Curves of genus 2, continued fractions, and Somos sequences.
Dedekind sums and continued fractions
Démonstration de la périodicité des fractions continues, engendrées par les racines d'une équation du deuxième degré
Des mots en arithmétique
Développement en fraction continue à l'entier le plus proche, idéaux α-réduits et un problème d'Eisenstein
Discrete planes, -actions, Jacobi-Perron algorithm and substitutions
We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of -actions by rotations.
Dyadic Fractions with Small Partial Quotients.
Dynamique des nombres et physique des oscillateurs
Nous présentons un modèle mathématique permettant de reproduire le spectre expérimental des fréquences dans un composant électronique appelé boucle ouverte. Le spectre semble s’organiser suivant une contrainte de nature diophantienne sur les fréquences. Sa structure peut donc se comprendre via une étude de l’ensemble des fractions continues en fonction de leur longueur et de la taille des quotients partiels.
Einheitenberechnung in kubischen Körpern mittels des Jacobi-Perronschen Algorithmus aus der Rechenanlage.
Elasticity in certain block monoids via the Euclidean table
Elementarmathematik und Didaktik. Reguläre Kettenbrüche und quadratische diophantische Probleme.
Elementary proof of Yu. V. Nesterenko expansion of the number zeta(3) in continued fraction.
Elliptic curves and continued fractions.
End-symmetric continued fractions and quadratic congruences
We show that for a fixed integer n ≠ ±2, the congruence x² + nx ± 1 ≡ 0 (mod α) has the solution β with 0 < β < α if and only if α/β has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number α/β > 1 in lowest terms has a symmetric continued fraction precisely when β² ≡ ±1(mod α ).
Episturmian morphisms and a Galois theorem on continued fractions
We associate with a word on a finite alphabet an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of . Then when we deduce, using the sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.
Episturmian morphisms and a Galois theorem on continued fractions
We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.
Ergodicity of ℤ² extensions of irrational rotations
Let = [0,1) be the additive group of real numbers modulo 1, α ∈ be an irrational number and t ∈ . We study ergodicity of skew product extensions T : × ℤ² → × ℤ², .
Evaluations of the Rogers-Ramanujan continued fraction R(q) by modular equations
Explicit evaluations of the Rogers-Ramanujan continued fraction.
Fractions continues normales dans un corps de fonctions hyperelliptiques