In this study the derivation of an objective metrics to appreciate the degree of structuring of written and spoken texts is presented. The proposed metrics is based on the scoring of recurrences inside a text by means of the application of Recurrence Quantification Analysis (RQA), a non linear technique widely used in other fields of sciences. The adopted approach allowed us to create a ranking...

Let Rn (n = 0, 1, 2, . . .) be a second order linear recursive sequence of rational integers defined by Rn = ARn−1+BRn−2 for n > 1, where A and B are integers and the initial terms are R0 = 0, R1 = 1. It is known, that if α, β are the roots of the equation x2 −Ax−B = 0 and |α| > |β|, then Rn+1/Rn −→ α as n −→ ∞. Approximating α with the rational number Rn+1/Rn, it was shown that ∣∣∣α− Rn+1 Rn ∣...

The mechanism of the exponential transient statistics of Poincaré recurrences in the presence of chaos border with its critical structure is studied using two simple models: separatrix map and the kicked rotator ('microtron'). For the exponential transient to exist the two conditions have been shown to be crucial: fast (ballistic) relaxation, and a small measure of the critical structure. The l...

Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the firs...

Three term recurrence relations yn+1+bnyn+anyn−1 = 0 can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of n with respec...

Fredman and Knuth have treated certain recurrences, such as M(O) 1 and M(n + 1) min (aM(k) + 3M(nk)) O 1, by means of auxiliary recurrences such as J" 0 if 0 _< x < 1, h(x) l+h(x/cz)+h(x//) ifl_<x< The asymptotic behavior of h(x) as x oo with a and 3 fixed depends on whether log a/logf is rational or irrational. The solution of Fredman and Knuth used analytic methods in bo...

The purpose of this paper is to outline an application of number theory in medicine. More specifically, a linear second-order recurrence relation is utilized in a technique for the diagnosis of breast cancer. This continues a tradition in this journal of applications of second-order recurrences. Indeed, the very first issue contained an article by Basin [1] on the Fibonacci sequence in art and ...

Fix a prime/?. We say that a set S forms a complete residue system modulop if, for all i such that 0 < i < p 1 , there exists s GS such that s = J (mod /?). We say that a set S forms a reduced residue system modulo p if, for all / such that 1 < i < p -1, there exists s GS such that s = i (mod p). In [9], Shah showed that, ifp is a prime and p = 1,9 (mod 10), then the Fibonacci sequence does not...

This study investigates the presence of deterministic dependencies in international stock markets using recurrence plots and recurrence quantification analysis (RQA). The results are based on a large set of free float-adjustedmarket capitalization stock indices, covering a period of 15 years. The statistical tests suggest that the dynamics of stock prices in emerging markets is characterized by...

We present a short argument that for a wide class of recurrence relations of the form xn+1 = h(xn, . . . , xn−k) and for an arbitrary fixed (forbidden) sequence (bn) the sequence (xn) avoids (bn) (i.e., ∀n ∈ N xn 6= bn) for almost all initial values (x−k, . . . , x0) ∈ R, excluding a set of Lebesgue measure 0. These sequences include algebraic recurrences and our result also generalizes the aff...