Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. We study the connection between self-similar algebras with run-length encoding and rational languages. In particular, we provide a curious relationship between the eigenvalues of a sequence of matrices related to a specific self-similar algebra and the smooth words over a 2-letter alphabet. We also consider the language $L(s)$ of words $u$ in $(\Sigma\times \Sigma)^*$ where $\Sigma=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$

More Kolakoski Sequences

Our goal in this article is to review the known properties of the mysterious Kolakoski sequence and at the same time look at generalizations of it over arbitrary two letter alphabets. Our primary focus will here be the case where one of the letters is odd while the other is even, since in the other cases the sequences in question can be rewritten as (well-known) primitive substitution sequences. We will look at word and letter frequencies, squares, palindromes and complexity.Comment: 17 pages, 3 tables, 1 figur

Vertical representation of C∞C^{\infty}-words

We present a new framework for dealing with $C^{\infty}$-words, based on their left and right frontiers. This allows us to give a compact representation of them, and to describe the set of $C^{\infty}$-words through an infinite directed acyclic graph $G$. This graph is defined by a map acting on the frontiers of $C^{\infty}$-words. We show that this map can be defined recursively and with no explicit references to $C^{\infty}$-words. We then show that some important conjectures on $C^{\infty}$-words follow from analogous statements on the structure of the graph $G$.Comment: Published in Theoretical Computer Scienc

Stable set of self map

The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal in the general case. Here we determine whether or not the equality holds in several particular cases, among which are the case of a dense range continuous function on an Hilbert space, and the case of a substitution over left infinite words

Smooth Words on a 2-letter alphabets having same parity

International audienceIn this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b − 1 + 1)

連長圧縮操作で閉じた文字列集合の性質に関する研究

Tohoku University篠原