408 research outputs found
Computing (or not) Quasi-Periodicity Functions of Tilings
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 tiling. We prove that the tilings by a tileset that admits only
quasi-periodic tilings have a recursively (and uniformly) bounded
quasi-periodicity function. This corrects an error from [6, theorem 9] which
stated the contrary. Instead we construct a tileset for which any
quasi-periodic tiling has a quasi-periodicity function that cannot be
recursively bounded. We provide such a construction for 1-dimensional effective
subshifts and obtain as a corollary the result for tilings of the plane via
recent links between these objects [1, 10].Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010
C*-algebras of Penrose's hyperbolic tilings
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit,
up to affine transformations a finite number of prototiles. In this paper, we
give a complete description of the C*-algebras and of the K-theory for such
tilings. Since the continuous hull of these tilings have no transversally
invariant measure, these C*-algebras are traceless. Nevertheless, harmonic
currents give rise to 3-cyclic cocycles and we discuss in this setting a
higher-order version of the gap-labelling.Comment: 36 pages. v2: some mistakes corrected, a section on topological
invariants of the continuous hull of the Penrose hyperbolic tilings adde
50 Years of the Golomb--Welch Conjecture
Since 1968, when the Golomb--Welch conjecture was raised, it has become the
main motive power behind the progress in the area of the perfect Lee codes.
Although there is a vast literature on the topic and it is widely believed to
be true, this conjecture is far from being solved. In this paper, we provide a
survey of papers on the Golomb--Welch conjecture. Further, new results on
Golomb--Welch conjecture dealing with perfect Lee codes of large radii are
presented. Algebraic ways of tackling the conjecture in the future are
discussed as well. Finally, a brief survey of research inspired by the
conjecture is given.Comment: 28 pages, 2 figure
Proceedings of JAC 2010. Journées Automates Cellulaires
The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku.
The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume.
The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible.
These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast
Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings
This paper provides a bridge between the classical tiling theory and the
complex neighborhood self-assembling situations that exist in practice. The
neighborhood of a position in the plane is the set of coordinates which are
considered adjacent to it. This includes classical neighborhoods of size four,
as well as arbitrarily complex neighborhoods. A generalized tile system
consists of a set of tiles, a neighborhood, and a relation which dictates which
are the "admissible" neighboring tiles of a given tile. Thus, in correctly
formed assemblies, tiles are assigned positions of the plane in accordance to
this relation. We prove that any validly tiled path defined in a given but
arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of
microtiles. A ribbon is a special kind of polyomino, consisting of a
non-self-crossing sequence of tiles on the plane, in which successive tiles
stick along their adjacent edge. Finally, we extend this construction to the
case of traditional tilings, proving that we can simulate
arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving
some of their essential properties.Comment: Submitted to Theoretical Computer Scienc
Self-similarity under inflation and level statistics: a study in two dimensions
Energy level spacing statistics are discussed for a two dimensional
quasiperiodic tiling. The property of self-similarity under inflation is used
to write a recursion relation for the level spacing distributions defined on
square approximants to the perfect quasiperiodic structure.
New distribution functions are defined and determined by a combination of
numerical and analytical calculations.Comment: Latex, 13 pages including 6 EPS figures, paper submitted to PR
- …