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shorttalksfebruary2021 [2022/10/30 23:04] – old revision restored (2021/01/26 19:13) 139.124.146.3shorttalksfebruary2021 [2022/10/30 23:04] (current) – old revision restored (2020/12/09 12:11) 139.124.146.3
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-======  Day of Short Talks on Combinatorics on Words, February 22, 2021 ======+======  Day of Short Talks on Combinatorics on Words ======
  
 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.
  
-The event will consist of short talks in 25-minute slots, including your new results, discussions, and open questions.+The event will consist of short talks in 25-minute slots, including your new results, discussions, and open questions. If you are interested in giving such a talk, please send your title and abstract to Anna Frid (anna.e.frid@gmail.com) as soon as possible.
  
 The abstracts (with eventual references to arxiv.org), slides and some videos of talks will be published at this page. The abstracts (with eventual references to arxiv.org), slides and some videos of talks will be published at this page.
  
-The deadline for submissions was January 24, 2021+Deadline for submissions: January 24, 2021
  
-==== Speakers ====+==== Interested speakers ====
  
 ** Mélodie Andrieu** //A Rauzy fractal unbounded in all directions of the plane// ** Mélodie Andrieu** //A Rauzy fractal unbounded in all directions of the plane//
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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 of the composition of a $2$-tape automaton and a weighted automaton// +** Anna Frid** //The semigroup of trimmed morphisms//
- +
-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.  +
- +
-This is a joint work with Émilie Charlier and Manon Stipulanti based on two recent papers: [[https://arxiv.org/abs/2012.04969|arXiv:2012.04969]] and [[https://arxiv.org/abs/2006.11126|arXiv:2006.11126]]. +
- +
- +
-**Lubka Dvořáková** //On balanced sequences with the minimal asymptotic critical exponent// +
- +
-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 $E^*(d)$, $d\leq 15$. +
- +
-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 $\beta>1$. The graph has the following property: If $\Gamma$ does not contain an oriented cycle, then any balanced sequence arising by $y$ and $y‘$ has the asymptotic critical exponent at least $\beta$. +
-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  $p$.  The program we have implemented finds among them the balanced sequence with the minimal asymptotic critical exponent.   +
- +
-This is a joint work with Edita Pelantová.+
  
 ** Štěpán Holub** //Formalization of Combinatorics on Words in Isabelle/HOL// ** Štěpán Holub** //Formalization of Combinatorics on Words in Isabelle/HOL//
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 To achieve these results, we introduce a novel data-structure, called Simon-Tree, which allows us to construct a natural representation of the equivalence classes induced by $\sim_k$ on the set of suffixes of a word, for all $k\geq 1$. We show that such a tree can be constructed for an input word in linear time. Then, when working with two words $s$ and $t$, we compute their respective Simon-Trees and efficiently build a correspondence between the nodes of these trees. This correspondence, which can also be constructed in linear time $O(|s|+|t|)$, allows us to retrieve the largest $k$ for which $s\sim_k t$. To achieve these results, we introduce a novel data-structure, called Simon-Tree, which allows us to construct a natural representation of the equivalence classes induced by $\sim_k$ on the set of suffixes of a word, for all $k\geq 1$. We show that such a tree can be constructed for an input word in linear time. Then, when working with two words $s$ and $t$, we compute their respective Simon-Trees and efficiently build a correspondence between the nodes of these trees. This correspondence, which can also be constructed in linear time $O(|s|+|t|)$, allows us to retrieve the largest $k$ for which $s\sim_k t$.
  
 +** Jeffrey  Shallit** //Robbins and Ardila meet Berstel//
  
- +In this short talk I show how to obtain a result of Robbins and (later) 
-** Christophe Reutenauer ** //An arithmetical characterization of Christoffel words// +Ardila on partitions of integers in Fibonacci numbers, and much more, 
- +using a 2001 transducer created by Jean Berstel.   The "fun" part is 
-Christoffel word have many combinatorial characterizations. +that once you have Berstel's transducer, the rest follows by purely 
-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 +computational means.
-$p(a_1,\ldots,a_{2n}) - p(a_2,\ldots,a_{2n-1}) < 3 p(a_2,\ldots,a_{2n})$. +
- +
-Here $p(x_1,\ldots,x_k)$ denotes the continuant polynomials, defined recursively by $p()=1$, $p(x_1)=x_1$, and +
-$p(x_1,\ldots,x_k)=x_1p(x_2,\ldots,x_k)+p(x_3,\ldots,x_k)$. +
- +
-These polynomials are well-known in the theory of continued fractions. One has for example $p(x,y)=xy+1$, $p(x,y,z)=xyz+x+z$, $p(x,y,z,t)=xyzt+xy+xt+zt+1$. +
- +
-The characterization above is motivated by the theory of Markoff. As intermediate result, I prove another characterization: +
-a primitive word $w$ on the alphabet $\{a<b\}$ is a Christoffel word if and only for any conjugate $aub$ (resp. $bua$) of $w$, one has $u \geq_{lex} \tilde u$ (resp. $u \leq_{lex} \tilde u$). This condition must be called the finitary Markoff condition, due to its similarity to Markoff’s condition on bi-infinite words, which appears in his articles of 1880. +
- +