It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log 2 (n)⌉ + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈logφ(n)⌉ + 1, where φ denotes the gol...

We present a results about convergence of products of row-stochastic matrices which are infinite to the left and all have positive diagonals. This is regarded as in inhomogeneous consensus process where confidence weights may change in every time step but where each agent has a little bit of self confidence. The positive diagonal leads to a fixed zero pattern in certain subproducts of the infin...

The arithmetical complexity of infinite words, defined by Avgustinovich, Fon-Der-Flaass and the author in 2000, is the number of words of length n which occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infinite word of length n. In this paper, we show that...

To any infinite word t over a finite alphabet A we can associate two infinite words min(t) and max(t) such that any prefix of min(t) (resp. max(t)) is the lexicographically smallest (resp. greatest) amongst the factors of t of the same length. We say that an infinite word t over A is fine if there exists an infinite word s such that, for any lexicographic order, min(t) = as where a = min(A). In...

We study optical bistability of stationary light transmission in nonlinear periodic structures of finite and semi-infinite length. For finite-length structures, the system exhibits instability mechanisms typical for dissipative dynamical systems. We construct a LeraySchauder stability index and show that it equals the sign of the Evans function in λ = 0. As a consequence, stationary solutions w...

In this paper, we build upon previous work concerning inequalities characterizing Sturmian and episturmian words – see [19, 29, 30, 14, 16]. First let us recall from [29] the following notion relating to lexicographic order. Let A be a totally ordered finite alphabet consisting of at least two letters. To any infinite word x over A, we can associate two infinite words min(x) and max(x) such tha...

in this paper we consider the group algebra $r(c_2times‎ ‎d_infty)$‎. ‎it is shown that $r(c_2times d_infty)$ can be‎ ‎represented by a $4times 4$ block circulant matrix‎. ‎it is also‎ ‎shown that $mathcal{u}(mathbb{z}_2(c_2times d_infty))$ is‎ ‎infinitely generated‎.