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This mini-event, organized by [[start|One World Combinatorics on Words Seminar]], is designed to take the place of a small and very informal conference in our field, since many of us miss direct communication of this kind. | This mini-event, organized by [[start|One World Combinatorics on Words Seminar]], is designed to take the place of a small and very informal conference in our field, since many of us miss direct communication of this kind. | ||
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The study of the pairs of abelianized factors of Arnoux-Rauzy words disproves this belief. | The study of the pairs of abelianized factors of Arnoux-Rauzy words disproves this belief. | ||
- | **Célia Cisternino** //Two applications | + | ** Anna Frid** //The semigroup |
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- | Starting with a $2$-tape automaton ${\mathcal A}$ and a weighted automaton ${\mathcal B}$, we can obtain a new $K$-automaton using an ad hoc operation that can be considered as the composition ${\mathcal B}\circ {\mathcal A}$. First, I will define and illustrate this operation. Next, I will present an application of this operation on automata in terms of synchronized relation and formal series. In particular, using the characterization of synchronized sequences in terms of synchronized relations and that of regular sequences in terms of formal series, I will present two consequences of this composition of automata in the theory of regular sequences. | + | |
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- | This is a joint work with Émilie Charlier and Manon Stipulanti based on two recent papers: [[https:// | + | |
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- | **Francesco Dolce** //On morphisms | + | |
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- | A finite word is said to be rich if it has maximal number of palindromic factors (this is known to be the length of the word plus one). This definition can be naturally extended to infinite words.Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while strict episturmian words and words coding symmetric interval exchange transformations give us other examples on larger alphabets. | + | |
- | In this talk we study homomorphisms of the free monoid which allow to construct new rich words from already known rich words. In particular, we focus on two types of morphisms: Arnoux-Rauzy morphisms and morphisms from Class $P_{ret}$. These morphisms contain Sturmian morphisms as a subclass. We show that Arnoux-Rauzy morphisms preserve the set of all rich words. | + | |
- | We also characterize $P_{ret}$ morphisms which preserve richness on binary alphabet. | + | |
- | This is a joint work with Edita Pelantová. | + | |
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- | **Lubka Dvořáková** //On balanced sequences with the minimal asymptotic critical exponent// | + | |
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- | We study balanced sequences over a $d$-letter alphabet, $d\geq 3$. Any balanced sequence is defined by a Sturmian sequence and two periodic constant gap sequences $y$ and $y‘$. The minimal value $E(d)$ of the critical exponent of balanced sequences over a $d$-letter alphabet has been recently determined by Baranwal, Rampersad, Shallit, and Vandomme for $d\leq 8$. We focus on the minimal asymptotic critical exponent $E^*(d)$. We show that $E(d)$ and $E^*(d)$ coincide if $d\leq 5$ and differ if $6\leq d\leq 8$. We determine $E^*(6) $ and give an upper bound on $E^*(d)$, $d\leq 15$. | + | |
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- | Our method is based on the construction of a finite directed labeled graph $\Gamma$ which depends on the period lengths of $y$ and $y'$ and on a parameter $\beta> | + | |
- | On the other hand, if $\Gamma$ contains an oriented cycle labeled by $p$, then we inspect all balanced sequences associated to Sturmian sequences whose slopes have the continued fraction expansion with the period | + | |
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- | This is a joint work with Edita Pelantová. | + | |
** Štěpán Holub** // | ** Štěpán Holub** // | ||
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To achieve these results, we introduce a novel data-structure, | To achieve these results, we introduce a novel data-structure, | ||
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- | **Jane D. Palacio** //Coverable bi-infinite substitution shifts// | ||
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- | A finite or infinite word w is said to be coverable if it can be formed by | ||
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- | overlapping or adjacent occurrences of some finite subword u of w. Coverabil- | ||
- | ity of finite words was introduced by A. Apostolico and A. Ehrenfeucht in the | ||
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- | context of text algorithms in 1993. In 2004, S. Marcus extended the concept | ||
- | of coverability to (one-sided) infinite sequences. In this presentation, | ||
- | define the notion of coverable bi-infinite shifts. | ||
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- | In general, coverability of subshifts is not topologically invariant. Never- | ||
- | theless, we show that a special type of coverability of substitutive shifts is | ||
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- | invariant under topological conjugacy. To better understand coverability of | ||
- | bi-infinite shifts, we aim to identify properties of substitutions that ensure | ||
- | coverability or non-coverability of its associated shifts. | ||
- | This is joint work with Manuel Joseph Loquias and Eden Delight Miro. | ||
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- | ** Christophe Reutenauer ** //An arithmetical characterization of Christoffel words// | ||
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- | Christoffel word have many combinatorial characterizations. | ||
- | I propose a new one, of arithmetical nature: a primitive word on the alphabet $11$, $22$ is a Christoffel word if and only if for any of its conjugates, $u=a_1\cdots a_{2n}$ say, one has | ||
- | $p(a_1, | ||
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- | Here $p(x_1, | ||
- | $p(x_1, | ||
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- | These polynomials are well-known in the theory of continued fractions. One has for example $p(x, | ||
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- | The characterization above is motivated by the theory of Markoff. As intermediate result, I prove another characterization: | ||
- | a primitive word $w$ on the alphabet $\{a< | ||
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** Jeffrey | ** Jeffrey |