Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The Continued Logarithm Algorithm –CL for short– introduced by Gosper in 1978 computes the gcd of two integers; it is seemingly very efficient, as it only performs shifts and subtractions. Shallit has studied its worst-case complexity in 2016 and showed it to be linear. In this talk I present our average-case analysis of the algorithm: we study its main parameters (number of iterations, total number of shifts) and obtain precise asymptotics for their mean values.

Abstract. We present some background and also recent advances due to J. Bourgain, K. Ford, S. Konyagin and the speaker.
These new results are based on a combination of various techniques including the distribution of smooth numbers, distribution of elements of multiplicative subgroups of residue rings, bound of Heilbronn exponential sums and a large sieve inequality with square moduli. These techniques will be briefly explained as well.

Let $T$ and $S$ be contractions on a Banach space $X$. Elements of $(I-T)X$
are called coboundaries of $T$; the elements of $(I-T)X \cap (I-S)X$ are called
joint coboundaries of $T$ and $S$.  If $T$ and $S$ commute, then obviously
the elements of $(I-T)(I-S)X$, called double coboundaries, are joint coboundaries.
It is natural to ask if there exist joint coboundaries which are not double coboundaries.

We propose a new stochastic block model that focuses on the analysis of interaction lengths in dynamic networks. The model does not rely on a discretization of the time dimension and may be used to analyze networks that evolve continuously over time. The framework relies on a clustering structure on the nodes, whereby two nodes belonging to the same latent group tend to create interactions and non-interactions of similar lengths.

La simulation directe du phénomène de propagation des vagues à l'aide des équations d'Euler ou de Navier-Stokes à surface libre est complexe et coûteuse numériquement. Certains phénomène aux grandes échelles sont bien décrit par des modèles réduits plus simples à simuler numériquement; toutefois, ces modèles nécessitent des techniques plus avancées pour imposer les conditions aux limites.

The goal of this presentation is to discuss the reliable sequential detection of transient changes in a multi-parameter exponential distribution. The sequentially observed data are represented by a sequence of independent random vectors with the exponentially distributed components.

We first consider a multidimensional diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a Lévy process without Gaussian component, whose drift coefficient depends on a multidimensional unknown parameter. We prove the local asymptotic normality (LAN) property from high-frequency discrete observations with increasing observation window by assuming some hypotheses on the coefficients of the equation, the ergodicity of the solution and the integrability of the Lévy measure.

Firstly, I will present the definition of spectral disjointness of two automorphisms and show a criterion which can be used  to distinguish this property. Secondly, I will show how to see that powers of a generic interval exchange transformation of three intervals satisfy the asumptions of the criterion.

The result is based on a joint work with Adam Kanigowski.

Nous allons donner des conditions suffisantes pour la loi des logarithmes itérés bornée
pour des champs aléatoires, en considérant la sommation sur des rectangles. Nous nous concentrons sur
deux classes de champs aléatoires strictement stationnaires : les martingales pour l'ordre lexicographique
et les orthomartingales.