb{Z}^d$ and on discrete point sets of finite local complexity. Such sets include many standard examples of aperiodic order such as model sets or certain…
Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It…
We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric…
Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they…
We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1,…
We show that diffraction features of $1D$ quasicrystals can be retrieved from a single topological quantity, the \v{C}ech cohomology group, $\check{H}^{1}\cong\mathbb{Z}^2$, which encodes all relevant combinatorial information of tilings.…
We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.
The single electron spectrum and wavefunctions in quasicrystals continue to be a fascinating problem, with few known solutions, especially in two and higher dimensions. This paper investigates the energy spectra and gap structures in…