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 ==== Upcoming talks ====  ==== Upcoming talks ==== 
  
-**June 9 2026:  [[https://kmlinux.fjfi.cvut.cz/~balkolub/| Ľubomíra Dvořáková]]** //Attractors of sequences coding beta-integers// 
  
-String attractor is an intensively studied object in Combinatorics on Words. 
-In our talk, we will recall known results and also some previously used techniques. 
-We will then describe minimal string attractors of prefixes of simple Parry sequences. 
-These sequences form a coding of distances between consecutive beta-integers in numeration systems with a real base beta. 
-Simple Parry sequences have been recently studied from this point of view and (not necessarily minimal) attractors of their prefixes have been described and a conjecture that attractors of alphabet size should be sufficient was stated. We prove this conjecture. Moreover, we provide attractors of prefixes of some particular form of binary non-simple Parry sequences. 
  
 +**July 7 2026: Delaram Moradi**
 +
 +==== Past talks 2026 ====
 **June 23 2026: [[https://arxiv.org/search/math?searchtype=author&query=Cloitre,+B|Benoit Cloitre]]** //The Thue-Morse Transform// **June 23 2026: [[https://arxiv.org/search/math?searchtype=author&query=Cloitre,+B|Benoit Cloitre]]** //The Thue-Morse Transform//
 +
 +{{ seminar2026:20260623cloitre.pdf |slides}}
 +
 +{{ seminar2026:20260623cloitre.mp4 |video of the talk}}
 +
 +
  
 We define a transform $T$ on binary words. Given a binary word, we use the positions of its zeros and ones to build a new binary word. Applied to the alternating word $a_0 = 0101\ldots$, the transform gives the Thue-Morse word. We then study the orbit $a_m = T^m(a_0)$, together with the sequences $u_m$ and $v_m$ giving the positions of the ones and the zeros in $a_m$. We obtain an explicit formula for $a_m(n)$ in terms of the binary digits of $n$ and $m-1$. From this formula we derive Prouhet-Tarry-Escott identities, composition formulas that generalize the identities for evil and odious numbers, and a recurrence formula for the factor complexity of $a_m$. We end with a few directions, such as applying the transform to the Fibonacci word, which yields the Fibonacci-Thue-Morse word.  We define a transform $T$ on binary words. Given a binary word, we use the positions of its zeros and ones to build a new binary word. Applied to the alternating word $a_0 = 0101\ldots$, the transform gives the Thue-Morse word. We then study the orbit $a_m = T^m(a_0)$, together with the sequences $u_m$ and $v_m$ giving the positions of the ones and the zeros in $a_m$. We obtain an explicit formula for $a_m(n)$ in terms of the binary digits of $n$ and $m-1$. From this formula we derive Prouhet-Tarry-Escott identities, composition formulas that generalize the identities for evil and odious numbers, and a recurrence formula for the factor complexity of $a_m$. We end with a few directions, such as applying the transform to the Fibonacci word, which yields the Fibonacci-Thue-Morse word. 
  
-**July 7 2026: Delaram Moradi** 
  
-==== Past talks 2026 ====+**June 9 2026:  [[https://kmlinux.fjfi.cvut.cz/~balkolub/| Ľubomíra Dvořáková]]** //Attractors of sequences coding beta-integers// 
 + 
 +{{ seminar2026:20260609dvorakova.pdf |slides}} 
 + 
 +{{ seminar2026:20260609dvorakova.mp4 |video of the talk}} 
 + 
 + 
 + 
 +String attractor is an intensively studied object in Combinatorics on Words. 
 +In our talk, we will recall known results and also some previously used techniques. 
 +We will then describe minimal string attractors of prefixes of simple Parry sequences. 
 +These sequences form a coding of distances between consecutive beta-integers in numeration systems with a real base beta. 
 +Simple Parry sequences have been recently studied from this point of view and (not necessarily minimal) attractors of their prefixes have been described and a conjecture that attractors of alphabet size should be sufficient was stated. We prove this conjecture. Moreover, we provide attractors of prefixes of some particular form of binary non-simple Parry sequences.