Related papers: Translational tilings: structured or wild?
Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles,…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local…
This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented…
We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a…
Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view,…
Word embeddings are powerful representations that form the foundation of many natural language processing architectures, both in English and in other languages. To gain further insight into word embeddings, we explore their stability (e.g.,…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
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…
The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in $\mathbb R$, and recently it was…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
Translations between different nonmonotonic formalisms always have been an important topic in the field, in particular to understand the knowledge-representation capabilities those formalisms offer. We provide such an investigation in terms…
We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…
We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
The identification of structuring effects of transportation infrastructure on territorial dynamics remains an open research problem. This issue is one of the aspects of approaches on complexity of territorial dynamics, within which…
We study possibilities for semantic and syntactic rigidity, i.e., the rigidity with respect to automorphism group and with respect to definable closure. Variations of rigidity and their degrees are studied in general case, for special…
We analyze the role played by local translational symmetry in the context of gauge theories of fundamental interactions. Translational connections and fields are introduced, with special attention being paid to their universal coupling to…