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Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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Cited by 8 (5 self)
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
Hairpin structures defined by DNA trajectories
 C. MAO AND T. YOKOMORI (EDS), PROCEEDINGS OF DNA COMPUTING 12, LNCS 4287
, 2006
"... We examine scattered hairpins, which are structures formed when a single strand folds into a partially hybridized stem and a loop. To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin. ..."
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Cited by 3 (0 self)
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We examine scattered hairpins, which are structures formed when a single strand folds into a partially hybridized stem and a loop. To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin. DNA trajectories have previously been used to describe bonding between separate strands. We are interested in the mathematical properties of scattered hairpins described by DNA trajectories. We examine the complexity of set of hairpinfree words described by a set of DNA trajectories. In particular, we consider the closure properties of language classes under sets of DNA trajectories of differing complexity. We address decidability of recognition problems for hairpin structures.
Acta Informatica manuscript No. (will be inserted by the editor)
"... Abstract Let π(w) denote the minimum period of the word w. Let w be a primitive word with period π(w) < w, and z a prefix of w. It is shown that if π(wz) = π(w), then z  < π(w)−gcd(w,z). Detailed improvements of this result are also proven. As a corollary we give a short proof of the fact th ..."
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Abstract Let π(w) denote the minimum period of the word w. Let w be a primitive word with period π(w) < w, and z a prefix of w. It is shown that if π(wz) = π(w), then z  < π(w)−gcd(w,z). Detailed improvements of this result are also proven. As a corollary we give a short proof of the fact that if u,v,w are primitive words such that u 2 is a prefix of v 2, and v 2 is a prefix of w 2, then w > 2u. Finally, we show that each primitive word w has a conjugate w ′ = vu, where w = uv, such that π(w ′ ) = w ′  and u  < π(w). 1
On the maximal number of highly periodic runs in a string ⋆
, 907
"... Abstract. A run is a maximal occurrence of a repetition v with a period p such that 2p ≤ v. The maximal number of runs in a string of length n was studied by several authors and it is known to be between 0.944n and 1.029n. We investigate highly periodic runs, in which the shortest period p satisfi ..."
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Abstract. A run is a maximal occurrence of a repetition v with a period p such that 2p ≤ v. The maximal number of runs in a string of length n was studied by several authors and it is known to be between 0.944n and 1.029n. We investigate highly periodic runs, in which the shortest period p satisfies 3p ≤ v. We show the upper bound 0.5n on the maximal number of such runs in a string of length n and construct a sequence of words for which we obtain the lower bound 0.406n. 1
From Biideals to Periodicity
"... The necessary and sufficient conditions are extracted for periodicity of biideals. By the way two proper subclasses of uniformly recurrent words are introduced. 1 ..."
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The necessary and sufficient conditions are extracted for periodicity of biideals. By the way two proper subclasses of uniformly recurrent words are introduced. 1
On the Maximal Sum of Exponents of Runs in a String
, 1003
"... Abstract. A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ v. The exponent of a run is defined as v/p and is ≥ 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1n is ..."
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Abstract. A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ v. The exponent of a run is defined as v/p and is ≥ 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1n is better than the best previously known proven bound of 5.6n by Crochemore & Ilie (2008). The lower bound of 2.035n, obtained using a family of binary words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the maximal sum of exponents of runs in a string of length n is smaller than 2n. 1