Related papers: An Integer Linear Programming Model for Tilings
In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
It is well known that if a finite set $A\subset\mathbb{Z}$ tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group.…
An algorithm is provided to tile the plane with the aperiodic monotile Tile(1,1) recently discovered by Smith et al. (2023). Their geometric construction guidelines are expanded into a numerical MATLAB algorithm. The intention is to remove…
We consider finite sets $A\subset\mathbb{Z}$ tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. Building on por previous work, we prove…
We consider translational integer tilings by finite sets $A\subset\mathbb{Z}$. We introduce a new method based on \emph{splitting}, together with a new combinatorial interpretation of some of the main tools from our earlier work. We also…
Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…
In Mathematical Music theory, the Aperiodic Tiling Complements Problem consists in finding all the possible aperiodic complements of a given rhythm $A$. The complexity of this problem depends on the size of the period $n$ of the canon and…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We present a Mixed Integer Linear Program (MILP) approach in order to model the nonlinear problem of minimizing the tire noise. We first take more industrial constraints into account than in a former work of the authors. Then, we associate…
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…
This paper presents a novel approach for the identification of linear time-periodic (LTP) systems in continuous time. This method is based on harmonic modeling and consists in converting any LTP system into an equivalent LTI system with…
The aim of this paper is to investigate the use of an entropic projection method for the iterative regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both…
In this paper we address the numerical solution of nonlinear ill-posed systems by iterative regularization methods in the classes of Levenberg-Marquardt, trust-region and adaptive quadratic regularization procedures. Both with exact and…
We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to tile the integers under the assumption…
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…
This paper presents a novel approach to the joint optimization of job scheduling and data allocation in grid computing environments. We formulate this joint optimization problem as a mixed integer quadratically constrained program. To…
In this paper we present a new algorithm for solving linear programs that requires only $\tilde{O}(\sqrt{rank(A)}L)$ iterations to solve a linear program with $m$ constraints, $n$ variables, and constraint matrix $A$, and bit complexity…
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…
We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic…