Related papers: The Symbolic Dynamics of Tiling the Integers

Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…

Our earlier article proved that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. We re-prove this…

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The…

Let K be a number field with euclidean ring of integers O. Let G be a finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite, geometrically defined spanning set of the top dimensional integral cohomology of G by generalizing…

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…

In combinatorial game theory, the winning player for a position in normal play is analyzed and characterized via algebraic operations. Such analyses define a value for each position, called a game value. A game (ruleset) is called universal…

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

Does the interaction graph of a finite dynamical system can force this system to have a "complex" dynamics ? In other words, given a finite interval of integers $A$, which are the signed digraphs $G$ such that every finite dynamical system…

In this paper we extend counting of traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be done in linear time with…

We present a suite of algorithmic techniques for handling substitution tilings by treating a tile's hierarchy of supertiles in a purely combinatorial fashion using finite state automata. The resulting techniques are very convenient for…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda$ in such a way that each point of the space gets covered exactly $k$ times,…

Let $X$ be a topological space and $f:X\to X$ a bijection. Let ${\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\mathcal C}(X,f)$ if and only if the bijection $f^n:X\to X$ is continuous. A subset $S$ of…

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…

We show that silting modules are closely related with localisations of rings. More precisely, every partial silting module gives rise to a localisation at a set of maps between countably generated projective modules and, conversely, every…

We present a type system that combines, in a controlled way, first-order polymorphism with intersectiontypes, union types, and subtyping, and prove its safety. We then define a type reconstruction algorithm that issound and terminating.…

Formally verifying properties of programs that manipulate arrays in loops is computationally challenging. In this paper, we focus on a useful class of such programs, and present a novel property-driven verification method that first infers…

Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these…

An $SL_2$-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 by 2 submatrix has determinant 1. Such tilings are infinite analogues of Conway-Coxeter friezes, and they have strong links to cluster algebras,…

We define bicolored tilings as a disk with a collection of smooth curves with a coloring map on the tiles that these curves delimit. Using two transformations, we define an equivalence on tilings. We then define the Scott map which creates…