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Optimal word chains for the Thue-Morse word

1989
*
Information and Computation
*

We study

doi:10.1016/0890-5401(89)90056-4
fatcat:spvvj5ncung5dngptzvxozgrxa
*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. ...##
###
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]

2013
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-40579-2_23
fatcat:hrtnb7uwfjeldjvcsu2hwenlfu
*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*. ...##
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Another generalization of abelian equivalence: Binomial complexity of infinite words

2015
*
Theoretical Computer Science
*

We study

doi:10.1016/j.tcs.2015.07.025
fatcat:7hhgzbqxbndrtlpm7rwotnna7e
*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*. ...##
###
The origins of combinatorics on words

2007
*
European journal of combinatorics (Print)
*

These considerations gave rise to early results such as those

doi:10.1016/j.ejc.2005.07.019
fatcat:hgw2rs5ve5g6ral7o3ftmtsney
*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. ...##
###
On the structure of compacted subword graphs of Thue–Morse words and their applications

2012
*
Journal of Discrete Algorithms
*

In particular we show a simple algorithm that checks, for a given string w, if w is a subword

doi:10.1016/j.jda.2011.01.001
fatcat:zwcdbja6azbulpp3d42pkdb2fe
*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*. ...##
###
Exploring Predictive States via Cantor Embeddings and Wasserstein Distance
[article]

2022
*
arXiv
*
pre-print

We

arXiv:2206.04198v1
fatcat:sexbqqc3gfgp3ar7wcesfv4am4
*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. ...##
###
Streams are Forever

2013
*
Bulletin of the European Association for Theoretical Computer Science
*

In this paper, we describe a

dblp:journals/eatcs/EndrullisHK13
fatcat:dpfouwrtwvejfa4ugtivffsgt4
*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? ...##
###
Quasicrystals, model sets, and automatic sequences

2014
*
Comptes rendus. Physique
*

We survey mathematical properties

doi:10.1016/j.crhy.2013.09.002
fatcat:4ttqrigmebbzffitprga5kfdca
*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. ...##
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Computations for symbolic substitutions
[article]

2017
*
arXiv
*
pre-print

We implement a check to verify that no counterexample exists to

arXiv:1705.11130v1
fatcat:ifrgdsibv5ad3dk4miepkxroni
*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. ...##
###
Gaps in the Thue–Morse word
[article]

2021
*
arXiv
*
pre-print

We give an explicit method to

arXiv:2102.01018v2
fatcat:sdsm7fnmdvhkbc75ewtfugxauu
*compute**the*discrepancy*of**the**number**of*occurrences*of**the*block 01 in*the**Thue*–*Morse*sequence. ...*The**Thue*–*Morse*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. ...##
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Exact Regularity and the Cohomology of Tiling Spaces
[article]

2018
*
arXiv
*
pre-print

If Ȟ^d(Ω_T, Q) = Q^k, then there exist k patches whose appearance govern

arXiv:1004.2281v2
fatcat:4ktc5yehyfhyrkqfsppddfpleq
*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. ...##
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Exact regularity and the cohomology of tiling spaces

2011
*
Ergodic Theory and Dynamical Systems
*

If H d ( , Q) = Q k , then there exist k patches each

doi:10.1017/s0143385710000611
fatcat:unh6jvlhfveodan3d743ibjy6i
*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. ...##
###
Quantum walks with sequential aperiodic jumps
[article]

2020
*
arXiv
*
pre-print

We analyze a set

arXiv:1910.02254v2
fatcat:arqs2z6mjzd3xhj56pwnvuiyhu
*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. ...
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