Svetlana Puzynina

On abelian versions of critical factorization theorem

In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.

FINE AND WILF'S THEOREM FOR k-ABELIAN PERIODS

Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. This leads to a hierarchy of equivalence relations on words which lie p...

A regularity lemma and twins in words

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,@S)=min{f(S):S is of length n, over alphabet @S}. Here, it is shown that2f(n,{0,1})=n-o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length o(n). A similar result is proven for k identical subwords of a word over an alphabet with at most k letters.

On k-Abelian Palindromic Rich and Poor Words

A word is called a palindrome if it is equal to its reversal. In the paper we consider a k-abelian modification of this notion. Two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. We say that a word is a k-abelian palindrome if it is k-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length n can contain at most n + 1 distinct palindromes as its factors; such words are called rich. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called poor. It is easy to see that there are no abelian poor words, and there exist words containing Θ(n 2) distinct abelian palindromes. We analyze these notions with respect to k-abelian equivalence. We show that in the k-abelian case there exist poor words containing finitely many distinct k-abelian palindromic factors, and there exist rich words containing Θ(n 2) distinct k-abelian palindromes as their factors. Therefore, for poor words the situation resembles normal words, while for rich words it is similar to the abelian case.

Self-shuffling words

In this paper we introduce and study a new property of infinite words which is invariant under the action of a morphism: We say an infinite word

Weak Abelian Periodicity of Infinite Words

x\in \mathbb{A}^{\mathbb N},

Infinite self-shuffling words

defined over a finite alphabet

Aperiodic pseudorandom number generators based on infinite words

\mathbb{A}

Subword Complexity and Decomposition of the Set of Factors

, is self-shuffling if x admits factorizations:

Infinite Words with Well Distributed Occurrences

x=\prod_{i=1}^\infty U_iV_i=\prod_{i=1}^\infty U_i=\prod_{i=1}^\infty V_i