arXiv subjects

Benoît Rittaud

Publications and source records attributed to Benoît Rittaud.

9 recordsLinked to original sources

arXiv2026

Reflective Numeration Systems I: a Global Standpoint

We present a framework to generalize the standard b-ary Gray code to get the k-bonacci ones obtained in [5] as well as many others by using theoretical tools that allow to make calculations on lists. We introduce the notion of Z-Gray product, from which we deduce sequences of lists of finite words avoiding a predefinite list Z of factors and which satisfy a power-associativity property as well a generalizations of the classical flipping digit property.

cs.DM↗

arXiv2022

Designing videos for undergraduate mathematics

This article presents some technical and pedagogical considerations about a series of videos made by the authors to teach elementary theory of differential equations. Intended to be a pedagogical support for undergraduate students of scientific curricula, these videos provide rigorous mathematical contents in a form that takes into account the specificities of the tool. Videos for teaching mathematics should be seen as a specific way for providing rigorous scientific content to learners, and should not be reduced to oral manuals or standard classroom lessons. Research Contribution: Arij Bouzelmate and Beno{î}t Rittaud are searcher and teacher in mathematics. Arij Bouzelmate is a specialist in partial differential equations and director of LaR2A. Beno{î}t Rittaud is a specialist in discrete mathematics and is involved in popularization of mathematics.

math.HO↗

arXiv2022

The Field Q and the Equality 0.999 . . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics

We reconsider the classical equality 0.999. .. = 1 with the tool of circular words, that is: finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the xviii th century, who appear to be the firsts to assert the equality 0.999. .. = 1. Their level of understanding show links with Dubinsky et al.'s apos theory in mathematics education. Eventually, we rebuilt the field Q from circular words, and provide an original proof of the fact that an algebraic integer is either an integer or an irrational number.

math.HO↗

arXiv2022

Fibonacci-like sequences for variants of the tower of Hanoi, and corresponding graphs and gray codes

We modify the rules of the classical Tower of Hanoi puzzle in a quite natural way to get the Fibonacci sequence involved in the optimal algorithm of resolution, and show some nice properties of such a variant. In particular, we deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary words without the factor 11, which has some properties intersting for itself and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we briefly extend the study to some natural generalizations.

cs.DM↗

arXiv2011

Circular words and applications

We define the notion of circular words, then consider on such words a constraint derived from the Fibonacci condition. We give several results on the structure of these circular words, then mention possible applications to various situations: periodic expansion of numbers in numeration systems, "gcd-property" of integer sequences, partition of the prefix of the fixed point of the Fibonacci substitution, spanning trees of a wheel. Eventually, we mention some open questions.

cs.FL↗

arXiv2011

Dynamics of $λ$-continued fractions and $β$-shifts

For a real number $0<λ<2$, we introduce a transformation $T_λ$ naturally associated to expansion in $λ$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_λ$ provides an algorithm to expand any positive real number in $λ$-continued fraction. We prove the conjugacy between $T_λ$ and some $β$-shift, $β>1$. Some properties of the map $λ\mapstoβ(λ)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.

math.PR↗

arXiv2008

Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |λg_{n-1} +/- g_{n-2} |. When λ>=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When λ= λ_k = 2 cos(π/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-λ_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author.

math.PR↗

arXiv2008

Almost-sure Growth Rate of Generalized Random Fibonacci sequences

We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = λF_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |λ\widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le 1$). Our main result is that, when $λ$ is of the form $λ_k = 2\cos (π/k)$ for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le 1$, and of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive and given by $$ \int_0^\infty \log x dν_{k, ρ} (x), $$ where $ρ$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ν_{k, ρ}$ is an explicit probability distribution on $\RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\le 1$ in the easier case $λ\ge 2$. Finally, we study the variations of the exponent as a function of $p$.

math.PR↗

arXiv2006

How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x dν_α(x), $$ where $α$ is an explicit function of $p$ depending on the case we consider, and $ν_α$ is an explicit probability distribution on $\RR_+$ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.

math.PR↗