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17
The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse 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 ProuhetThueMorse sequence appears to be som ..."
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Cited by 59 (9 self)
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We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse 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 ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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Cited by 22 (3 self)
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
The KomornikLoreti constant is transcendental
, 2000
"... ively 1) if the sum of the binary digits of n is even (respectively odd). This number q can be then obtained as the unique positive solution of 1 = P 1 n=1 ffi n q \Gamman . It is equal to 1:787231650::: In the electronic abstract of [4], the authors ask whether the number q = 1:787231650::: in ..."
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Cited by 13 (5 self)
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ively 1) if the sum of the binary digits of n is even (respectively odd). This number q can be then obtained as the unique positive solution of 1 = P 1 n=1 ffi n q \Gamman . It is equal to 1:787231650::: In the electronic abstract of [4], the authors ask whether the number q = 1:787231650::: in Theorem 1 above is irrational. The purpose of this note is to prove, as a simple consequence of a result of Mahler, that q is transcendental. Theorem 2 The number q = 1:787231650::: defined as the smallest number in (1; 2) for which there exists a unique expansion of 1 as 1 = P 1 n=1 ffi n q<F1
NONREPETITIVE COLORINGS OF GRAPHS
"... A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long no ..."
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Cited by 9 (0 self)
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A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thue’s sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and ∆(G). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ c∆(G) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n − 3, and π(T) ≤ 4(∆(T) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.
ThueLike Sequences and Rainbow Arithmetic Progressions
 ELECTRONIC J. COMBINATORICS
, 2002
"... A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive seq ..."
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Cited by 6 (0 self)
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A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set fa; b; cg. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this
Doppler resilient Golay complementary pairs for radar,” presented at the
 IEEE Statist. Signal Process. Workshop (SSP
, 2007
"... We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of th ..."
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Cited by 6 (1 self)
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We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of the Taylor expansion of the composite ambiguity function. The ProuhetThueMorse sequence plays a key role in the construction of Doppler resilient sequences of Golay pairs. We extend this construction to multiple dimensions. In particular, we consider radar polarimetry, where the dimensions are realized by two orthogonal polarizations. We determine a sequence of twobytwo Alamouti matrices, where the entries involve Golay pairs and for which the matrixvalued composite ambiguity function vanishes at small Doppler shifts. 1.
The Complexity of Agent Design Problems: Determinism and History Dependence
, 2001
"... The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the ..."
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Cited by 3 (3 self)
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The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the agent design problem for tasks that are of the form "achieve this state of affairs" or "maintain this state of affairs". We consider three general formulations of these problems (in both nondeterministic and deterministic environments) that differ in the nature of what is viewed as an "acceptable" solution: in the least restrictive formulation, no limit is placed on the number of actions an agent is allowed to perform in attempting to meet the requirements of its specified task. We show that the resulting decision problems are intractable, in the sense that these are nonrecursive (but recursively enumerable) for achievement tasks, and nonrecursively enumerable for maintenance tasks. In the second formulation, the decision problem addresses the existence of agents that have satisfied their specified task within some given number of actions. Even in this more restrictive setting the resulting decision problems are either PSPACEcomplete or NPcomplete. Our final formulation requires the environment to be history independent and bounded. In these cases polynomial time algorithms exist: for deterministic environments the decision problems are NLcomplete; in nondeterministic environments, Pcomplete.
Counting ordered patterns in words generated by morphisms
, 2007
"... We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, ..."
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Cited by 2 (1 self)
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We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non)inversions, squares and prepetitions. We give recurrence formulas in the general case, then deducing exact formulas for particular families of morphisms. Many (classical or new) examples are given illustrating the techniques and showing their interest.