Combinatorics On Sturmian Words. Amy Glen. Major Review Seminar. February 27, 2004 DISCIPLINE OF PURE MATHEMATICS
|
|
- Carmella Jefferson
- 5 years ago
- Views:
Transcription
1 Combinatorics On Sturmian Words Amy Glen Major Review Seminar February 27, 2004 DISCIPLINE OF PURE MATHEMATICS
2 OUTLINE Finite and Infinite Words Sturmian Words Mechanical Words and Cutting Sequences Infinite Words and Morphisms The Fibonacci Word & Characteristic Sturmian Words Decompositions of Sturmian Words into Palindrome Words Powers of Factors of Sturmian Words The Tribonacci Sequence Future Endeavours Application to Continued Fractions 1
3 FINITE AND INFINITE WORDS Combinatorial properties have become significantly important Applications in physics, biology, maths & computer science Family of Sturmian words a very active subject of research Named after Charles François Sturm ( ) Morse and Hedlund (1940): symbolic dynamics Largest developments in last twenty years Points of view: geometrically, combinatorially, algebraically Combinatorial approach is most common 2
4 WHAT ARE FINITE AND INFINITE WORDS? Alphabet: A denotes a finite set of symbols called letters. Finite/Infinite word: concatenation of letters from A. Let u, w denote finite words & x, y denote infinite words. If x = uwy w is a factor of x, u is a prefix of x, y is a suffix of x. The length of w is number of letters it contains, denoted by w. The empty word of length 0 is denoted by ε. The set A of all finite words over A is a monoid (Id = ε). 3
5 WHAT IS A STURMIAN WORD? An infinite word s with exactly n + 1 distinct factors of length n, n 0 = s is over a two-letter alphabet, say A = {a, b}. Sturmian words are not eventually periodic they are aperiodic words of minimal complexity many characterizations and numerous properties of Sturmian words Applications: symbolic dynamics, continued fraction expansion, physics (crystallography), computer science (pattern recognition) Equivalent definitions: mechanical words & cutting sequences 4
6 IRRATIONAL MECHANICAL WORDS An infinite word s over A is Sturmian an irrational α (0, 1) & ρ R such that s is either: s α,ρ = s(0)s(1)s(2) or s α,ρ = s (0)s (1)s (2), where s(n) = a if (n + 1)α + ρ nα + ρ = 0, s(n) = b otherwise; s (n) = a if (n + 1)α + ρ nα + ρ = 0, s (n) = b otherwise. Note: denotes floor function & denotes ceiling function α is called the slope & ρ is the intercept ρ = 0 s α,0 = ac α & s α,0 = bc α where c α is the characteristic Sturmian word of slope α 5
7 CUTTING SEQUENCES An interpretation of mechanical words Line: y = βx + ρ with irrational β > 0, ρ R consider this ray in the positive quadrant of R 2 overlay quadrant with an integer grid & construct infinite word K β,ρ vertical grid-line crossed: label intersection with a horizontal grid-line crossed: label intersection with b Labels: x 0, x 1, x 2,... K β,ρ = x 0 x 1 x 2 is a Sturmian word, i.e. K β,ρ = s β/(1+β),ρ/(1+β). Note: c α = K β,0 α = β/(1 + β) 6
8 y = x Fibonacci word b a a b b a a b a a Fibonacci word: f = K ( 5 1)/2,0 = abaababaabaababaababaab special example of a characteristic Sturmian word of slope α = Properties of f may be extended to Sturmian words 7
9 INFINITE WORDS & MORPHISMS Let (u n ) n 0 be a sequence of words from A s.t. u n is a proper prefix of u n+1. This gives obvious meaning to lim n u n as an infinite word. A morphism on A is a map ψ : A A such that ψ(uv) = ψ(u)ψ(v), u, v A. If ψ(c) = cw, c A, w A ψ n (c) is a proper prefix of ψ n+1 (c) and (ψ n (c)) n 0 converges to a unique infinite word: We say that x is generated by ψ. x = lim n ψn (c). 8
10 PALINDROME WORDS Palindrome: a finite word that reads the same backwards as forwards Examples: aa, abaaba, aba, ababa Objects of great interest in computer science Important tools used in the study of factors of Sturmian words Question: Where exactly do palindromes occur in a Sturmian word? Example: f = ab(aa)bab(aa)b(aa)bab(aa)bab(aa), where aa occurs at positions: 2, 7, 10, 15, 20,... Distances: 5, 3, 5, 5,... Fibonacci word over the alphabet {5, 3} I have described precisely where palindromes occur in a Sturmian word 9
11 FINITE FIBONACCI WORDS Define the Fibonacci numbers (F n ) n 1 by F 1 = F 0 = 1, F n = F n 1 + F n 2. Let f n denote the prefix of f of length F n and set f 1 = b. Then f 0 = a, f n = f n 1 f n 2 and f = lim n f n. E.g. f 1 = ab, f 2 = aba, f 3 = abaab Singular words of f : w 2n 1 = af 2n 1 b 1, w 2n = bf 2n a 1 Examples: w 0 = b, w 1 = aa, w 2 = bab, w 3 = aabaa palindromes Wen & Wen (1994): f = aw 0 w 1 w 2 w 3 = a b aa bab aabaa 10
12 CONTINUED FRACTIONS Combinatorial structure of c α has close relationship with the continued fraction (CF) of its slope α Recall: every irrational α (0, 1) has a unique CF expansion 1 α = [0; a 1, a 2, a 3,...] = 1 a 1 + a 2 + where all the a i Z + are called partial quotients. 1 a 3 +, 11
13 CHARACTERISTIC STURMIAN WORDS Let α (0, 1) be irrational with α = [0; 1 + d 1, d 2, d 3,...]. Define standard sequence (s n ) n 1 of words by: s 1 = b, s 0 = a, s n = s d n n 1 s n 2. n 0, s n is a prefix of s n+1 lim n s n is a well-defined infinite word in fact, each s n is a prefix of c α and c α = lim n s n Fibonacci word: α = (3 5)/2 = [0; 2, 1, 1, 1,...] Singular words of c α : w 2n 1 = as 2n 1 b 1, w 2n = bs 2n a 1 Melançon (1999): defined palindrome words v n such that c α = av 0 v 1 v 2 v 3 v 4 12
14 EXAMPLE Consider c α with α = ( 3 1)/2 = [0; 2, 1, 2, 1, 2, 1, 2,...]. Standard sequence: s 1 = ab, s 2 = s 1 s 0 = aba, s 3 = s 2 2 s 1 = abaabaab,... c α = abaabaababaabaabaababaabaabaab Singular words: w 0 = b, w 1 = aa, w 2 = bab, w 3 = aabaabaa,... Decomposition in terms of w 2 : c α = abaabaa(bab)aabaabaa(bab)aabaabaa(bab)aabaa(bab)aabaabaa = abaabaa(bab)z 1 (bab)z 2 (bab)z 3 (bab)z 4, where z 1 z 2 z 3 is the characteristic Sturmian word of slope β = [0; 1, 2] = 3 1 over the alphabet {aabaa, w 3 } 13
15 CONJUGATES OF f AND c α Let x, x be infinite words such that x = wx, w finite, w = k. Then x is called the k th conjugate of x; i.e. x is obtained from x by deleting the first k letters of x. Levé and Séébold (2003): a factorization of each conjugate of f as an infinite concatenation of generalized singular words I have extended this result to conjugates of c α for α = [0; 2, r, r, r,...] and α = [0; 1, 1, r, r, r,...] Paper accepted for publication (European J. Combinatorics) 14
16 POWERS IN STURMIAN WORDS Interested in finite words w and integers p Z + such that w p = www w (p times) is a factor of c α Application: spectral properties of associated quantum mechanical quasi-crystal models (Physics) Damanik and Lenz (2003): explicitly determined all integer powers of words occurring in c α also considered the # of distinct squares of words contained in s n combinatorial method: based on properties of the building blocks s n 15
17 EXACT NUMBER OF SQUARES IN f n A square is a word of the form uu; u a finite word Example: f 4 = abaababa contains 4 squares: (aba) 2, (ab) 2, (ba) 2, a 2 Fraenkel and Simpson (1999): exact # of distinct squares in f n is 2(F n 2 1), n 5 Recently extended to the case of the factors s n of c α u p (p 2) is a factor of f = u = F n for some n I have extended the above results to the Tribonacci sequence paper in preparation 16
18 TRIBONACCI SEQUENCE ξ Infinite word over A = {a, b, c}: ξ = abacabaabacababacabaabacab a generalization of the Fibonacci word Define the Tribonacci numbers (T n ) n 1 by T 1 = T 0 = 1, T 1 = 2, T n = T n 1 + T n 2 + T n 3. Let A n denote the prefix of ξ of length T n and set A 1 = c. Then A 0 = a, A 1 = ab, A n = A n 1 A n 2 A n 3 and ξ = lim n A n. E.g. A 2 = (ab)ac, A 3 = (abac)(ab)a, A 4 = (abacaba)(abac)(ab) Yet to determine exactly where palindromes occur in ξ 17
19 EXACT NUMBER OF SQUARES IN A n Define P 0 = ε, P n = A n 1 A n 2 A 1 A 0, n 1 E.g. P 1 = a, P 2 = aba, P 3 = abacaba palindrome prefixes of ξ Exact # of distinct squares in A n (n 6): n 3 m=0 ( P m + 1) + P n 5 + P n Example: A 5 = abacabaabacababacabaabac contains 7 squares: aa, abab, baba, abaaba, (abacab) 2, (bacaba) 2, (abacaba) 2 A 6 has (0 + 1) + (1 + 1) + (3 + 1) + (7 + 1) = 17 squares 18
20 SQUARES AND HIGHER POWERS IN ξ Let u be a factor of ξ with T n u < T n+1. u p # u s.t. u p in ξ T n 2 T n T n + T n 1 2 P n T n 3 P n T n + T n I am now extending this to episturmian words Episturmian words: a generalization of Sturmian words to an arbitrary finite alphabet 19
21 FUTURE ENDEAVOURS The study of episturmian words is a new area of research I hope to extend my results to the family of episturmian words Application to continued fractions and transcendence: γ R is transcendental if it is not a zero of a polynomial with integer coefficients Allouche et al. (2001): Let a, b Z + (a b) & let x = (x n ) n 0 be an infinite sequence over {a, b}. Then γ := [0; x 0, x 1, x 2,...] is transcendental if x is a Sturmian word over {a, b}. Question: What if the partial quotients form an infinite episturmian word over a k-letter alphabet? 20
22 SOME INTERESTING READING Allouche et al. (2001): Transcendence of Sturmian or morphic continued fractions Berstel and Séébold (2002): Chapter 2 of Algebraic Combinatorics on Words Damanik and Lenz (2003): Powers in Sturmian sequences Justin and Pirillo (2002): Episturmian words & episturmian morphisms Melançon (1999): Lyndon words & singular factors of Sturmian words Wen and Wen (1994): Some properties of the Fibonacci word 21
Sturmian Words, Sturmian Trees and Sturmian Graphs
Sturmian Words, Sturmian Trees and Sturmian Graphs A Survey of Some Recent Results Jean Berstel Institut Gaspard-Monge, Université Paris-Est CAI 2007, Thessaloniki Jean Berstel (IGM) Survey on Sturm CAI
More informationOn Sturmian and Episturmian Words, and Related Topics
On Sturmian and Episturmian Words, and Related Topics by Amy Glen Supervisors: Dr. Alison Wolff and Dr. Robert Clarke A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy
More informationConference on Diophantine Analysis and Related Fields 2006 in honor of Professor Iekata Shiokawa Keio University Yokohama March 8, 2006
Diophantine analysis and words Institut de Mathématiques de Jussieu + CIMPA http://www.math.jussieu.fr/ miw/ March 8, 2006 Conference on Diophantine Analysis and Related Fields 2006 in honor of Professor
More informationSpecial Factors and Suffix and Factor Automata
Special Factors and Suffix and Factor Automata LIAFA, Paris 5 November 2010 Finite Words Let Σ be a finite alphabet, e.g. Σ = {a, n, b, c}. A word over Σ is finite concatenation of symbols of Σ, that is,
More informationON THE LEAST NUMBER OF PALINDROMES IN AN INFINITE WORD
ON THE LEAST NUMBER OF PALINDROMES IN AN INFINITE WORD GABRIELE FICI AND LUCA Q. ZAMBONI ABSTRACT. We investigate the least number of palindromic factors in an infinite word. We first consider general
More informationPalindromic complexity of infinite words associated with simple Parry numbers
Palindromic complexity of infinite words associated with simple Parry numbers Petr Ambrož (1)(2) Christiane Frougny (2)(3) Zuzana Masáková (1) Edita Pelantová (1) March 22, 2006 (1) Doppler Institute for
More informationRECENT RESULTS IN STURMIAN WORDS 1. LITP, IBP, Universite Pierre et Marie Curie, 4, place Jussieu. F Paris Cedex 05, France
RECENT RESULTS IN STURMIAN WORDS 1 JEAN BERSTEL LITP, IBP, Universite Pierre et Marie Curie, 4, place Jussieu F-75252 Paris Cedex 05, France e-mail: Jean.Berstel@litp.ibp.fr ABSTRACT In this survey paper,
More informationTwo-dimensional Total Palindrome Complexity
Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity ISSN 1584-4536, vol 6, 2008, pp. 3 12. Two-dimensional Total Palindrome Complexity Mira-Cristiana Anisiu (Cluj-Napoca)
More informationThe free group F 2, the braid group B 3, and palindromes
The free group F 2, the braid group B 3, and palindromes Christian Kassel Université de Strasbourg Institut de Recherche Mathématique Avancée CNRS - Université Louis Pasteur Strasbourg, France Deuxième
More informationPERIODS OF FACTORS OF THE FIBONACCI WORD
PERIODS OF FACTORS OF THE FIBONACCI WORD KALLE SAARI Abstract. We show that if w is a factor of the infinite Fibonacci word, then the least period of w is a Fibonacci number. 1. Introduction The Fibonacci
More informationA Fine and Wilf s theorem for pseudoperiods and Justin s formula for generalized pseudostandard words
A Fine and Wilf s theorem for pseudoperiods and Justin s formula for generalized pseudostandard words A. Blondin Massé 1,2, G. Paquin 2 and L. Vuillon 2 1 Laboratoire de Combinatoire et d Informatique
More informationFactorizations of the Fibonacci Infinite Word
2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (205), Article 5.9.3 Factorizations of the Fibonacci Infinite Word Gabriele Fici Dipartimento di Matematica e Informatica Università di Palermo Via Archirafi
More informationarxiv: v2 [math.co] 24 Oct 2012
On minimal factorizations of words as products of palindromes A. Frid, S. Puzynina, L. Zamboni June 23, 2018 Abstract arxiv:1210.6179v2 [math.co] 24 Oct 2012 Given a finite word u, we define its palindromic
More informationON THE DECIMAL EXPANSION OF ALGEBRAIC NUMBERS
Fizikos ir matematikos fakulteto Seminaro darbai, Šiaulių universitetas, 8, 2005, 5 13 ON THE DECIMAL EXPANSION OF ALGEBRAIC NUMBERS Boris ADAMCZEWSKI 1, Yann BUGEAUD 2 1 CNRS, Institut Camille Jordan,
More informationGeneralizing the Wythoff Game
Generalizing the Wythoff Game Cody Schwent Advisor: Dr David Garth 1 Introduction Let c be a positive integer In the Wythoff Game there are two piles of tokens with two players who alternately take turns
More informationA characterization of balanced episturmian sequences
A characterization of balanced episturmian sequences Geneviève Paquin Laurent Vuillon Submitted: Nov 21, 2006; Accepted: April 23, 2007 Mathematics Subject Classification: 68R15 Abstract It is well-known
More informationAbout Duval Extensions
About Duval Extensions Tero Harju Dirk Nowotka Turku Centre for Computer Science, TUCS Department of Mathematics, University of Turku June 2003 Abstract A word v = wu is a (nontrivial) Duval extension
More informationCombinatorics on Finite Words and Data Structures
Combinatorics on Finite Words and Data Structures Dipartimento di Informatica ed Applicazioni Università di Salerno (Italy) Laboratoire I3S - Université de Nice-Sophia Antipolis 13 March 2009 Combinatorics
More informationWords with the Smallest Number of Closed Factors
Words with the Smallest Number of Closed Factors Gabriele Fici Zsuzsanna Lipták Abstract A word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal
More informationOpen and closed factors of Arnoux-Rauzy words arxiv: v1 [math.co] 12 Oct 2018 Olga Parshina 1,2 and Luca Zamboni 1
Open and closed factors of Arnoux-Rauzy words arxiv:18.05472v1 [math.co] 12 Oct 2018 Olga Parshina 1,2 and Luca Zamboni 1 1 Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan,
More informationarxiv: v1 [math.ds] 18 Jan 2016
The structure of palindromes in the Fibonacci sequence and some applications Huang Yuke 1,2 Wen Zhiying 3,4 arxiv:1601.04391v1 [math.ds] 18 Jan 2016 ABSTRACT Let P be the set of palindromes occurring in
More informationFIXED POINTS OF MORPHISMS AMONG BINARY GENERALIZED PSEUDOSTANDARD WORDS
#A21 INTEGERS 18 (2018) FIXED POINTS OF MORPHISMS AMONG BINARY GENERALIZED PSEUDOSTANDARD WORDS Tereza Velká Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
More informationarxiv: v1 [math.co] 22 Jan 2013
A Coloring Problem for Sturmian and Episturmian Words Aldo de Luca 1, Elena V. Pribavkina 2, and Luca Q. Zamboni 3 arxiv:1301.5263v1 [math.co] 22 Jan 2013 1 Dipartimento di Matematica Università di Napoli
More informationCrochemore factorization of Sturmian and other infinite words
WoWA,7june2006 1 Crochemore factorization of Sturmian and other infinite words Jean Berstel and Alessandra Savelli Institut Gaspard Monge Université de Marne la Vallée and CNRS(UMR 8049) Workshop on Words
More informationThe Difference Sequences of the Columns of Wythoff s Array
The Difference Sequences of the Columns of Wythoff s Array 1 Introduction Halleh Peterman Advisor: Dr. David Garth The Fibonacci numbers are defined by F 0 = 0,F 1 = 1, and F n = F n 1 +F n for n. For
More informationFrom Christoffel words to braids
From Christoffel words to braids Christian Kassel Institut de Recherche Mathématique Avancée CNRS - Université Louis Pasteur Strasbourg, France Colloquium at Rutgers 13 April 2007 Joint work With Christophe
More informationarxiv: v1 [cs.dm] 13 Feb 2010
Properties of palindromes in finite words arxiv:1002.2723v1 [cs.dm] 13 Feb 2010 Mira-Cristiana ANISIU Valeriu ANISIU Zoltán KÁSA Abstract We present a method which displays all palindromes of a given length
More informationCombinatorics, Automata and Number Theory
Combinatorics, Automata and Number Theory CANT Edited by Valérie Berthé LIRMM - Université Montpelier II - CNRS UMR 5506 161 rue Ada, F-34392 Montpellier Cedex 5, France Michel Rigo Université de Liège,
More informationSturmian words. Lecture notes
Sturmian words Lecture notes Sturmian words are a challenging topic, which is a bridge between combinatorics on words, number theory and dynamical systems. Sturmian words have been widely studied for their
More informationDiscrete Mathematics
Discrete Mathematics 310 (2010) 109 114 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Total palindrome complexity of finite words Mira-Cristiana
More informationEuropean Journal of Combinatorics
European Journal of Combinatorics 33 (202) 54 536 Contents lists available at SciVerse ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc A standard correspondence
More information#A36 INTEGERS 12 (2012) FACTOR FREQUENCIES IN LANGUAGES INVARIANT UNDER SYMMETRIES PRESERVING FACTOR FREQUENCIES
#A36 INTEGERS 2 (202) FACTOR FREQUENCIES IN LANGUAGES INVARIANT UNDER SYMMETRIES PRESERVING FACTOR FREQUENCIES L ubomíra Balková Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,
More informationTheoretical Computer Science. Completing a combinatorial proof of the rigidity of Sturmian words generated by morphisms
Theoretical Computer Science 428 (2012) 92 97 Contents lists available at SciVerse ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Note Completing a combinatorial
More informationINITIAL POWERS OF STURMIAN SEQUENCES
INITIAL POWERS OF STURMIAN SEQUENCES VALÉRIE BERTHÉ, CHARLES HOLTON, AND LUCA Q. ZAMBONI Abstract. We investigate powers of prefixes in Sturmian sequences. We obtain an explicit formula for ice(ω), the
More informationGames derived from a generalized Thue-Morse word
Games derived from a generalized Thue-Morse word Aviezri S. Fraenkel, Dept. of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel; fraenkel@wisdom.weizmann.ac.il
More informationA generalization of Thue freeness for partial words. By: Francine Blanchet-Sadri, Robert Mercaş, and Geoffrey Scott
A generalization of Thue freeness for partial words By: Francine Blanchet-Sadri, Robert Mercaş, and Geoffrey Scott F. Blanchet-Sadri, R. Mercas and G. Scott, A Generalization of Thue Freeness for Partial
More informationNotes on Rauzy Fractals
Notes on Rauzy Fractals It is a tradition to associate symbolic codings to dynamical systems arising from geometry or mechanics. Here we introduce a classical result to give geometric representations to
More informationPalindromic continued fractions. 1. Introduction
Palindromic continued fractions Boris ADAMCZEWSKI (Lyon) & Yann BUGEAUD * (Strasbourg) 1. Introduction An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion
More informationComplexité palindromique des codages de rotations et conjectures
Complexité palindromique des codages de rotations et conjectures Sébastien Labbé Laboratoire d Informatique, de Robotique et de Microélectronique de Montpellier Université Montpellier 2 Laboratoire de
More informationANNALESDEL INSTITUTFOURIER
ANNALESDEL INSTITUTFOURIER ANNALES DE L INSTITUT FOURIER Petr AMBROŽ, Zuzana MASÁKOVÁ, Edita PELANTOVÁ & Christiane FROUGNY Palindromic complexity of infinite words associated with simple Parry numbers
More informationThe number of runs in Sturmian words
The number of runs in Sturmian words Paweł Baturo 1, Marcin Piatkowski 1, and Wojciech Rytter 2,1 1 Department of Mathematics and Computer Science, Copernicus University 2 Institute of Informatics, Warsaw
More informationOn Aperiodic Subtraction Games with Bounded Nim Sequence
On Aperiodic Subtraction Games with Bounded Nim Sequence Nathan Fox arxiv:1407.2823v1 [math.co] 10 Jul 2014 Abstract Subtraction games are a class of impartial combinatorial games whose positions correspond
More informationReversals and palindromes in continued fractions
Theoretical Computer Science 380 (007) 0 37 www.elsevier.com/locate/tcs Reversals and palindromes in continued fractions Boris Adamczewski a, Jean-Paul Allouche b, a CNRS, Université Claude Bernard Lyon,
More informationMathematische Annalen
Math. Ann. (2010) 346:107 116 DOI 10.1007/s00208-009-0391-z Mathematische Annalen On the expansion of some exponential periods in an integer base Boris Adamczewski Received: 25 September 2008 / Revised:
More informationWorkshop on Fibonacci Words
Tero Harju Juhani Karhumäki (Eds.) Proceedings of the Workshop on Fibonacci Words Turku, September 2006 Turku Centre for Computer Science TUCS General Publication No 43, June 2007 Proceedings of the Workshop
More informationDENSITY OF CRITICAL FACTORIZATIONS
DENSITY OF CRITICAL FACTORIZATIONS TERO HARJU AND DIRK NOWOTKA Abstract. We investigate the density of critical factorizations of infinte sequences of words. The density of critical factorizations of a
More informationBOUNDS ON ZIMIN WORD AVOIDANCE
BOUNDS ON ZIMIN WORD AVOIDANCE JOSHUA COOPER* AND DANNY RORABAUGH* Abstract. How long can a word be that avoids the unavoidable? Word W encounters word V provided there is a homomorphism φ defined by mapping
More informationA connection between palindromic and factor complexity using return words
A connection between palindromic and factor complexity using return words Michelangelo Bucci Alessandro De Luca Amy Glen Luca Q. Zamboni June 8, 2008 Abstract In this paper we prove that for any infinite
More informationON SUBSTITUTION INVARIANT STURMIAN WORDS: AN APPLICATION OF RAUZY FRACTALS
ON SUBSTITUTION INVARIANT STURMIAN WORDS: AN APPLICATION OF RAUZY FRACTALS V. BERTHÉ, H. EI, S. ITO AND H. RAO Abstract. Sturmian words are infinite words that have exactly n + 1 factors of length n for
More informationEXTREMALPROPERTIES OF (EPI)STURMIAN SEQUENCES AND DISTRIBUTION MODULO 1 1. INTRODUCTION
L Enseignement Mathématique (2) 56 (2010), 365 401 EXTREMALPROPERTIES OF (EPI)STURMIAN SEQUENCES AND DISTRIBUTION MODULO 1 by Jean-Paul ALLOUCHE and Amy GLEN ABSTRACT. Starting from a study of Y.Bugeaud
More informationGENERALIZED PALINDROMIC CONTINUED FRACTIONS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number 1, 2018 GENERALIZED PALINDROMIC CONTINUED FRACTIONS DAVID M. FREEMAN ABSTRACT. In this paper, we introduce a generalization of palindromic continued
More informationA Generalization of Sturmian Sequences; Combinatorial Structure and Transcendence
A Generalization of Sturmian Sequences; Combinatorial Structure and Transcendence Rebecca N. Risley Department of Mathematics University of North Texas Denton, TX 76203-5116 rnr0002@jove.acs.unt.edu Luca
More informationGeneralized Thue-Morse words and palindromic richness
Generalized Thue-Morse words and palindromic richness arxiv:1104.2476v3 [math.co] 10 Jul 2011 Štěpán Starosta Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00
More informationON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS
ON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS B. ADAMCZEWSKI AND N. RAMPERSAD Abstract. We prove that every algebraic number contains infinitely many occurrences of 7/3-powers in its binary expansion.
More informationPALINDROMIC RICHNESS
PALINDROMIC RICHNESS Amy Glen Jacques Justin Steve Widmer Luca Q. Zamboni June 8, 2008 Abstract In this paper, we study combinatorial and structural properties of a new class of finite and infinite words
More informationA generalization of Thue freeness for partial words
A generalization of Thue freeness for partial words F. Blanchet-Sadri 1 Robert Mercaş 2 Geoffrey Scott 3 September 22, 2008 Abstract This paper approaches the combinatorial problem of Thue freeness for
More information#A7 INTEGERS 11B (2011) SUBWORD COMPLEXITY AND LAURENT SERIES
#A7 INTEGERS 11B (2011) SUBWORD COMPLEXITY AND LAURENT SERIES Alina Firicel Institut Camille Jordan, Université Lyon 1, Villeurbanne, France firicel@math.univ-lyon1.fr Received: 10/18/10, Revised: 4/4/11,
More informationCoding of irrational rotation: a different view
Coding of irrational rotation: a different view Shigeki Akiyama, Niigata University, Japan April 20, 2007, Graz Joint work with M. Shirasaka. Typeset by FoilTEX Let A = {0, 1,..., m 1} be a finite set
More informationSome improvments of the S-adic Conjecture
Some improvments of the S-adic Conjecture Julien Leroy Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens
More informationarxiv: v1 [math.co] 27 May 2012
Fundamenta Informaticae XX (2012) 1 9 1 IOS Press arxiv:1205.5946v1 [math.co] 27 May 2012 Some characterizations of Sturmian words in terms of the lexicographic order Michelangelo Bucci Department of Mathematics,
More informationCombinatorics on Words:
Combinatorics on Words: Applications to Number Theory and Ramsey Theory Narad Rampersad Department of Mathematics and Statistics University of Winnipeg 9 May 2008 Narad Rampersad (University of Winnipeg)
More informationSupplementary Examples for
Supplementary Examples for Infinite Interval Exchange Transformations from Shifts ([LN15]) Luis-Miguel Lopez Philippe Narbel Version 1.0, April, 2016 Contents A The Shifted Intervals Preservation (Lemma
More informationOn different generalizations of episturmian words
Theoretical Computer Science 393 (2008) 23 36 www.elsevier.com/locate/tcs On different generalizations of episturmian words Michelangelo Bucci a, Aldo de Luca a,, Alessandro De Luca a, Luca Q. Zamboni
More informationJulien Leroy 1 LAMFA, CNRS UMR 6410, Univ. de Picardie Jules Verne, Amiens, France
#A5 INTEGERS 13 (2013) A COMBINATORIAL PROOF OF S-ADICITY FOR SEQUENCES WITH LINEAR COMPLEXITY Julien Leroy 1 LAMFA, CNRS UMR 6410, Univ. de Picardie Jules Verne, Amiens, France julien.leroy@uni.lu Gwenaël
More informationto 1D Periodic Approximants Aperiodic Hamiltonians Sponsoring Jean BELLISSARD CRC 701, Bielefeld, Germany
Sponsoring Periodic Approximants to 1D Aperiodic Hamiltonians CRC 701, Bielefeld, Germany Jean BELLISSARD Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics e-mail: jeanbel@math.gatech.edu
More informationSums of Digits, Overlaps, and Palindromes
Sums of Digits, Overlaps, and Palindromes Jean-Paul Allouche, Jeffrey Shallit To cite this version: Jean-Paul Allouche, Jeffrey Shallit Sums of Digits, Overlaps, and Palindromes Discrete Mathematics and
More informationSUBSTITUTION DYNAMICAL SYSTEMS: CHARACTERIZATION OF LINEAR REPETITIVITY AND APPLICATIONS
SUBSTITUTION DYNAMICAL SYSTEMS: CHARACTERIZATION OF LINEAR REPETITIVITY AND APPLICATIONS DAVID DAMANIK 1 AND DANIEL LENZ 2 1 Department of Mathematics 253 37, California Institute of Technology, Pasadena,
More informationOn Strong Alt-Induced Codes
Applied Mathematical Sciences, Vol. 12, 2018, no. 7, 327-336 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8113 On Strong Alt-Induced Codes Ngo Thi Hien Hanoi University of Science and
More informationAperiodic Hamiltonians
Spectrum Approximation for Aperiodic Hamiltonians Sponsoring Jean BELLISSARD Westfälische Wilhelms-Universität, Münster Department of Mathematics Georgia Institute of Technology, Atlanta School of Mathematics
More informationGeneralized Thue-Morse words and palindromic richness extended abstract
arxiv:1104.2476v2 [math.co] 26 Apr 2011 1 Introduction Generalized Thue-Morse words and palindromic richness extended abstract Štěpán Starosta Department of Mathematics, FNSPE, Czech Technical University
More informationOn infinite permutations
On infinite permutations Dmitri G. Fon-Der-Flaass, Anna E. Frid To cite this version: Dmitri G. Fon-Der-Flaass, Anna E. Frid. On infinite permutations. Stefan Felsner. 2005 European Conference on Combinatorics,
More informationSIMPLE ALGORITHM FOR SORTING THE FIBONACCI STRING ROTATIONS
SIMPLE ALGORITHM FOR SORTING THE FIBONACCI STRING ROTATIONS Manolis Christodoulakis 1, Costas S. Iliopoulos 1, Yoan José Pinzón Ardila 2 1 King s College London, Department of Computer Science London WC2R
More informationOne-sided shift spaces over infinite alphabets
Reasoning Mind West Coast Operator Algebra Seminar October 26, 2013 Joint work with William Ott and Mark Tomforde Classical Construction Infinite Issues New Infinite Construction Shift Spaces Classical
More informationHW 3 Solutions. Tommy November 27, 2012
HW 3 Solutions Tommy November 27, 2012 5.1.1 (a) Online solution: S 0S1 ɛ. (b) Similar to online solution: S AY XC A aa ɛ b ɛ C cc ɛ X axb aa b Y by c b cc (c) S X A A A V AV a V V b V a b X V V X V (d)
More informationFrom p-adic numbers to p-adic words
From p-adic numbers to p-adic words Jean-Éric Pin1 1 January 2014, Lorentz Center, Leiden References I S. Eilenberg, Automata, Languages and Machines, Vol B, Acad. Press, New-York (1976). M. Lothaire,
More informationPascal Ochem 1 and Elise Vaslet Introduction REPETITION THRESHOLDS FOR SUBDIVIDED GRAPHS AND TREES
Theoretical Informatics and Applications Informatique Théorique et Applications Will be set by the publisher REPETITION THRESHOLDS FOR SUBDIVIDED GRAPHS AND TREES Pascal Ochem 1 and Elise Vaslet 2 Abstract.
More informationRECOGNIZABLE SETS OF INTEGERS
RECOGNIZABLE SETS OF INTEGERS Michel Rigo http://www.discmath.ulg.ac.be/ 1st Joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies, June 2012, Liège In the Chomsky s hierarchy,
More information#A26 INTEGERS 9 (2009), CHRISTOFFEL WORDS AND MARKOFF TRIPLES. Christophe Reutenauer. (Québec) Canada
#A26 INTEGERS 9 (2009), 327-332 CHRISTOFFEL WORDS AND MARKOFF TRIPLES Christophe Reutenauer Département de mathématiques, Université du Québec à Montréal, Montréal (Québec) Canada Reutenauer.Christophe@uqam.ca.
More information1 Alphabets and Languages
1 Alphabets and Languages Look at handout 1 (inference rules for sets) and use the rules on some examples like {a} {{a}} {a} {a, b}, {a} {{a}}, {a} {{a}}, {a} {a, b}, a {{a}}, a {a, b}, a {{a}}, a {a,
More informationClosure Properties of Regular Languages
Closure Properties of Regular Languages Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Sep 21, 2016 Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages
More informationarxiv: v1 [math.nt] 4 May 2012
ON THE EXPANSION OF SOME EXPONENTIAL PERIODS IN AN INTEGER BASE arxiv:1205.0961v1 [math.nt] 4 May 2012 by Boris Adamczewski Abstract. We derive a lower bound for the subword complexity of the base-b expansion
More informationOn Shyr-Yu Theorem 1
On Shyr-Yu Theorem 1 Pál DÖMÖSI Faculty of Informatics, Debrecen University Debrecen, Egyetem tér 1., H-4032, Hungary e-mail: domosi@inf.unideb.hu and Géza HORVÁTH Institute of Informatics, Debrecen University
More informationS-adic sequences A bridge between dynamics, arithmetic, and geometry
S-adic sequences A bridge between dynamics, arithmetic, and geometry J. M. Thuswaldner (joint work with P. Arnoux, V. Berthé, M. Minervino, and W. Steiner) Marseille, November 2017 REVIEW OF PART 1 Sturmian
More informationBordered Conjugates of Words over Large Alphabets
Bordered Conjugates of Words over Large Alphabets Tero Harju University of Turku harju@utu.fi Dirk Nowotka Universität Stuttgart nowotka@fmi.uni-stuttgart.de Submitted: Oct 23, 2008; Accepted: Nov 14,
More informationarxiv: v2 [math.nt] 25 Jan 2017
GENERALIZED PALINDROMIC CONTINUED FRACTIONS DAVID M FREEMAN arxiv:160406755v2 [mathnt] 25 Jan 2017 ABSTRACT In this paper we introduce a generalization of palindromic continued fractions as studied by
More informationOn the complexity of algebraic number I. Expansions in integer bases.
On the complexity of algebraic number I. Expansions in integer bases. Boris Adamczewsi, Yann Bugeaud To cite this version: Boris Adamczewsi, Yann Bugeaud. On the complexity of algebraic number I. Expansions
More informationFABER Formal Languages, Automata. Lecture 2. Mälardalen University
CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 2 Mälardalen University 2010 1 Content Languages, g Alphabets and Strings Strings & String Operations Languages & Language Operations
More informationarxiv:math/ v3 [math.nt] 23 May 2008
ON THE CONTINUED FRACTION EXPANSION OF A CLASS OF NUMBERS arxiv:math/0409233v3 [math.nt] 23 May 2008 DAMIEN ROY Au Professeur Wolfgang Schmidt, avec mes meilleurs vœux et toute mon admiration. 1. Introduction
More informationHow many double squares can a string contain?
How many double squares can a string contain? F. Franek, joint work with A. Deza and A. Thierry Algorithms Research Group Department of Computing and Software McMaster University, Hamilton, Ontario, Canada
More informationGolden Ratio Sinusoidal Sequences and the Multimode Pulsation of the δ Sct Star V784 Cassiopeiae
Chin. J. Astron. Astrophys. Vol. 4 (2004), No. 4, 343 348 ( http: /www.chjaa.org or http: /chjaa.bao.ac.cn ) Chinese Journal of Astronomy and Astrophysics Golden Ratio Sinusoidal Sequences and the Multimode
More informationThe Formal Inverse of the Period-Doubling Sequence
2 3 47 6 23 Journal of Integer Sequences, Vol. 2 (28), Article 8.9. The Formal Inverse of the Period-Doubling Sequence Narad Rampersad Department of Mathematics and Statistics University of Winnipeg 55
More informationAROUND THE FIBONACCI NUMERATION SYSTEM. Marcia Ruth Edson. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS
AROUND THE FIBONACCI NUMERATION SYSTEM Marcia Ruth Edson Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS May 2007 APPROVED: Luca Zamboni, Major Professor William
More informationSemigroup presentations via boundaries in Cayley graphs 1
Semigroup presentations via boundaries in Cayley graphs 1 Robert Gray University of Leeds BMC, Newcastle 2006 1 (Research conducted while I was a research student at the University of St Andrews, under
More informationSORTING SUFFIXES OF TWO-PATTERN STRINGS.
International Journal of Foundations of Computer Science c World Scientific Publishing Company SORTING SUFFIXES OF TWO-PATTERN STRINGS. FRANTISEK FRANEK and WILLIAM F. SMYTH Algorithms Research Group,
More informationA Short Proof of the Transcendence of Thue-Morse Continued Fractions
A Short Proof of the Transcendence of Thue-Morse Continued Fractions Boris ADAMCZEWSKI and Yann BUGEAUD The Thue-Morse sequence t = (t n ) n 0 on the alphabet {a, b} is defined as follows: t n = a (respectively,
More information#A2 INTEGERS 11B (2011) MULTIDIMENSIONAL EUCLIDEAN ALGORITHMS, NUMERATION AND SUBSTITUTIONS
#A2 INTEGERS 11B (2011) MULTIDIMENSIONAL EUCLIDEAN ALGORITHMS, NUMERATION AND SUBSTITUTIONS Valerie Berthé Laboratoire d Informatique Algorithmique : Fondements et Applications Université Paris Diderot,
More informationUnambiguous Morphic Images of Strings
Unambiguous Morphic Images of Strings Daniel Reidenbach, University of Kaiserslautern A joint work with: Dominik D. Freydenberger, University of Kaiserslautern Johannes C. Schneider, University of Kaiserslautern
More informationarxiv: v1 [cs.dm] 16 Jan 2018
Embedding a θ-invariant code into a complete one Jean Néraud, Carla Selmi Laboratoire d Informatique, de Traitemement de l Information et des Systèmes (LITIS), Université de Rouen Normandie, UFR Sciences
More informationOn the negative base greedy and lazy representations
On the negative base greedy and lazy representations Tomáš Hejda,, Zuzana Masáková, Edita Pelantová June 01 Abstract We consider positional numeration systems with negative real base β, where β > 1, and
More information