Related papers: Gaps in the Thue--Morse word
In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of…
Incoherent exciton dynamics in one-dimensional perfect lattices with traps at sites arranged according to aperiodic deterministic sequences is studied. We focus our attention on Thue-Morse and Fibonacci systems as canonical examples of…
Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called \emph{feasible region}. We show that this feasible region is a…
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences…
This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we…
Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realized by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free…
The main topic of this present thesis is the study of the asymptotic behaviour of sequences modulo 1. In particular, by using ergodic and dynamical methods, a new insight to problems concerning the asymptotic behaviour of multidimensional…
Instruction sequence is a key concept in practice, but it has as yet not come prominently into the picture in theoretical circles. This paper concerns instruction sequences, the behaviours produced by them under execution, the interaction…
The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note we observe…
Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and…
We introduce Morse, a recurrent encoder-decoder model that produces morphological analyses of each word in a sentence. The encoder turns the relevant information about the word and its context into a fixed size vector representation and the…
Let $q$ be a positive integer. Consider an infinite word $\omega=w_0w_1w_2\cdots$ over an alphabet of cardinality $q$. A finite word $u$ is called an arithmetic factor of $\omega$ if $u=w_cw_{c+d}w_{c+2d}\cdots w_{c+(|u|-1)d}$ for some…
The three gap theorem was originally a conjecture by Steinhaus, who asserted that there are at most three distinct gap lengths in the fractional parts of the sequence {\alpha},{2}{\alpha},{\cdots},{N}{\alpha} for any integer {N} and real…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known…
In the theory of programming languages, type inference is the process of inferring the type of an expression automatically, often making use of information from the context in which the expression appears. Such mechanisms turn out to be…
For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$ called the $p$th untwisting number which counts the number of null-homologous twists on at most $2p$ strands required to convert the knot to the unknot. We show…