Related papers: Tilings of space and superhomogeneous point proces…
Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zumino's N=2 model in four dimensions, provides manifestly…
This work is devoted to the study of the symmetries of (quasi)periodic architectured materials. For this purpose, the weaker symmetry criterion of indistinguishability is used. It relies on a statistical description of the mesostructure and…
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 cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier…
It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^n$ tiles $\mathbb{Z}^d$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots,k\} \subset \mathbb{Z}$…
For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in…
One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…
We develop a general framework of Euclidean patterns and pattern spaces of translational finite local complexity (FLC), analogues of translational tiling spaces. The notion of a self affine substitution of tilings is extended to both…
We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the…
By means of constructing a new edge-bending algorithm, we prove that every locally polyhedral tiling of $\mathbb{R}^3$ can be completely softened. A weaker form of this statement, for polyhedral space tilings, was conjectured by Domokos,…
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two…
We prove a purely Borel/measureless version of Dowker's ratio ergodic theorem, from which we derive a strengthening of Dowker's original theorem with a precise identification of the limit of local ergodic ratios. This is done by…
Tile B-splines in $\mathbb{R}^d$ are defined as autoconvolutions of the indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $\mathbb{R}^d$. These functions are not piecewise-polynomial,…
The existence of stationary finitely dependent processes on combinatorial models like $\mathbb Z^d$ subshifts can be quite mysterious. For instance, Holroyd and Liggett constructed such processes on proper $4$-colorings of $\mathbb Z^d$ for…
We introduce a new extragradient iterative process, motivated and inspired by [S. H. Khan, A Picard-Mann Hybrid Iterative Process, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2013-69], for finding a common element of the set…
An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential…
Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group…
We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_0\in S$, the spherical projection of $S\backslash\{\pm s_0\}$ to the hyperplane orthogonal to $\pm s_0$ is isometric to…
The tubular geometry (T-geometry) is a generalization of the proper Euclidean geometry, founded on the property of sigma-immanence. The proper Euclidean geometry can be described completely in terms of the world function $\sigma =\rho…
Van der Waals materials enable the construction of atomically sharp interfaces between compounds with distinct crystal and electronic properties. This is dramatically exploited in moir\'e systems, where a lattice mismatch or twist between…