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

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.Comment: 7 pages, 4 figure

What was your fracture risk evaluated by FRAX® the day before your osteoporotic fracture?

Osteoporotic fracture (OF) is one of the major causes of morbidity and mortality in industrialized countries. Switzerland is among the countries with the greatest risk. Our aim was (1) to calculate the FRAX® in a selected Swiss population the day before the occurrence of an OF and (2) to compare the results with the proposed Swiss FRAX® thresholds. The Swiss Association Against Osteoporosis proposed guidelines for the treatment of osteoporosis based on age-dependent thresholds. To identify a population at a very high risk of osteoporotic fracture, we included all consecutive patients in the active OF pathway cohort from the Lausanne University Hospital, Switzerland. FRAX® was calculated with the available data the day before the actual OF. People with a FRAX® body mass index (BMI) or a FRAX® (bone mineral density) BMD lower than the Swiss thresholds were not considered at high risk. Two-hundred thirty-seven patients were included with a mean age of 77.2years, and 80% were female. Major types of fracture included hip (58%) and proximal humerus (25%) fractures. Mean FRAX® BMI values were 28.0, 10.0, 13.0, 26.0, and 37.0% for age groups 50-59, 60-69, 70-79, and 80-89years old, respectively. Fifty percent of the population was not considered at high risk by the FRAX® BMI. FRAX® BMD was available for 95 patients, and 45% had a T score < −2.5 standard deviation. Only 30% of patients with a normal or osteopenic BMD were classified at high risk by FRAX® BMD. The current proposed Swiss thresholds were not able to classify at high risk in 50 to 70% of the studied population the day before a major O

“Inflammaging” and bone in the OsteoLaus cohort

In this large population-based cohort, we did not find a relation between the coined concept of “inflammaging” (hs-CRP, IL-6, IL-1β and TNF-α within the normal range) and bone parameters, measured in terms of prevalent and incident fractures, bone mass density and trabecular bone score

Are the Collatz and abc conjectures related?

17 pages, 2 figures. Inaccurate reference fixedThe 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

Étude numérique des motifs dunaires associés au mécanisme de digitation granulaire

Whether on a beach or in the midst of a desert, aeolian dunes are the result of a complex interaction between a granular medium and the air flow. They can take various forms depending on present and past wind conditions. This study focuses on dune patterns generated by a bimodal wind regime on a non-erodible bed. Such patterns tend to elongate in a direction close to the resultant of sediment transport under the effect of a growth mechanism called digitation. When sand supply is sufficient, the elongation may occur over huge distances, as shown by the vast linear dunefields present in some of the major terrestrial deserts and on various planetary bodies. We study the necessary conditions for these dunes to elongate and for their stability. Our approach is based on a numerical model consisting of two coupled cellular automata, in order to simulate the sediment transport processes and the wind flow. This model allows to reproduce different growth mechanisms such as bed instability, digitation or coalescence. These mechanisms depend both on the wind regime and on the availability of sand. We show the existence of a stationary state for dunes elongating from a sedimentary source and describe the elongation dynamics derived from transport laws. These results are consistent with observations made using satellite images in the Ténéré desert. Finally, we study an example of coexistence of two dune patterns in a complex dunefield. This example supports the idea that dunes are formed from a small number of elementary dune patterns.Que ce soit sur une plage ou en plein désert, les dunes éoliennes résultent de l'interaction complexe qui se déroule entre un milieu granulaire et un écoulement d'air. Elles peuvent prendre des formes variées selon les conditions de vent présentes et passées. L'objet de cette étude concerne les motifs dunaires engendrés par un régime de vent bimodal sur un sol non-érodable. De tels motifs tendent à s'allonger dans une direction proche de la résultante des transports sédimentaires, sous l'effet d'un mécanisme appelé digitation. Lorsque l'apport en sable est suffisant, cette élongation peut se poursuivre sur de très longues distances, comme en attestent les vastes champs de dunes linéaires présents dans la plupart des grands déserts terrestres et sur divers corps planétaires. Nous étudions les conditions nécessaires à l'élongation de ces dunes et à leur stabilité. Notre approche s'appuie sur un modèle numérique constitué de deux automates cellulaires couplés, afin de simuler les processus de transport sédimentaire et le flux éolien. Ce modèle permet de reproduire différents mécanismes de croissance tels que l'instabilité de lit plat, la digitation ou encore la coalescence. Ces mécanismes dépendent à la fois du régime de vent et de la disponibilité en sable. Nous montrons alors l'existence d'un état stationnaire pour les dunes s'allongeant depuis une source sédimentaire et décrivons la dynamique d'élongation issue des lois de transport. Ces résultats correspondent aux observations effectuées à l'aide d'images satellitaires dans le désert du Ténéré. Enfin, nous étudions un exemple de coexistence de deux motifs dunaires dans un champ de dunes complexes. Cet exemple conforte l'idée que les dunes sont formées d'un petit nombre de motifs dunaires élémentaires

Étude numérique des motifs dunaires associés au mécanisme de digitation granulaire

Whether on a beach or in the midst of a desert, aeolian dunes are the result of a complex interaction between a granular medium and the air flow. They can take various forms depending on present and past wind conditions. This study focuses on dune patterns generated by a bimodal wind regime on a non-erodible bed. Such patterns tend to elongate in a direction close to the resultant of sediment transport under the effect of a growth mechanism called digitation. When sand supply is sufficient, the elongation may occur over huge distances, as shown by the vast linear dunefields present in some of the major terrestrial deserts and on various planetary bodies. We study the necessary conditions for these dunes to elongate and for their stability. Our approach is based on a numerical model consisting of two coupled cellular automata, in order to simulate the sediment transport processes and the wind flow. This model allows to reproduce different growth mechanisms such as bed instability, digitation or coalescence. These mechanisms depend both on the wind regime and on the availability of sand. We show the existence of a stationary state for dunes elongating from a sedimentary source and describe the elongation dynamics derived from transport laws. These results are consistent with observations made using satellite images in the Ténéré desert. Finally, we study an example of coexistence of two dune patterns in a complex dunefield. This example supports the idea that dunes are formed from a small number of elementary dune patterns.Que ce soit sur une plage ou en plein désert, les dunes éoliennes résultent de l'interaction complexe qui se déroule entre un milieu granulaire et un écoulement d'air. Elles peuvent prendre des formes variées selon les conditions de vent présentes et passées. L'objet de cette étude concerne les motifs dunaires engendrés par un régime de vent bimodal sur un sol non-érodable. De tels motifs tendent à s'allonger dans une direction proche de la résultante des transports sédimentaires, sous l'effet d'un mécanisme appelé digitation. Lorsque l'apport en sable est suffisant, cette élongation peut se poursuivre sur de très longues distances, comme en attestent les vastes champs de dunes linéaires présents dans la plupart des grands déserts terrestres et sur divers corps planétaires. Nous étudions les conditions nécessaires à l'élongation de ces dunes et à leur stabilité. Notre approche s'appuie sur un modèle numérique constitué de deux automates cellulaires couplés, afin de simuler les processus de transport sédimentaire et le flux éolien. Ce modèle permet de reproduire différents mécanismes de croissance tels que l'instabilité de lit plat, la digitation ou encore la coalescence. Ces mécanismes dépendent à la fois du régime de vent et de la disponibilité en sable. Nous montrons alors l'existence d'un état stationnaire pour les dunes s'allongeant depuis une source sédimentaire et décrivons la dynamique d'élongation issue des lois de transport. Ces résultats correspondent aux observations effectuées à l'aide d'images satellitaires dans le désert du Ténéré. Enfin, nous étudions un exemple de coexistence de deux motifs dunaires dans un champ de dunes complexes. Cet exemple conforte l'idée que les dunes sont formées d'un petit nombre de motifs dunaires élémentaires