Related papers: Dynamic and Programmatic Analysis of Fibonacci Wor…

Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of…

Understanding how words change their meanings over time is key to models of language and cultural evolution, but historical data on meaning is scarce, making theories hard to develop and test. Word embeddings show promise as a diachronic…

Many languages' inflectional morphological systems are replete with irregulars, i.e., words that do not seem to follow standard inflectional rules. In this work, we quantitatively investigate the conditions under which irregulars can…

The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given…

The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…

Classically in combinatorics on words one studies unavoidable regularities that appear in sufficiently long strings of symbols over a fixed size alphabet. In this paper we take another viewpoint and focus on combinatorial properties of long…

This note contains a report of a proof by computer that the Fibonacci group F(2,9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that…

The initial non-repetitive complexity function of an infinite word x (first defined by Moothathu) is the function of n that counts the number of distinct factors of length n that appear at the beginning of x prior to the first repetition of…

For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite…

Word meaning has different aspects, while the existing word representation "compresses" these aspects into a single vector, and it needs further analysis to recover the information in different dimensions. Inspired by quantum probability,…

Word $W$ is an instance of word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V) = W$. For example, taking $\phi$ such that $\phi(c)=fr$, $\phi(o)=e$ and $\phi(l)=zer$, we see that "freezer" is…

Despite the recent popularity of word embedding methods, there is only a small body of work exploring the limitations of these representations. In this paper, we consider one aspect of embedding spaces, namely their stability. We show that…

Following Inoue et al., we define a word to be a repetition if it is a (fractional) power of exponent at least 2. A word has a repetition factorization if it is the product of repetitions. We study repetition factorizations in several…

We investigate the lengths and starting positions of the longest monochromatic arithmetic progressions for a fixed difference in the Fibonacci word. We provide a complete classification for their lengths in terms of a simple formula. Our…

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this paper, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded…

This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$,…

The focus of this note is to formulate the algorithms and give the examples used by Fibonacci in Liber Abaci to expand any fraction into a sum of unit fractions. The description in Liber Abaci is all verbal and the examples are numbers…

In this paper, we propose and investigate algorithms for the structured orthogonal dictionary learning problem. First, we investigate the case when the dictionary is a Householder matrix. We give sample complexity results and show…

This thesis investigates how the sub-structure of words can be accounted for in probabilistic models of language. Such models play an important role in natural language processing tasks such as translation or speech recognition, but often…