A simple formula for the series of constellations and quasi-constellations with boundaries

We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. The formula naturally extends to $p$-constellations and quasi-$p$-constellations with boundaries (the case $p=2$ corresponding to bipartite maps).Comment: 23 pages, full paper version of v1, with results extended to constellations and quasi constellation

Left- vs right-handed badminton slice shots: opposite shuttlecock spinning and Magnus effect

The chiral nature of a badminton shuttlecock is responsible for its anti-clockwise spinning as it naturally propagates through the air. This induces a dissymmetry between left-handed and right-handed players and the resulting trajectories of the shuttlecock, which were captured in real condition on the badminton court and in slow-motion at 3700 fps. The videos clearly evidence this dissymmetry as slice shots performed by righties accelerate the natural anti-clockwise spinning, while the one performed by lefties induces a clockwise to anti-clockwise spinning, making trajectories of shuttlecocks different. The slow-motion videos also caught a brief Magnus effect, often neglected in badminton, lifting up the shuttlecock for both lefties and righties and affecting the effectiveness of the slice shot.Comment: Videos are available here: doi 10.5281/zenodo.1001287

A bijection for plane graphs and its applications

International audienceThis paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations. 1 Introduction A planar graph is a graph that can be embedded in the plane (drawn in the plane without edge crossing). A pla-nar map is an embedding of a connected planar graph considered up to deformation. The enumeration of pla-nar maps has been the subject of intense study since the seminal work of Tutte in the 60's [20] showing that many families of planar maps have beautiful counting formulas. Starting with the work of Cori and Vauquelin [10] and then Schaeffer [18, 19], bijective constructions have been discovered that provide more transparent proofs of such formulas. The enumeration of planar graphs has also been the focus of a lot of efforts, culminating with the asymptotic counting formulas obtained by Giménez and Noy [16]. In this paper we focus on simple planar maps (planar maps without loops nor multiple edges), which are also called plane graphs. This family of planar maps has, quite surprisingly, not been considered until fairly recently. This is probably due to the fact that loops and multiple edges are typically allowed in studies about planar maps, whereas they are usually forbidden in studies about planar graphs. At any rate, the first result about plane graphs was an exact algebraic expressio

On the distance-profile of random rooted plane graphs

International audienceWe study the distance-profile of the random rooted plane graph Gn with n edges (by a plane graph we mean a planar map with no loops nor multiple edges). Our main result is that the profile and radius of Gn (with respect to the root-vertex), rescaled by (2n) 1/4 , converge to explicit distributions related to the Brownian snake. A crucial ingredient of our proof is a bijection we have recently introduced between rooted outer-triangular plane graphs and rooted eulerian triangulations, combined with ingredients from Chassaing and Schaeffer (2004), Bousquet-Mélou and Schaeffer (2000), and Addario-Berry and Albenque (2013). We also show that the result for plane graphs implies similar results for random rooted loopless maps and general maps

Connected component trees for multivariate image processing applications in astronomy

International audienceIn this paper, we investigate the possibilities offered by the extension of the connected component trees (cc-trees) to multivariate images. We propose a general framework for image processing using the cc-tree based on the lattice theory and we discuss the possible applications depending on the properties of the underlying ordered set. This theoretical reflexion is illustrated by two applications in mul-tispectral astronomical imaging: source separation and object detection

From hyperconnections to hypercomponent tree: Application to document image binarization

International audienceIn this paper, we propose an extension of the component tree based on at zones to hyperconnections (h-connections). The tree is dened by a special order on the h-connection and allows non at nodes. We apply this method to a particular fuzzy h-connection and we give an ecient algorithm to transform the component tree into the new fuzzy h-component tree. Finally, we propose a method to binarize document images based on the h-component tree and we evaluate it on the DIBCO 2009 benchmarking dataset: our novel method places rst or second according to the dierent evaluation measures. Hierarchical and tree based representations have become very topical in image processing. In particular, the component tree (or Max-Tree) has been the subject of many studies and practical works. Nevertheless, the component tree inherits the weaknesses of the at zone approach, namely its high sensitivity to noise, gradients and diculty to manage disconnected objects. Even if some solutions have been proposed to preserve the component tree [5, 4], it seems that a more general framework for grayscale component tree [1] based on non at zones becomes necessary. In this article, we propose a method to design grayscale component tree based on h-connections. The h-connection theory has been proposed in [7] and developed in [1, 3, 4, 8, 9]. It provides a general denition of the notion of connected component in arbitrary lattices. In Sec. 2, we present the h-connection theory and a method to generate a related hierarchical representation. This method is applied to a fuzzy h-connection in Sec. 3 where an algorithm is given to transform a Max-Tree into the new grayscale component tree. In Sec. 4, we illustrate the interest of this tree with an application on document image binarization. 2 H-component Tree This section presents the basis of the h-connection theory [7, 1] and gives a denition of the h-component tree. The construction of the tree is based on the z-zones [1] of the h-connection, together with a special partial ordering. Z-zones are particular regions where all points generate the same set of hyperconnected (h-connected) components and the entire image can be divided into such zones. Under a given condition, the Hasse diagram obtained in this way is acyclic and provides a tree representation. Let L be a complete lattice furnished with the partial ordering ≤, the inmum , the supremum. The least element of L is denoted by ⊥ = L. We assume the existence of a sup-generatin

Femtosecond photoswitching dynamics and microsecond thermal conversion driven by laser heating in FeIII spin-crossover solids.

International audienceIn this paper we review time-resolved studies of ultrafast light-induced spin-state switching, triggered by a femtosecond laser flash,and the following out-of-equilibrium dynamics in FeIII spincrossover crystals. The out-of-equilibrium dynamics involves several steps, resulting fromthe ultrafast molecular photoswitchingof low-spin (LS) to high-spin (HS) states in solids. First, the transient HS state is reached within 200 femtoseconds, and mayrapidly decayinto the stable LS state of the system. A second process at longer delay,associated with volume expansion, drives additional conversion to the HS state during the so-called elastic step occurring at nanosecond time scale. Finally,the laser heating process induces a temperature jump in the crystal that may result in a significant thermal population of the HS state on microsecond time scale. The photoswitching mechanism is of local nature and has linear dependenceon the excitation fluence, whereas the heating effect can macroscopically perturb the LS/HS equilibrium. We discuss similarities and differences between photoswitching dynamics in solution and in different crystals for which the thermal spin conversion is of more or less pronounced cooperative nature