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Theorem (J.Kari and K. Culik, 1996) . .. . . . . . The Kari-Culik tileset is aperiodic. . Sketch of proof. . . . . . . . . Suppose there is a periodic tiling. Then each line has an average. ... If a substitutive square is found in any n × n-square of any tiling, then the entropy of the tiles is positive. ... Is there a sub-shift of finite type A, with positive entropy, such that any subshift of finite type ⊂ A also has positive entropy? ...doi:10.1016/j.tcs.2016.12.013 fatcat:gk37isccqjhoxaq2ia7fo3ml3e
Lecture Notes in Computer Science
In this paper we present a construction of Kari-Culik aperiodic tile set - the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. ... Aperiodic sequences and tilings A tile is an unit square with colored sides. ... All tilings with this tileset are aperiodic. ...doi:10.1007/978-3-319-09698-8_15 fatcat:joqxmsoujjdx3blcvxznuyo32m
Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic ... tiles. ... Acknowledgements We are grateful to Rathindra Nath Das, Emmanuel Floratos, Haye Hinrichsen and Ronny Thomale for useful discussions. ...arXiv:2205.05693v1 fatcat:pmdu4hj5ejhghpo7azi64mbixa
Next, we show that for a second class of tilings, the octahdral and tetrahedral tilings have equal entropy. ... In this paper, Goodman-Strauss presents a new and remarkably small and simple set of aperiodic tiles: there are only two tiles and eight translation classes. ...
Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively. ... It has been shown, that diffraction patterns of aperiodic structures (quasicrystals and modulated structures) consists of periodic series of peaks [1,2]. ... Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively. ...doi:10.1107/s2053273314088020 fatcat:vhmbw3tq75fb5akq5343cxefny
-Experimental results on diffuse scattering and phason elasticity do not seem to be in complete agreement with an entropy term arising from a tile configuration alone. ... The two models which are frequently opposed are referred to as the energy and entropy stabilised QC. ...doi:10.1080/14786430500419411 fatcat:yvfgriseczg7vl7dox7c7gjshy
Lecture Notes in Computer Science
We show that such words have zero entropy. However, contrarily to the one-dimensional case, they may not be uniformly recurrent. ... This notion was previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend several results to two dimensions. ... We will now prove that µ(w) is aperiodic for any aperiodic bidimensional word w. First, we need a technical lemma about our tiles. Figure 4 ). ...doi:10.1007/978-3-319-15579-1_31 fatcat:a2nejzxrzvhfrbrsggrctm3hcy
Question 3: Aperiodic order as a form of disorder. What can ndings in the quasicrystal community teach us about theory of (spin-) glasses ? How (dis)ordered can tilings and tiling models be? ... In particular, we consider some spectral characterisations, and shortly review what is known about the occurrence of aperiodic order in lattice models at zero and nonzero temperatures. ... Acknowledgments I wish to thank the conference organisers of ICQ12 for inviting me, and my co-authors for all they taught me. ...doi:10.12693/aphyspola.126.621 fatcat:77yful7245fx7cbrdo7aoxmz4i
There are very close the dynamical systems arising from projection method aperiodic connections between such sets and tilings of R?. The work of H. patterns. ... The metric Quas, Anthony (1-MEMP; Memphis, TN) arises in the usual way from considering the Hausdorff metric on Entropy along convex shapes, random tilings and shifts of finite type. increasing bounded ...
Here we discuss some issues related to the notion and occurrence of aperiodic order in equilibrium statistical mechanics. ... In particular, we consider some spectral characterisations,and shortly review what is known about the occurrence of aperiodic order in lattice models at zero and non-zero temperatures. ... Acknowledgements: I wish to thank the conference organisers of ICQ12 for inviting me, and my coauthors for all they taught me. ...arXiv:1310.0267v2 fatcat:ybbgczo3e5dzxd52c36pfesnxa
Given a random distribution of impurities on a periodic crystal, an equivalent uniquely ergodic tiling space is built, made of aperiodic, repetitive tilings with finite local complexity, and with configurational ... entropy close to the entropy of the impurity distribution. ... Kellendonk and L. Sadun for comments and useful new references. ...arXiv:0906.2997v5 fatcat:x7gbtr2ptjbntaonocbzpklxey
The authors consider a random tiling model whose tiles are rect- angles and triangles, the rectangles of side lengths | and 2 sin(z/5) and the triangles of side lengths 1, 1 and 2 sin(z/5) (there are natu ... As the temperature T goes to zero, the asymptotic behavior of the entropy and the specific heat are (Jo/2T) exp(—Jo/2T) and (Jj /4T*) exp(—Jo/2T ), respectively.” ...
We consider the structure of aperiodic points in Z 2 -subshifts, and in particular the positions at which they fail to be periodic. ... Another consequence is that Z 2 -subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some Z-subshift. ... Acknowledgements The authors wish to thank anonymous reviewers for many helpful remarks and improvements. ...doi:10.4230/lipics.icalp.2018.128 dblp:conf/icalp/GrandjeanMV18 fatcat:54xjss4fxrfglnspjn75yssbl4
Two different relaxed versions of Gummelt's aperiodic cluster covering rules are considered. ... The entropy density of this covering ensemble is determined by Monte Carlo simulations, using entropic sampling techniques. ... Acknowledgments We would like to thank Petra Gummelt for useful discussions on the relationships between the different relaxed coverings and random tiling ensembles. ...doi:10.1016/s0925-8388(02)00168-8 fatcat:ezvgff43o5h5tnh7mthyanel24
Given a finite set of tiles, can one cover the plane with copies of these tiles such that there are no gaps and no overlaps? Are there sets that force the tiling to be aperiodic? ... Penrose, who constructed a set of two tiles together with rules to put the tiles together that produced such an aperiodic tiling, later called the Penrose tiling. ...doi:10.1107/s0108767312033715 pmid:23075609 fatcat:jqhwbo7qgzfizjp5chtknido3i
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