Related papers: Dynamic and Programmatic Analysis of Fibonacci Wor…
For a given prime $p$, we determine the limit, as $\lambda \to \infty$, of the density of residues modulo $p^\lambda$ attained by the Fibonacci sequence. In particular, we show that this limiting density is related to zeros in the sequence…
This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the…
We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant $V$, this measure is exact-dimensional and the almost everywhere value $d_V$ of the local scaling exponent is a…
We present three simple algorithms to uniformly generate `Fibonacci words' (i.e., some words that are enumerated by Fibonacci numbers), Schr{\"o}der trees of size $n$ and Motzkin left factors of size $n$ and final height $h$. These…
A finite word $u$ is called closed if its longest repeated prefix has exactly two occurrences in $u,$ once as a prefix and once as a suffix. We study the function $f_x^c:\mathbb N \rightarrow \mathbb N$ which counts the number of closed…
We study random words in a weighted regular language that achieve the maximal free energy using thermodynamics formalism. In particular, typical words in the language are algorithmically generated which have applications in computer…
Approximation theory is concerned with the ability to approximate functions by simpler and more easily calculated functions. The first question we ask in approximation theory concerns the {\it possibility of approximation}. Is the given…
Recent methods for learning vector space representations of words have succeeded in capturing fine-grained semantic and syntactic regularities using vector arithmetic. However, these vector space representations (created through large-scale…
This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval…
An $n$-length binary word is $q$-decreasing, $q\geq 1$, if every of its length maximal factor of the form $0^a1^b$ satisfies $a=0$ or $q\cdot a > b$.We show constructively that these words are in bijection with binary words having no…
This article presents a combinatorial result on indexed languages which was inspired by an attempt to understand the structure of groups with indexed language word problem. We show that a sufficiently long word in an indexed language can be…
We make an asymptotic analysis via singularity analysis of generating functions of a number sequence that involves the Fibonacci numbers and generalizes the binomial coefficients.
In previous papers, for an arithmetical function $f_0$, we defined functions $f_m$ and $c_m$ and designated numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem for…
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our…
A dictionary is a database of standard vectors, so that other vectors / signals are expressed as linear combinations of dictionary vectors, and the task of learning a dictionary for a given data is to find a good dictionary so that the…
The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\dots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$…
This work lists and describes the main recent strategies for building fixed-length, dense and distributed representations for words, based on the distributional hypothesis. These representations are now commonly called word embeddings and,…
Continuous word representation (aka word embedding) is a basic building block in many neural network-based models used in natural language processing tasks. Although it is widely accepted that words with similar semantics should be close to…
We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two…
Word Embeddings are used widely in multiple Natural Language Processing (NLP) applications. They are coordinates associated with each word in a dictionary, inferred from statistical properties of these words in a large corpus. In this paper…