arXiv subjects

Olivier Rozier

Publications and source records attributed to Olivier Rozier.

7 recordsLinked to original sources

arXiv2026

Paradoxical behavior in Collatz sequences

On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial cycle $(1,2)$. In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting with an integer $n\geq2$ is sufficient to determine its stopping time, namely, the number of iterations needed to descend below $n$. However, when iterating beyond the stopping time, there exist "paradoxical" sequences of finite length whose first term is unexpectedly exceeded, given the proportion of odd terms. In the present study, we show that this non-typical behavior is closely related to the Collatz conjecture. Furthermore, we find that it most likely occurs finitely many times, thus lending support to Terras' conjecture.

math.GM↗

arXiv2025

Are the Collatz and abc conjectures related?

The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.

math.NT↗

arXiv2025

Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding

In this paper, we consider the one-to-one correspondence between a 2-adic integer and its parity sequence under iteration of the so-called "3x+1" map. First, we prove a new formula for the inverse transform. Next, we briefly review what is known about the induced automorphism and study its dynamics on the 2-adic integers. We find that it is ergodic on many small odd invariant sets, and that it has two odd cycles of period 2 in addition to its two odd fixed points. Finally, a plane embedding is presented, for which we establish affine self-similarity by using functional equations.

math.DS↗

arXiv2020

Incipient bedforms in a bidirectional wind regime

Most terrestrial sand seas form at `horse' latitudes, where the wind direction exhibits seasonal variation. Here, we extend the two-dimensional linear stability analysis of a flat sand bed associated with a unidirectional wind to the three-dimensional case in order to account for multidirectional wind regimes. Focusing on the simplest case of bidirectional flow regimes, we show that the transition from transverse to oblique or longitudinal patterns is controlled by the transport ratio and the divergence angle between the two flows. Our predictions agree with previous results for dune orientation, and also provide a wider range of possible alignments depending on flow strength, especially when the two winds are perpendicular, at which the transition occurs. This analysis also predicts the selected pattern wavelength, which either decreases close to the transition angle for strong winds, due to a geometric effect, or increases at low winds, when the bed slope affects the transport. This theoretical analysis is complemented by analogous subaqueous experiments, where bedforms are submitted to alternate water flows. For transverse bedforms, the experimental data validate the model at strong flows, providing evidence for the predicted geometric effect, but also for the increase of the wavelength close to the transport threshold. For longitudinal bedforms, a discrepancy is observed, which we interpret as the sign of enhanced nonlinearities induced by the development of slip faces when the flow alternately blows on both sides of the dune.

physics.flu-dyn↗

arXiv2020

Periodicity in fields of elongating dunes

Dune fields are commonly associated with periodic patterns that are among the most recognizable landscapes on Earth and other planetary bodies. However, in zones of limited sediment supply, where periodic dunes elongate and align in the direction of the resultant sand flux, there has been no attempt to explain the emergence of such a regular pattern. Here, we show, by means of numerical simulations, that the elongation growth mechanism does not produce a pattern with a specific wavelength. Periodic elongating dunes appear to be a juxtaposition of individual structures, the arrangement of which is due to regular landforms at the border of the field acting as boundary conditions. This includes, among others, dune patterns resulting from bed instability, or the crestline reorganization induced by dune migration. The wavelength selection in fields of elongating dunes therefore reflects the interdependence of dune patterns over the course of their evolution.

physics.geo-ph↗

arXiv2016

The 3x+1 problem: a lower bound hypothesis

Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two. In this paper, we formulate a new hypothesis asserting that the first terms of those sequences have a lower bound which depends on the binary entropy of the "ones-ratio". It is in agreement with all computations so far. Furthermore it implies accurate upper bounds for the total stopping time and the maximum excursion of an integer. Theses results are consistent with two previous stochastic models of the 3x+1 problem.

math.NT↗

arXiv2014

Dynamique du problème $3x+1$ sur la droite réelle

The 3x+1 problem is a difficult conjecture dealing with quite a simple algorithm on the positive integers. A possible approach is to go beyond the discrete nature of the problem, following M. Chamberland who used an analytic extension to the half-line $\mathbb{R}^{+}$. We complete his results on the dynamic of the critical points and obtain a new formulation the 3x+1 problem. We clarify the links with the question of the existence of wandering intervals. Then, we extend the study of the dynamic to the half-line $\mathbb{R}^{-}$, in connection with the 3x-1 problem. Finally, we analyze the mean behaviour of real iterations near infinity and present a method based on a heuristic argument by R. E. Crandall in the discrete case. It follows that the average growth rate of the iterates is close to $(2+\sqrt{3})/4$ under a condition of uniform distribution modulo 2.

math.DS↗