## The ubiquitous Prouhet-Thue-Morse sequence (1999)

Venue: | Sequences and their applications, Proceedings of SETA’98 |

Citations: | 58 - 8 self |

### BibTeX

@INPROCEEDINGS{Allouche99theubiquitous,

author = {Jean-Paul Allouche and Jeffrey Shallit},

title = {The ubiquitous Prouhet-Thue-Morse sequence},

booktitle = {Sequences and their applications, Proceedings of SETA’98},

year = {1999},

pages = {1--16},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We discuss a well-known binary sequence called the Thue-Morse sequence, or the Prouhet-Thue-Morse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The Prouhet-Thue-Morse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.

### Citations

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Citation Context ...e nal result! No expression for Q in terms of known constants is currently known, nor is it known if Q is transcendental or even irrational. This number rst occurred in a paper of Flajolet and Martin =-=[36]-=-, who studied a class of probabilistic counting algorithms for estimating the number of distinct elements in a large collection of data. Their asymptotic analysis involves the constant ' = :77351 give... |

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Citation Context ...hly speaking, a sequence is k-automatic if its n'th term is generated by a nite-state machine which takes as input the base-k expansion of n. For more about this class of sequences, see, for example, =-=[29,33, 4]-=-. For the general subject of combinatorics on words, see [43]. 4 Di erential geometry The Prouhet-Thue-Morse sequence has the nice property that it exhibits regularity without being ultimately periodi... |

82 |
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Citation Context ...alysis 7.1 Iteration of continuous functions Iterating a unimodal continuous function from [0� 1] into [0� 1] yields various behaviors going from convergent orbits to chaos� see the general reference =-=[30]-=-. M. Cosnard and the rst author proved the following theorem [8] (also see [3, 31,39]). Let f be a continuous map from [0� 1] into [0� 1], that is unimodal, i.e., increasing for x between 0 and some c... |

71 |
Recurrent geodesics on a surface of negative curvature
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(Show Context)
Citation Context ... The Prouhet-Thue-Morse sequence has the nice property that it exhibits regularity without being ultimately periodic. Morse rediscovered the sequence t in 1921 in connection with di erential geometry =-=[48]-=-. He proved the following: Theorem 5 (Morse). On a surface ofnegative curvature, having at least two di erent normal segments, there exists a set of geodesics that are recurrent without being periodic... |

60 |
Quadratische Körper im Gebiete der höheren
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Citation Context ...d by Christol [27], and more details are given in the paper of Christol, Kamae, Mendes France and Rauzy [28]. There is a theory of continued fractions for Laurent series with coe cients in a nite eld =-=[16]-=-. In particular the continued fraction expansion of a quadratic series is ultimately periodic (see [47]� see [61] for a careful study when the ground eld is not nite). The continued fraction expansion... |

47 |
Suites algébriques, automates et substitutions
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Citation Context ...e eld Fp(X) if and only if the sequence of its coe cients is p-automatic. This theorem was proved by Christol [27], and more details are given in the paper of Christol, Kamae, Mendes France and Rauzy =-=[28]-=-. There is a theory of continued fractions for Laurent series with coe cients in a nite eld [16]. In particular the continued fraction expansion of a quadratic series is ultimately periodic (see [47]�... |

47 | Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen - Mahler - 1929 |

44 | Unique developments in non-integer bases
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Citation Context ...n some respects from the usual base-k expansions where k is an integer. For example, some numbers may have multiple representations. However, Komornik and Loreti recently proved the following theorem =-=[41]-=-: Theorem 9 (Komornik, Loreti). There exists a smallest real number in the interval P (1� 2), for which there exists a unique -expansion of 1 in the form 1= 1 n=1 n ;n , with n 2f0� 1g. Furthermore, f... |

37 |
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Citation Context ... aformalpower series with coe cients in Fp, wherep is a prime number, is algebraic over the eld Fp(X) if and only if the sequence of its coe cients is p-automatic. This theorem was proved by Christol =-=[27]-=-, and more details are given in the paper of Christol, Kamae, Mendes France and Rauzy [28]. There is a theory of continued fractions for Laurent series with coe cients in a nite eld [16]. In particula... |

36 |
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Citation Context ...oves (Nb1-c3, Nb8-c6, Nc3-b1, Nc6-b8). The resulting sequence of moves represents a legal in nite game of chess, and no draw occurs under the German rule. Later, Morse rediscovered the same technique =-=[49, 51]-=-. 3.3 Morphisms of the free monoid Let A be an alphabet, i.e., a nite set of symbols. The set of words over A (i.e., blocks | or strings | of symbols of A) equipped with the operation of concatenation... |

17 |
Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set
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(Show Context)
Citation Context ...graphical order, and 0=1, 1=0) satis es wn = tn for n 1. The second author observed that this last result was previously stated in a slightly di erent formby the rst author and M. Cosnard in [8]. See =-=[9]-=- and Section 7.1 below. 6 Semigroup and group theory The Prouhet-Thue-Morse sequence t (or one of its variants) occurs in the solution of the Burnside problem for groups: Is every group with a nite nu... |

14 |
Trancendance du nombre de Thue-Morse
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- 1977
(Show Context)
Citation Context ...e-Morse sequence.s5.4 Algebraicity of formal power series in positive characteristic The Prouhet-Thue-Morse real number P n 0 tn2 ;n was proved transcendental by Mahler [44, p. 363]� also see Dekking =-=[32]-=-. What can be said about the formal power series P1 n 0 tnX ;n ? This series is transcendental over Q(X), as noted, for example, in [32]. But, considering this series modulo 2, we havethe following pr... |

13 |
Itérations de fonctions unimodales et suites engendrées par automates
- Allouche, Cosnard
- 1983
(Show Context)
Citation Context ...is lexicographical order, and 0=1, 1=0) satis es wn = tn for n 1. The second author observed that this last result was previously stated in a slightly di erent formby the rst author and M. Cosnard in =-=[8]-=-. See [9] and Section 7.1 below. 6 Semigroup and group theory The Prouhet-Thue-Morse sequence t (or one of its variants) occurs in the solution of the Burnside problem for groups: Is every group with ... |

13 |
A characterization of overlap-free morphisms
- Berstel, Séébold
- 1993
(Show Context)
Citation Context ...quence that is both overlap-free and generated by a morphism? The next theorem, due to Seebold [62], answers this question negatively. Another proof of this result was given by Berstel and Seebold in =-=[21]-=-. Theorem 4 (Seebold). If an overlap-free binarysequence isa xedpoint of a non-trivial morphism, then it is either equal to t, theProuhet-Thue-Morse sequence, or its complement t =(tn)n 0 = 1001011001... |

11 |
Approximations diophantiennes dans un corps local
- Mathan
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(Show Context)
Citation Context ... [28]. There is a theory of continued fractions for Laurent series with coe cients in a nite eld [16]. In particular the continued fraction expansion of a quadratic series is ultimately periodic (see =-=[47]-=-� see [61] for a careful study when the ground eld is not nite). The continued fraction expansion of P n 0 tnX ;n is ultimately periodic with a pleasantly short period. It is given by X n 0 tnX ;n =[0... |

9 |
Generalised Morse sequences
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- 1968
(Show Context)
Citation Context ... 1 be integers. The generalized Prouhet-ThueMorse sequence tk�m is overlap-free if and only if m k. Several other generalizations of the Prouhet-Thue-Morse sequence have been studied, see for example =-=[40,45, 46, 53, 69]-=-. 10 Conclusion The Prouhet-Thue-Morse sequence occurs in various elds, so that many apparently unrelated de nitions of this sequence are equivalent. For example, Proposition 1, Theorem 2, Proposition... |

8 |
The Prouhet-Tarry-Escott Problem Revisited
- Borwein, Ingalls
- 1994
(Show Context)
Citation Context ...gic cubes, Prouhet sequences, and the Prouhet-Tarry-Escott problem was given by Adler and Li [2]. For the state of art about the Prouhet-Tarry-Escott problem the reader can look at one of the surveys =-=[24, 60]-=-. j2Js5.2 Curious in nite products Woods asked [68], what is the limit of the sequence 1=2� 1 2 =3 4 � 1 5 2 6 = 3 7 4 8 � ::: ? Robbins [58] proved that this limit is p 2 . More precisely, wehavethe ... |

7 |
A relative of the Thue–Morse sequence, Discrete Math
- Allouche, Arnold, et al.
- 1995
(Show Context)
Citation Context ...long to A. Hence the rst few elements of A are 0 1 3 4 5 7 9 11 12 13 15 16 17 19 20 21 23 It is not hard to see that A and 2A = f2x : x 2 Ag form a partition of the set of non-negative integers� see =-=[26, 5, 64]-=-. An unexpected connection with the Thue-Morse sequence, proved in [5], is given below. Theorem 8 (Allouche, Arnold, Berstel, Brlek, Jockusch, Plou e, Sagan). Let (an)n 0 = 0 1 3 4 5 7 9 11 12 13 15 1... |

7 |
Mengentheoretische Betrachtungen über das Schachspiel
- Euwe
- 1929
(Show Context)
Citation Context ...r, the Dutch chess grandmaster Machgielis (Max) Euwe (1901{1981, world champion 1935{1937) independently discovered the Prouhet-Thue-Morse sequence t and applied it to an interesting problem in chess =-=[35]-=-. The so-called German rule states that a draw occurs if the same sequence of moves occurs three times in succession. Euwe proved, using the cube-free property oft, that under such a rule in nite game... |

6 |
Finite Automata and Morphisms in Assisted Musical
- Allouche, Johnson
- 1995
(Show Context)
Citation Context ...ue-Morsessequence is a simple example, are also useful because they are both \simple to generate" and \non-trivial": in physicsasmentioned above, but also in other elds, such asmusic (see for example =-=[11]-=-). Searching for the many occurrences of the Prouhet-Thue-Morse sequence in the literature can be used as a pretext to take adelightful stroll through many fascinating areas of mathematics. Acknowledg... |

6 |
Spectrum and extended states in a harmonic chain with controlled disorder: effects of the Thue-Morse symmetry
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- 1989
(Show Context)
Citation Context ...l quasicrystal is the Fibonacci sequence, i.e., the xed point of the Fibonacci morphism 0 ! 01, 1 ! 0. Hence a large number of papers in physics study the Prouhet-Thue-Morse sequence. We only mention =-=[17,18]-=-, and the papers given in the bibliography of[12]. 9 Generalizations The alternative de nitions of the Prouhet-Thue-Morse sequence given in Section 2 each suggest possible approaches to generalize the... |

6 |
The Tarry-Escott problem
- Lehmer
- 1947
(Show Context)
Citation Context ...a partition of [0�q N ; 1] into q sets I1�I2�:::�Iq such that the q sums P i2Ij ik (with j = 1� 2�:::�q and k =0� 1�:::�N ; 1) do not depend on j. Hegave the following solution (for a proof see e.g., =-=[42]-=-): for each q 2 de ne the sequence Tq = (Tq(n))n 0 by Tq(n) =sq(n) modq.ThenletIj = f0 i q N ;1 : Tq(i) =jg. For an occurrence of related sequences, see [59]. A very nice relationship between magic cu... |

5 |
Magic Cubes and Prouhet Sequences
- Adler, Li
- 1977
(Show Context)
Citation Context ...imately periodic. The sequence that Morse gives is exactly t, where 0's have been replaced by 1's and 1's by 2's. 5 Number theory 5.1 The Prouhet-Tarry-Escott problem As already noted by Adler and Li =-=[2]-=-, the sequence t appears implicitly in an 1851 paper of Prouhet [55]. Prouhet was interested in a problem that was also studied more than fty years later by Tarry and Escott, and which isnowknown as t... |

5 |
Automata and automatic sequences, in Beyond Quasicrystals
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(Show Context)
Citation Context ...xed point of the Fibonacci morphism 0 ! 01, 1 ! 0. Hence a large number of papers in physics study the Prouhet-Thue-Morse sequence. We only mention [17,18], and the papers given in the bibliography of=-=[12]-=-. 9 Generalizations The alternative de nitions of the Prouhet-Thue-Morse sequence given in Section 2 each suggest possible approaches to generalize the sequence. For example, Proposition 1 suggests st... |

5 |
Identités de Thue–Morse dans les groupes
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(Show Context)
Citation Context ... and subject to the relation w 2 = z for all w 2 S. Related questions were discussed by Brzozowski, Culik, and Gabrielian [25]. There is another occurrence of the Thue-Morse sequence in group theory, =-=[22]-=-, as follows: Theorem 10 (Bo a, Point). De ne the Thue-Morse group identities In by: I0(x� y) if x = y� In+1(x� y) is In(xy� yx). Then a nite group satis es a ThueMorse identity if and only if it is a... |

4 |
Some recent results on squarefree words
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- 1984
(Show Context)
Citation Context ...rs appeared in an obscure Norwegian journal, and for a long time were not widely known or appreciated. His original results were rediscovered by several di erent authors, including Marston Morse� see =-=[38, 19]-=-.sAlthough there are uncountably many overlap-free sequences on two symbols, the Prouhet-Thue-Morse sequence is, roughly speaking, the \canonical" example. For example, if in addition to being overlap... |

3 |
Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tidskrift 15
- Hedlund
- 1967
(Show Context)
Citation Context ...rs appeared in an obscure Norwegian journal, and for a long time were not widely known or appreciated. His original results were rediscovered by several di erent authors, including Marston Morse� see =-=[38, 19]-=-.sAlthough there are uncountably many overlap-free sequences on two symbols, the Prouhet-Thue-Morse sequence is, roughly speaking, the \canonical" example. For example, if in addition to being overlap... |

3 |
Periodic orbits and kneading invariants
- Jonker
- 1979
(Show Context)
Citation Context ...nction from [0� 1] into [0� 1] yields various behaviors going from convergent orbits to chaos� see the general reference [30]. M. Cosnard and the rst author proved the following theorem [8] (also see =-=[3, 31,39]-=-). Let f be a continuous map from [0� 1] into [0� 1], that is unimodal, i.e., increasing for x between 0 and some c, then decreasing for x between c and 1. Suppose that f(1) = 0. With the orbit (f (n)... |

2 |
Classi cation of non-counting events
- Brzozowski, Culik, et al.
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(Show Context)
Citation Context ...f0� 1� 2g [ fzg, where z is the zero element (i.e., wz = zw = z for all w 2 S) and subject to the relation w 2 = z for all w 2 S. Related questions were discussed by Brzozowski, Culik, and Gabrielian =-=[25]-=-. There is another occurrence of the Thue-Morse sequence in group theory, [22], as follows: Theorem 10 (Bo a, Point). De ne the Thue-Morse group identities In by: I0(x� y) if x = y� In+1(x� y) is In(x... |

2 |
Jr., Representations for a special sequence, Fibonacci Quart
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(Show Context)
Citation Context ...long to A. Hence the rst few elements of A are 0 1 3 4 5 7 9 11 12 13 15 16 17 19 20 21 23 It is not hard to see that A and 2A = f2x : x 2 Ag form a partition of the set of non-negative integers� see =-=[26, 5, 64]-=-. An unexpected connection with the Thue-Morse sequence, proved in [5], is given below. Theorem 8 (Allouche, Arnold, Berstel, Brlek, Jockusch, Plou e, Sagan). Let (an)n 0 = 0 1 3 4 5 7 9 11 12 13 15 1... |

2 |
codes and the Thue-Morse-Hedlund sequence
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- 1992
(Show Context)
Citation Context ...ce, is a =10111010 This sequence is called the period-doubling sequence. It is not hard to show that a is a xed point of the morphism de ned by (1) = 10, (0) = 11. For connections with Gray code, see =-=[37]-=-. The link between Theorem 11 and Proposition 7 above is easy [9]. Note also that Theorem 11 can be reformulated in number-theoretical terms: Corollary 2. Let ; 0 the set of real numbers de ned by ; 0... |

2 |
Generalized Morse sequences on n symbols
- Martin
- 1976
(Show Context)
Citation Context ... 1 be integers. The generalized Prouhet-ThueMorse sequence tk�m is overlap-free if and only if m k. Several other generalizations of the Prouhet-Thue-Morse sequence have been studied, see for example =-=[40,45, 46, 53, 69]-=-. 10 Conclusion The Prouhet-Thue-Morse sequence occurs in various elds, so that many apparently unrelated de nitions of this sequence are equivalent. For example, Proposition 1, Theorem 2, Proposition... |

2 |
The structure of generalized Morse minimal sets on n symbols
- Martin
- 1977
(Show Context)
Citation Context ... 1 be integers. The generalized Prouhet-ThueMorse sequence tk�m is overlap-free if and only if m k. Several other generalizations of the Prouhet-Thue-Morse sequence have been studied, see for example =-=[40,45, 46, 53, 69]-=-. 10 Conclusion The Prouhet-Thue-Morse sequence occurs in various elds, so that many apparently unrelated de nitions of this sequence are equivalent. For example, Proposition 1, Theorem 2, Proposition... |

2 |
On in nite periodic groups
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(Show Context)
Citation Context ...atisfying the identity x n =1 nite? The answer is yes (and well-known) if n = 2, since the group must be abelian in this case. But the answer is no for large odd n, since, as Novikov and Adian showed =-=[52]-=-, an in - nite group ; (m� n) onm generators and satisfying x n = 1 for all x 2 ; (m� n) exists for all m>1, and for all odd n with n 4381. Adian's book [1] gives more details about the result and its... |

1 |
The Burnside problem and identities in groups," Ergebnisse der Mathematik und ihrer Grenzgebiete 95
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(Show Context)
Citation Context ... odd n, since, as Novikov and Adian showed [52], an in - nite group ; (m� n) onm generators and satisfying x n = 1 for all x 2 ; (m� n) exists for all m>1, and for all odd n with n 4381. Adian's book =-=[1]-=- gives more details about the result and its history, and improves the constant 4381 to 665. One of the steps in the proof consists of nding a cube-free binary sequence (see [1, p. 5]). Actually the c... |

1 |
Th'eorie des Nombres et Automates," Th`ese d' ' Etat, Universit'e Bordeaux I
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- 1983
(Show Context)
Citation Context ...nction from [0� 1] into [0� 1] yields various behaviors going from convergent orbits to chaos� see the general reference [30]. M. Cosnard and the rst author proved the following theorem [8] (also see =-=[3, 31,39]-=-). Let f be a continuous map from [0� 1] into [0� 1], that is unimodal, i.e., increasing for x between 0 and some c, then decreasing for x between c and 1. Suppose that f(1) = 0. With the orbit (f (n)... |

1 |
Sealed Rechargeable Batteries, Application Manual”, Internal Data: Gates Energy Products (currently owned by Energizer Batteries a division of Eveready Batteries) Gainesville FL, copyright
- unknown authors
- 1989
(Show Context)
Citation Context ...hly speaking, a sequence is k-automatic if its n'th term is generated by a nite-state machine which takes as input the base-k expansion of n. For more about this class of sequences, see, for example, =-=[29,33, 4]-=-. For the general subject of combinatorics on words, see [43]. 4 Di erential geometry The Prouhet-Thue-Morse sequence has the nice property that it exhibits regularity without being ultimately periodi... |

1 |
Sridar Mahadevan and Jonathan Connell "Automatic Programming of Behavior-based Robots using Reinforcement Learning
- unknown authors
- 1991
(Show Context)
Citation Context ...nent are iteratively chosen so that the product thus far approximates as closely as possible at each step. The following conjecture of the second author [63] was proved by the rst author and H. Cohen =-=[6]-=-. Theorem 7 (Allouche, Cohen). De ne a sequence ofsigns( n)n 0 by 0 = 1, and, if 0� 1�:::� n are known, de ne n+1 by: n+1 = 8 < : +1 if ; 1 2 ;1 if ; 1 2 0 ; 3 4 0 ; 3 4 1 5 6 1 1 2n+1 2n+2 1 2n+1 2n+... |

1 |
Sridar Mahadevan "To Discount or Not to Discount in Reinforcement Learning: A Case Study Comparing R- Learning and QLearning
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(Show Context)
Citation Context ...n+1 2n+2 n n > p 2 2 � < p 2 2 : Then the sequence ( n)n 0 is equal to the Prouhet-Thue-Morse sequence onthe alphabet f;1� +1g, i.e., n =(;1) tn for all n 0. For generalizations of these results, see =-=[63, 6, 13, 7]-=-. 5.3 Partitions ofthesetofintegers Let A be the (lexicographically) smallest set of integers such that0and1belong to A, and for each n 1 that belongs to A, thenumber 2n does not belong to A. Hence th... |

1 |
Extremal in nite overlap-free binary words
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- 1998
(Show Context)
Citation Context ...r the following theorem of Berstel [20]. Theorem 2. The lexicographically largest overlap-free binary sequence beginning with 0 is t, theProuhet-Thue-Morse sequence. Recently, together with J. Currie =-=[10]-=-, we generalized this theorem, proving in particular the following result: Theorem 3. The lexicographically largest overlap-free binary sequence is the sequence 110110t. 3.2 The problem of in nite pla... |

1 |
nite products associated with counting blocks in binary strings
- Allouche, Shallit, et al.
- 1989
(Show Context)
Citation Context ...n+1 2n+2 n n > p 2 2 � < p 2 2 : Then the sequence ( n)n 0 is equal to the Prouhet-Thue-Morse sequence onthe alphabet f;1� +1g, i.e., n =(;1) tn for all n 0. For generalizations of these results, see =-=[63, 6, 13, 7]-=-. 5.3 Partitions ofthesetofintegers Let A be the (lexicographically) smallest set of integers such that0and1belong to A, and for each n 1 that belongs to A, thenumber 2n does not belong to A. Hence th... |

1 |
Sums of digits and overlap-free words, in preparation
- Allouche, Shallit
(Show Context)
Citation Context ... that the sequence t2�m has been studied by J.Tromp and the second author in [67], and that the sequence tq�q is the sequence Tq of Section 5.1. Very recently the authors proved the following theorem =-=[14]-=-, which generalizes the work of Thue: Theorem 13. Let k 2, m 1 be integers. The generalized Prouhet-ThueMorse sequence tk�m is overlap-free if and only if m k. Several other generalizations of the Pro... |

1 |
Arshon, Proof of the existence of asymmetric in nite sequences
- E
- 1937
(Show Context)
Citation Context ... cube-free binary sequence (see [1, p. 5]). Actually the cube-free binary sequence given there is not the Prouhet-Thue-Morse sequence, since it is not overlap-free. The author uses a result of Arshon =-=[15]-=- in order to construct a cube-free binary sequence, but in that paper Arshon actually gave a cube-free binary sequence that is equal to the Prouhet-Thue-Morse sequence on the alphabet f1� 2g (see [15,... |

1 |
Ultra high frequency follow on communications satellite system fact sheet, http://www.fafb.af.mil/uhf/fact sheet.html
- unknown authors
- 1986
(Show Context)
Citation Context ...l quasicrystal is the Fibonacci sequence, i.e., the xed point of the Fibonacci morphism 0 ! 01, 1 ! 0. Hence a large number of papers in physics study the Prouhet-Thue-Morse sequence. We only mention =-=[17,18]-=-, and the papers given in the bibliography of[12]. 9 Generalizations The alternative de nitions of the Prouhet-Thue-Morse sequence given in Section 2 each suggest possible approaches to generalize the... |

1 |
Thue's Papers on Repetitions in Words: a Translation," Publications du Laboratoire de Combinatoire et d'Informatique Mathematique 20, Universite duQuebec a Montreal
- Berstel, \Axel
- 1995
(Show Context)
Citation Context ...equence t. Let v =(vn)n 1. Hence v =21020121012 Thue proved, as a corollary of Theorem 1 above, the following: Corollary 1. The sequence v =(vn)n 1 is square-free. This work of Thue (see also Berstel =-=[20]-=- and references therein) was the starting point of an importantbranchofcombinatorics, now called combinatorics on words.Itisworth noting that Thue explained he had no particular application in mind, b... |

1 |
Etude de la classi cation topologique des fonctions unimodales, Ann. Inst. Fourier 35
- Cosnard
- 1985
(Show Context)
Citation Context ...nction from [0� 1] into [0� 1] yields various behaviors going from convergent orbits to chaos� see the general reference [30]. M. Cosnard and the rst author proved the following theorem [8] (also see =-=[3, 31,39]-=-). Let f be a continuous map from [0� 1] into [0� 1], that is unimodal, i.e., increasing for x between 0 and some c, then decreasing for x between c and 1. Suppose that f(1) = 0. With the orbit (f (n)... |

1 |
Le maximum de la fonction de
- Dubuc, Elqortobi
- 1990
(Show Context)
Citation Context ... for each word w to decide if it is in L.) Then, as the second author has observed, p = Y (1 ; 1 22i )= X (;1) tj 2j i 0 7.2 The Knopp function j 0 =2; 2 : The Knopp function (see the introduction of =-=[34]-=-) is de ned, for a 2 (0� 1) and b 2 N nf0g, by Ka�b(x) = 1X n=0 a n kb n xkswhere kyk is the distance from y to the nearest integer. In 1990, S. Dubuc and A. Elqortobi [34] came across the Prouhet-Thu... |

1 |
Combinatorics on words," 2nd ed
- Lothaire
- 1997
(Show Context)
Citation Context ...ed by a nite-state machine which takes as input the base-k expansion of n. For more about this class of sequences, see, for example, [29,33, 4]. For the general subject of combinatorics on words, see =-=[43]-=-. 4 Di erential geometry The Prouhet-Thue-Morse sequence has the nice property that it exhibits regularity without being ultimately periodic. Morse rediscovered the sequence t in 1921 in connection wi... |

1 |
Abstract 360: a solution of the problem of in nite play inchess
- Morse
- 1938
(Show Context)
Citation Context ...oves (Nb1-c3, Nb8-c6, Nc3-b1, Nc6-b8). The resulting sequence of moves represents a legal in nite game of chess, and no draw occurs under the German rule. Later, Morse rediscovered the same technique =-=[49, 51]-=-. 3.3 Morphisms of the free monoid Let A be an alphabet, i.e., a nite set of symbols. The set of words over A (i.e., blocks | or strings | of symbols of A) equipped with the operation of concatenation... |

1 |
All generalized Morse-sequences are loosely
- Nurnberg
- 1983
(Show Context)
Citation Context |