相关论文: Simultaneous Translational and Multiplicative Tili…

Classical Landau theory considers structural phase transitions and crystallization as a condensation of several critical density waves whose wave vectors are symmetrically equivalent. Analyzing the simplest nonequilibrium Landau potentials…

We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given,…

Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$…

This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I…

These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling.…

Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for…

We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in $\mathbb{R}^2$ with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the…

Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting…

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…

Suppose that we have a repetitive and aperiodic tiling $\bf T$ of $\mathbb{R}^n$, and two mass distributions $f_1$ and $f_2$ on $\mathbb{R}^n$, each pattern equivariant with respect to $\bf T$. Under what circumstances is it possible to do…

In this thesis, we take a journey through two different but not dissimilar stories with an underlying theme of combinatorics emerging from scattering amplitudes in quantum field theories. The first part tells the tale of the…

Surfaces in i-Al68Pd23Mn9 as observed with STM and LEED experiments show atomic terraces in a Fibonacci spacing. We analyze them in a bulk tiling model due to Elser which incorporates many experimental data. The model has dodecahedral…

This study explores the properties of the function which can tile the field $\mathbb{Q}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\mathbb{Q}_p$ is by translation uniformly locally constancy.…

Data augmentation has been proven to be an effective technique for developing machine learning models that are robust to known classes of distributional shifts (e.g., rotations of images), and alignment regularization is a technique often…

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore…

We give a $O(n)$-time algorithm for determining whether translations of a polyomino with $n$ edges can tile the plane. The algorithm is also a $O(n)$-time algorithm for enumerating all such tilings that are also regular, and we prove that…

In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $P_n$ inside the equilateral triangle $S_n$…

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each…