Related papers: On the Entropy of Random Fibonacci Words
The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution $\varepsilon$ within $T$ time units. It can then be formally defined as a limit of a limit…
We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to…
In 1989, Godreche and Luck introduced the concept of local mixtures of primitive substitution rules along the example of the well-known Fibonacci substitution and foreshadowed heuristic results on the topological entropy and the spectral…
We propose the concept of open network as an arbitrary selection of nodes of a large unknown network. Using the hypothesis that information of the whole network structure can be extrapolated from an arbitrary set of its nodes, we use Renyi…
We propose a new method for the calculation of the statistical properties, as e.g. the entropy, of unknown generators of symbolic sequences. The probability distribution $p(k)$ of the elements $k$ of a population can be approximated by the…
The word-frequency distribution provides the fundamental building blocks that generate discourse in language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf's law, at…
We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy…
We consider the problem of finding a fixed L-ary sequence in a stream of random L-ary data. It is known that the expected search time is a strictly increasing function of the lengths of the bifices of the pattern. In this paper we prove the…
We consider words as a network of interacting letters, and approximate the probability distribution of states taken on by this network. Despite the intuition that the rules of English spelling are highly combinatorial (and arbitrary), we…
For a word $\pi$ and integer $i$, we define $L^i(\pi)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(\pi):=\max_i L^i(\pi)$. In this paper we estimate the expected values of $L^1(\pi)$ and $L(\pi)$…
A targeted exponentiation algorithm computes a group exponentiation operation $a^k$ with a reversible circuit in such a way that the initial state of the circuit consists of only the base $a$ and fixed values, and the final state consists…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
Topologically constrained genome-like polymers often double-fold into tree-like configurations. Here we calculate the exact number of tightly double-folded configurations available to a ring polymer in ideal conditions. For this purpose, we…
Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent…
We show how generalized Gibbs-Shannon entropies can provide new insights on the statistical properties of texts. The universal distribution of word frequencies (Zipf's law) implies that the generalized entropies, computed at the word level,…
We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the…
Zipf's law states that if words of language are ranked in the order of decreasing frequency in texts, the frequency of a word is inversely proportional to its rank. It is very robust as an experimental observation, but to date it escaped…
We model the generation of words with independent unequal probabilities of occurrence of letters. We prove that the probability $p(r)$ of occurrence of words of rank $r$ has a power asymptotics. As distinct from the paper published earlier…
Fibonacci chains are special diatomic, harmonic chains with uniform nearest neighbour interaction and two kinds of atoms (mass-ratio $r$) arranged according to the self-similar binary Fibonacci sequence $ABAABABA...$, which is obtained by…
This paper introduces the concept of the Fibonacci Word Index $\operatorname{FWI}$, a novel topological index derived from the Albertson index, applied to trees constructed from Fibonacci words. Building upon the classical Fibonacci…