Related papers: Relation between powers of factors and recurrence …
A celebrated theorem by Coven and Hedlund (1973) states that Sturmian words are characterized by their abelian complexity: they are precisely the infinite words with rationally independent letter frequencies and constant abelian complexity…
Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an…
We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions
We characterize all quasiperiodic Sturmian words: a Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity.
In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term…
The present study provides another look on Lamperti's theorem on recurrence or transience of stochastic sequences. We establish connection between Lamperti's theorem and the recent result by the author [V. M. Abramov, Theor. Probab. Math.…
In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called "rich words" is that all complete returns…
We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. We show that any transfinite reduction can be…
Let $D^+$ be the first octant of the Euclidean space and consider the integral cube grid $G$ in $D^+$. The intersections of each line with $G$ form an infinite sequence of three letters which can be considered as an extension of well-known…
The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to…
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
Originally introduced and studied by the third and fourth authors together with J. Justin and S. Widmer in arXiv:0801.1656, rich words constitute a new class of finite and infinite words characterized by containing the maximal number of…
The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor…
We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…
We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic function is non-decreasing, submultiplicative and grows at least linearly. We prove that conversely, every…
An infinite word x is said to be quasiperiodic if there exists a finite word q such that x is covered by occurrences of q (such a q is called a quasiperiod of x). Using the notion of derivation, we show that this definition is not…
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…
We show that if the complexity difference function p(n+1)-p(n) of a infinite minimal shift is bounded, then the the automorphism group of the one-sided shift is finite, and the automorphism group of the corresponding two-sided shift "modulo…
The non-repetitive complexity $nr\mathcal{C}_{\bf u}$ and the initial non-repetitive complexity $inr\mathcal{C}_{\bf u}$ are functions which reflect the structure of the infinite word ${\bf u}$ with respect to the repetitions of factors of…
From Rauzy graph Rauzy Scheme can be obtaining by uniting sequence of vertices of ingoing and outgoing degree 1 by arches. This notion is a tool to describe Rauzy graph behavior. For morphic superword we prove periodicity of Rauzy schemes.…