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Optimal word chains for the Thue-Morse word

A. Arnold, S. Brlek
1989 Information and Computation  
We study word chains for the iterates $'(a) of the Thue-Morse word.  ...  The length of optimal word chains for #'(a) is proved to be 2n -1, and a conjecture on the enumeration of optimal word chains computing q+'(a) is proposed. 6  ...  The word chain is said to compute a word w if w belongs to the chain. The chain length of w is the smallest length of a word chain computing w.  ... 
doi:10.1016/0890-5401(89)90056-4 fatcat:spvvj5ncung5dngptzvxozgrxa

Page 2816 of Mathematical Reviews Vol. , Issue 91E [page]

1991 Mathematical Reviews  
In the case of a one-letter alphabet, word chains correspond to addition chains of natural numbers.  ...  The authors investigate word-chain complexity of iterates of the well-known Thue-Morse sequence—an infinite cube-free word over the alphabet {a,b}, which is the basis for constructing square-free words  ... 

Page 6944 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews  
Packel (1-LFC) 93m:68139 68R15 Bousquet-Mélou, Mireille (F-BORD-LB) The number of minimal word chains computing the Thue-Morse word. (English summary) Inform. Process. Lett. 44 (1992), no. 2, 57-64.  ...  The infinite word of Thue and Morse is usually defined as the result of infinite iteration of the morphism g(a) =a, 9(b) = b starting from the letter a.  ... 

Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words [chapter]

Michel Rigo, Pavel Salimov
2013 Lecture Notes in Computer Science  
We study the first properties of m-binomial equivalence. We compute the m-binomial complexity of the Sturmian words and of the Thue-Morse word.  ...  The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u.  ...  extension of Mahlers theorem on interpolation series, involving binomial coefficients on words.  ... 
doi:10.1007/978-3-642-40579-2_23 fatcat:hrtnb7uwfjeldjvcsu2hwenlfu

Another generalization of abelian equivalence: Binomial complexity of infinite words

M. Rigo, P. Salimov
2015 Theoretical Computer Science  
We study the first properties of m-binomial equivalence. We compute the m-binomial complexity of the Sturmian words and of the Thue-Morse word.  ...  The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u.  ...  extension of Mahlers theorem on interpolation series, involving binomial coefficients on words.  ... 
doi:10.1016/j.tcs.2015.07.025 fatcat:7hhgzbqxbndrtlpm7rwotnna7e

The origins of combinatorics on words

Jean Berstel, Dominique Perrin
2007 European journal of combinatorics (Print)  
These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century.  ...  We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration.  ...  The anonymous referee made a great number of concrete suggestions for improvements.  ... 
doi:10.1016/j.ejc.2005.07.019 fatcat:hgw2rs5ve5g6ral7o3ftmtsney

On the structure of compacted subword graphs of Thue–Morse words and their applications

Jakub Radoszewski, Wojciech Rytter
2012 Journal of Discrete Algorithms  
In particular we show a simple algorithm that checks, for a given string w, if w is a subword of a Thue-Morse word and computes its number of occurrences in nth Thue-Morse word in O(|w| + log n) time and  ...  The cdawgs of Thue-Morse words have regular and very simple structure, in particular they oer a powerful (exponential) compression of the set of all subwords in case of nite Thue-Morse words.  ...  [3, 11] A word u ∈ Sub(τ ), |u| > 4, is a bispecial factor of the innite Thue-Morse word if and only if ϕ −1 (u) is a (shorter) bispecial word.  ... 
doi:10.1016/j.jda.2011.01.001 fatcat:zwcdbja6azbulpp3d42pkdb2fe

Exploring Predictive States via Cantor Embeddings and Wasserstein Distance [article]

Samuel P. Loomis, James P. Crutchfield
2022 arXiv   pre-print
We compute Wasserstein distances between distributions over sequences ("predictions"), using a finite-dimensional embedding of sequences based on the Cantor for the underlying geometry.  ...  Recent progress on the self-supervised reconstruction of predictive states from time-series data focused on the use of reproducing kernel Hilbert spaces.  ...  The geometric approach is particularly insightful when computing the Wasserstein matrix between predictions estimated from Morse-Thue process data.  ... 
arXiv:2206.04198v1 fatcat:sexbqqc3gfgp3ar7wcesfv4am4

Streams are Forever

Jörg Endrullis, Dimitri Hendriks, Jan Willem Klop
2013 Bulletin of the European Association for Theoretical Computer Science  
In this paper, we describe a number of aspects of streams that we have encountered and studied during the last years. * This research has been funded by the Netherlands Organization for Scientific Research  ...  Here one can think of the streams of data queries flowing into search engines, or streams of financial data processed by financial organizations.  ...  Now the transformation of the Thue-Morse sequence M to the Toeplitz word T = T 101?  ... 
dblp:journals/eatcs/EndrullisHK13 fatcat:dpfouwrtwvejfa4ugtivffsgt4

Quasicrystals, model sets, and automatic sequences

Jean-Paul Allouche, Yves Meyer
2014 Comptes rendus. Physique  
We survey mathematical properties of quasicrystals, first from the point of view of harmonic analysis, then from the point of view of morphic and automatic sequences.  ...  Then σ ∞ (0... which is called the Thue-Morse (or Prouhet-Thue-Morse) sequence (see, e.g, [2] for a survey of numerous properties of this sequence). Theorem 2.4 A model set is a Meyer set.  ...  Example 2 The Fibonacci sequence and the Thue-Morse sequence are both morphic (see Example 1, and take B = A and ϕ = id). Furthermore the Thue-Morse sequence is 2-automatic.  ... 
doi:10.1016/j.crhy.2013.09.002 fatcat:4ttqrigmebbzffitprga5kfdca

Computations for symbolic substitutions [article]

Dan Rust, Scott Balchin
2017 arXiv   pre-print
We implement a check to verify that no counterexample exists to the so-called "strong coincidence conjecture" for a large number of substitutions on three and four letters.  ...  We study their subshifts (and related spaces) with an emphasis on examples of computations.  ...  This research used the SPECTRE High Performance Computing Facility at the University of Leicester.  ... 
arXiv:1705.11130v1 fatcat:ifrgdsibv5ad3dk4miepkxroni

Gaps in the Thue–Morse word [article]

Lukas Spiegelhofer
2021 arXiv   pre-print
We give an explicit method to compute the discrepancy of the number of occurrences of the block 01 in the ThueMorse sequence.  ...  The ThueMorse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science.  ...  Finally, we thank Michel Rigo for fruitful discussions on the topic and Clemens Müllner for pointing out that the case of arbitrary factors can be reduced to the case 01.  ... 
arXiv:2102.01018v2 fatcat:sdsm7fnmdvhkbc75ewtfugxauu

Exact Regularity and the Cohomology of Tiling Spaces [article]

Lorenzo Sadun
2018 arXiv   pre-print
If Ȟ^d(Ω_T, Q) = Q^k, then there exist k patches whose appearance govern the number of appearances of every other patch.  ...  If T is also one-dimensional, we put constraints on the measure of any cylinder set in Ω_T.  ...  I also thank Bill Kalahurka for figures of the equithirds tiling, and the anonymous referee for pointing out an error in an earlier version of Theorem 3.  ... 
arXiv:1004.2281v2 fatcat:4ktc5yehyfhyrkqfsppddfpleq

Exact regularity and the cohomology of tiling spaces

LORENZO SADUN
2011 Ergodic Theory and Dynamical Systems  
If H d ( , Q) = Q k , then there exist k patches each of whose appearances governs the number of appearances of every other patch.  ...  This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate.  ...  I also thank Bill Kalahurka for figures of the equithirds tiling, and the anonymous referee for pointing out an error in an earlier version of Theorem 3.  ... 
doi:10.1017/s0143385710000611 fatcat:unh6jvlhfveodan3d743ibjy6i

Quantum walks with sequential aperiodic jumps [article]

Marcelo A. Pires, Sílvio M. Duarte Queirós
2020 arXiv   pre-print
We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse and Rudin-Shapiro.  ...  Fingerprints of the aperiodicity in the hoppings are also found when additional distributional measures such as Shannon entropy, IPR, Jensen-Shannon dissimilarity, and kurtosis are computed.  ...  That measure [46] computes the number of nonidentical patterns in a sequence when scanned from t 0 to t max ; to that, we use the Kaspar-Schuster's method [47] to compute it, see the Appendix A.  ... 
arXiv:1910.02254v2 fatcat:arqs2z6mjzd3xhj56pwnvuiyhu
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