Coupling any number of balls in the infinite-bin model

The infinite-bin model, introduced by Foss and Konstantopoulos, describes the Markovian evolution of configurations of balls placed inside bins, obeying certain transition rules. We prove that we can couple the behaviour of any finite number of balls, provided at least two different transition rules are allowed. This coupling makes it possible to define the regeneration events needed by Foss and Zachary to prove convergence results for the distribution of the balls.Comment: 12 pages, 2 figure

The Foata correspondence, cycle lengths and anomalies

International audienceIn their study of the densest jammed configurations for theater models, Krapivsky and Luck observe that two classes of permutations have the same cardinalities and ask for a bijection between them. In this note we show that the Foata correspondence provides the desired bijection. Krapivsky and Luck (2019) introduced the theater model as a variant of directed random sequential adsorption, where spectators sequentially select a seat in a row of L seats, with the constraint that they cannot go past a cluster of b or more consecutive occupied seats. Configurations where all the seats are eventually occupied are parametrized by permutations σ of {1,. .. , L} such that for any i between 1 and L, one cannot find b consecutive integers j + 1,. .. , j + b with j + b σ(i) for all k between 1 and b. Krapivsky and Luck (2019) showed that the number D (b) L of such permutations satisfies a linear recurrence relation which implies that they have the same cardinality as the permutations of L elements with cycles of lengths at most b. The authors then asked for a bijective proof of this fact. The goal of this note is to show that the Foata correspondence provides such a bijection. In Section 1 we recall the Foata correspondence and in Section 2 we show that it provides the desired bijection. 1 The Foata correspondence Let S L be the group of permutations of {1,. .. , L}. We will represent permutations in S L by words with L distinct letters in {1,. .. , L}. For example 1423 denotes the permutation s ∈ S 4 such that s(1) = 1, s(2) = 4, s(3) = 2 and s(4) = 3. One can associate to every permutation in S L its cycle decomposition. Including the fixed points in that decomposition, the above s ∈ S 4 has cycle decomposition [1][243]. This way of writing is however not unique for two reasons: • each cycle of length d can be written in d different ways (one can freely choose what element to put first) ; • if a permutation has k cycles (including singletons corresponding to fixed points) one can have them appear in k! different orders

Laminations of a graph on a pair of pants

A lamination of a graph embedded on a surface is a collection of pair-wise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some graph on a pair of pants. This characterizes the image of a purely topological version of the spectral map for the vector bundle Laplacian for a flat connection on a pair of pants. The proof uses a graph exploration technique akin to the peeling of planar maps

Impact of deforestation on rodet distribution and on the prevalence of leptospirosis in Cambodia

La conversion de forêt en cultures agricoles entraîne des changements brusques et irréversibles des écosystèmes. De tels changements peuvent modifier la distribution des rongeurs et donc des pathogènes qu’ils hébergent. L’hypothèse que pose cette étude est que la déforestation modifie la population des rongeurs et favorise la transmission de leptospirose. L’étude se base sur un modèle de chronoséquence où trois zones à des temps différents de déforestation représentent trois étapes du processus de déforestation : forêt intacte ; forêt perturbée et champs récents. Les rongeurs sont capturés dans ces trois zones et testés par PCR en temps réel pour déterminer les individus excrétant des leptospires. Les leptospires pathogènes ont été séquencées et identifiées afin de déterminer le potentiel risque de leptospirose humaine au cours de la déforestation

Points and lines configurations for perpendicular bisectors of convex cyclic polygons

We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random

Splinting of Traumatized Teeth

Repositioning or replantation followed by stabilization with splint is the standard management of traumatized teeth. Numerous types of splints are being used and each type has its own advantages and disadvantages. Knowledge on the different types of splints and its effect on dental tissues so as to aid in the clinical decision on the type of splint to be used for managing a particular type of traumatic dental injury and its fixation period are important. Therefore, this chapter aims in explaining the different methods of splinting along with its advantages and disadvantages of each type along with the standard recommendation for duration of splinting