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A new geometric approach to Sturmian words

Kaisa Matomäki, Kalle Saari
2012 Theoretical Computer Science  
We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n ×n-grid and sets of finite Sturmian words of length n.  ...  We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.  ...  The purpose of this note is to introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n × n-grid and sets of finite Sturmian words of length n.  ... 
doi:10.1016/j.tcs.2012.01.040 fatcat:wwhqywl3u5eapa53h4xxre5m2e

A new geometric approach to Sturmian words [article]

Kaisa Matomäki, Kalle Saari
2012 arXiv   pre-print
We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n.  ...  We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.  ...  The purpose of this note is to introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n × n-grid and sets of finite Sturmian words of length n.  ... 
arXiv:1201.4468v1 fatcat:vsxd7652pjgyppkd5wjvxo3oza

Page 5108 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
The paper under review offers a new charac- terization of Sturmian words based on the counting function of palindromes: an infinite word is Sturmian if and only if for ev- ery integer n, it contains exactly  ...  Our algorithm uses a partition of the MST, similar to the sequential approach due to G. N. Frederickson [SIAM J.  ... 

A little more about morphic Sturmian words

Isabelle Fagnot
2006 RAIRO - Theoretical Informatics and Applications  
Among Sturmian words, some of them are morphic, i.e. fixed point of a non-identical morphism on words.  ...  Berstel and Séébold (1993) have shown that if a characteristic Sturmian word is morphic, then it can be extended by the left with one or two letters in such a way that it remains morphic and Sturmian.  ...  First, I wish to thank Jean Bertel for giving me the subject of this article.  ... 
doi:10.1051/ita:2006031 fatcat:qte3p5s47bfkbim2t2bgltd5wu

Elastodynamical properties of Sturmian structured media [article]

Mario Lázaro, Agnieszka Niemczynowicz, Artur Siemaszko, Luis M. García-Raffi
2021 arXiv   pre-print
The approach is presented in a general form so that it can be applied to waveguides of different nature, as long as they can be modeled with the transfer matrix method.  ...  We propose a methodology based on Sturmian sequences for the generation of structured mechanical systems from a given parameter.  ...  In particular, it will be shown later in Sec. 2.3 that Sturmian words defined as in Eq. (2) have an interesting geometrical interpretation.  ... 
arXiv:2105.02548v1 fatcat:cyn53jvryndjveledooiwg23ne

A geometrical approach of palindromic factors of standard billiard words

Jean-Pierre Borel
2007 Discrete Mathematics & Theoretical Computer Science  
We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.  ...  International audience Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension  ...  There are many ways to associate to ρ a family of Sturmian words, which are special words on the two letters alphabet A := {a 1 , a 2 }. Here we choose the geometrical one.  ... 
doi:10.46298/dmtcs.410 fatcat:leakulpicjeqnmcyvax2ir47iq

RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS

JEAN BERSTEL
2002 International journal of algebra and computation  
Among several extensions of Sturmian words to larger alphabets, the Arnoux-Rauzy words appear to share many of the properties of Sturmian words.  ...  Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years.  ...  Observe that, in the binary case, characteristic words were defined by a geometric property which appeared to be equivalent to the combinatorial definition.  ... 
doi:10.1142/s021819670200095x fatcat:skg3etiodncdxp2lne7ruex6cq

Various Properties of Sturmian Words

P. Baláži
2005 Acta Polytechnica  
This overview paper is devoted to Sturmian words. The first part summarizes different characterizations of Sturmian words.  ...  We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.  ...  The author is also very obliged to Edita Pelantová and Zuzana Masáková for their help and discussions on the paper.  ... 
doaj:5d7080b38c4b4cc6a403da2bd191c8e1 fatcat:uszccp35svdjllqp3axd7krciu

On the resonance spectra of particle-unstable light nuclei with a Sturmian approach that preserves the Pauli principle [article]

L. Canton, K. Amos, S. Karataglidis, G. Pisent, J.P. Svenne
2006 arXiv   pre-print
The formalism leads one to discuss a new concept, Pauli hindrance, which appears to be important especially to understand the structure of weakly-bound and unbound systems.  ...  Currently, to describe these systems, we use a macroscopic, collective model.  ...  Sturmians provide the solution of the scattering problems by matrix manipulation (hence the word "algebraic" in MCAS).  ... 
arXiv:nucl-th/0612062v1 fatcat:bfwfgsjtzvgbfinnbfxtehmbli

On k-abelian Equivalence and Generalized Lagrange Spectra [article]

Jarkko Peltomäki, Markus A. Whiteland
2019 arXiv   pre-print
Our geometric approach allows the study of k-abelian powers in Sturmian words by means of continued fractions.  ...  We study the set of k-abelian critical exponents of all Sturmian words. It has been proven that in the case k = 1 this set coincides with the Lagrange spectrum.  ...  Acknowledgments We thank the referee for a careful reading of the paper, which improved the presentation.  ... 
arXiv:1809.09047v2 fatcat:xst2p672z5g55axprhjhu5hijm

Episturmian words: a survey

Amy Glen, Jacques Justin
2009 RAIRO - Theoretical Informatics and Applications  
In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters.  ...  After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words.  ...  The authors would like to thank Jean Berstel and Pierre Arnoux for their helpful comments on a preliminary version of this paper.  ... 
doi:10.1051/ita/2009003 fatcat:b2fwwhlldjftjmgruabyj45xr4

Minimal complexity of equidistributed infinite permutations [article]

Sergey V. Avgustinovich, Anna E. Frid, Svetlana Puzynina
2016 arXiv   pre-print
Hence minimal complexity of aperiodic words is equal to n+1, and words with such complexity are called Sturmian.  ...  The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.  ...  Acknowledgements We are grateful to Pascal Hubert for suggesting a term "equidistributed permutation" instead of "ergodic", and to an anonimous referee for careful reading and valuable comments.  ... 
arXiv:1503.06188v3 fatcat:sbhvzgiesfdjljasg572owfodu

Ergodic Infinite Permutations of Minimal Complexity [chapter]

Sergey V. Avgustinovich, Anna E. Frid, Svetlana Puzynina
2015 Lecture Notes in Computer Science  
directly related to Sturmian words.  ...  Hence minimal complexity of aperiodic words is equal to n + 1, and words with such complexity are called Sturmian.  ...  A Christoffel word can be defined as a word of the form 0b1 or 1b0, where b is a bispecial factor of a Sturmian word s.  ... 
doi:10.1007/978-3-319-21500-6_5 fatcat:rhnkcbillfhtlexg3gigu7332u

Local Rules for Computable Planar Tilings

Thomas Fernique, Mathieu Sablik
2012 Electronic Proceedings in Theoretical Computer Science  
A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones.  ...  They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982).  ...  This bound is known to be at most one for Sturmian words (and only them). The words at distance at most one from a Sturmian word of slope α are however not all Sturmian.  ... 
doi:10.4204/eptcs.90.11 fatcat:pvcnb3ygofa5tdojoj4hgxvcba

New algebraic and geometric constructs arising from Fibonacci numbers

Fabio Caldarola, Gianfranco d'Atri, Mario Maiolo, Giuseppe Pirillo
2020 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family {C n : n ∈ N} of octagons that come very close to the regular octagon.  ...  The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case n = 5.  ...  The authors are indebted to the anonymous reviewers who, with their meticulous observations, have allowed them to significantly improve the readability of this paper.  ... 
doi:10.1007/s00500-020-05256-1 fatcat:bnfkz7gkpfafvd6fsvbojpeq5q
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