相关论文: Abelian Properties of Words (Extended abstract)
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite…
In this paper, we study the $2$-binomial complexity $b_{\mathbf{t}_{m},2}(n)$ of the generalized Thue-Morse words $\mathbf{t}_{m}$ for every integer $m\geq 3$. We obtain the exact value of $b_{\mathbf{t}_{m},2}(n)$ for every integer $n\geq…
If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at…
Carpi (1993) and Lepisto (1994) proved independently that smooth words are cube-free for the alphabet {1, 2}, but nothing is known on whether for the other 2-letter alphabets, smooth words are k-power-free for some suitable positive integer…
Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…
We say a finite word $x$ is a palindromic periodicity if there exist two palindromes $p$ and $s$ such that $|x| \geq |ps|$ and $x$ is a prefix of the word $(ps)^\omega = pspsps\cdots$. In this paper we examine the palindromic periodicities…
For every finite abelian group $G$, there are positive integers $n$ and $d$ such that $G$ is isomorphic to the multiplicative group of $d$-th powers of reduced residues modulo $n$.
This paper is the extended version of On the Complexity of Infinite Advice Strings (ICALP 2018). We investigate a notion of comparison between infinite strings. In a general way, if M is a computation model (e.g. Turing machines) and C a…
We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for…
In this work, we explore the following question: If two words in a finitely generated free group have identical images as word maps on every finite group, must they be endomorphic to each other? In this regard, we introduce weak profinite…
We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in…
An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of a same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words are known to be rigid. In the present paper,…
The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical…
(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal…
We consider a measure of similarity for infinite words that generalizes the notion of asymptotic or natural density of subsets of natural numbers from number theory. We show that every overlap-free infinite binary word, other than the…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
We show that every finite Abelian algebra A from congruence-permutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite…
Words whose three successive factors of the same length are all different i.e. 3-anti-power words are a natural extension of square-free words (two successive factors of the same length are different). We give a way to verify whether a…
We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by $10$ and this bound is optimal, extending a result of Rampersad, who proved that…