Related papers: Abelian Complexity and Synchronization
Motivated by the problems of computing sample covariance matrices, and of transforming a collection of vectors to a basis where they are sparse, we present a simple algorithm that computes an approximation of the product of two n-by-n real…
Generalizing the notion of automatic complexity of individual strings due to Shallit and Wang, we define the automatic complexity $A(E)$ of an equivalence relation $E$ on a finite set $S$ of strings. We prove that the problem of determining…
In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the…
Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in{\{\mathtt{0},\mathtt{1}\}}^n$. They also stated that Holger Petersen had informed them that the constant 13 can be reduced to 7. Here…
We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the…
In this article we consider finite automata networks (ANs) with two kinds of update schedules: the parallel one (all automata are updated all together) and the sequential ones (the automata are updated periodically one at a time according…
Fibonacci word is the archetype of the Sturmian word, and it is one of the most studied of combinatorics on words. We studied the properties of the Fibonacci word and found its density for limited value then by calculating the limit…
The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. The main result is twofold: (1) we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the…
We give an algorithm with singly exponential complexity for computing the barcodes up to dimension $\ell$ (for any fixed $\ell \geq 0$) of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our…
We propose an efficient and flexible method for solving Abel integral equation of the first kind, frequently appearing in many fields of astrophysics, physics, chemistry, and applied sciences. This equation represents an ill-posed problem,…
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of…
We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an…
The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any…
Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period $\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having abelian period $\mathcal{P}$. Our main result is an online algorithm that, given a word…
We present a rigorous and relatively fast method for the computation of the "complexity" of a natural number (sequence A005245), and answer some "old and new" questions related to the question in the title of this note. We also extend the…
In this paper we consider the problem of computing the longest common abelian factor (LCAF) between two given strings. We present a simple $O(\sigma~ n^2)$ time algorithm, where $n$ is the length of the strings and $\sigma$ is the alphabet…
We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APC). This class corresponds to level $3/2$ of the Straubing-Th{\'e}rien Hierarchy and includes the finite,…
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…
We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $\alpha$ with continued…
It is well-know that deciding consistency for normal answer set programs (ASP) is NP-complete, thus, as hard as the satisfaction problem for classical propositional logic (SAT). The best algorithms to solve these problems take exponential…