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We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size(More)
In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ∈ N1∪{+∞} where N1 denotes the set of positive integers. Two finite words u and v in A * are said to be k-abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the(More)
FGF signaling plays a pivotal role in regulating cell movements and lineage induction during gastrulation. Here we identify 44 microRNAs that are expressed in the primitive streak region of gastrula stage chicken embryos. We show that the primary effect of FGF signaling on microRNA abundance is to negatively regulate the levels of miR-let-7b, -9, -19b,(More)
In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z + ∪ {+∞}. Let k ∈ Z + ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A * are said to be k-Abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the number of(More)
We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on(More)
Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length k and, moreover, start and end with a same factor of length k − 1, respectively. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze(More)
We analyze and reprove the famous theorem of Hmelevskii, which states that the general solutions of constant-free equations on three unknowns are finitely parameterizable, that is expressible by a finite collection of formulas of word and numerical parameters. The proof is written, and simplified, by using modern tools of combinatorics on words. As a new(More)