Graph properties of graph associahedra

A graph associahedron is a simple polytope whose face lattice encodes the nested structure of the connected subgraphs of a given graph. In this paper, we study certain graph properties of the 1-skeleta of graph associahedra, such as their diameter and their Hamiltonicity. Our results extend known results for the classical associahedra (path associahedra) and permutahedra (complete graph associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction

Geometric realizations of the accordion complex of a dissection

Consider $2n$ points on the unit circle and a reference dissection $\mathrm{D}_\circ$ of the convex hull of the odd points. The accordion complex of $\mathrm{D}_\circ$ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of $\mathrm{D}_\circ$. In particular, this complex is an associahedron when $\mathrm{D}_\circ$ is a triangulation and a Stokes complex when $\mathrm{D}_\circ$ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection $\mathrm{D}_\circ$, generalizing known constructions arising from cluster algebras.Comment: 25 pages, 10 figures; Version 3: minor correction

Compatibility fans for graphical nested complexes

Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph $G$, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of $G$ and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of $G$. The constructions of M. Carr and S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are all based on the nested fan which coarsens the normal fan of the permutahedron. In view of the combinatorial and geometric variety of simplicial fan realizations of the classical associahedra, it is tempting to search for alternative fans realizing graphical nested complexes. Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's construction of compatibility fans from the former to the latter setting. For this, we define a compatibility degree between two tubes of a graph $G$. Our main result asserts that the compatibility vectors of all tubes of $G$ with respect to an arbitrary maximal tubing on $G$ support a complete simplicial fan realizing the nested complex of $G$. In particular, when the graph $G$ is reduced to a path, our compatibility degree lies in $\{-1,0,1\}$ and we recover F. Santos' Catalan many simplicial fan realizations of the associahedron.Comment: 51 pages, 30 figures; Version 3: corrected proof of Theorem 2

Compatibility fans realizing graphical nested complexes

Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra

Simple modeling of self-oscillation in Nano-electro-mechanical systems

We present here a simple analytical model for self-oscillations in nano-electro-mechanical systems. We show that a field emission self-oscillator can be described by a lumped electrical circuit and that this approach is generalizable to other electromechanical oscillator devices. The analytical model is supported by dynamical simulations where the electrostatic parameters are obtained by finite element computations.Comment: accepted in AP

La télédétection des infrastructures agro-écologiques : de la promesse aux méthodes opérationnelles (Tél-IAE)

Les infrastructures agro-écologiques comme les haies et les bandes enherbées sont des éléments paysagers clé pour la biodiversité dans les territoires agricoles. Les cartographier est une étape importante pour évaluer la qualité des paysages et prédire l’impact d’aménagements. La télédétection spatiale présente un potentiel important pour atteindre cet objectif à coût raisonnable et sur une surface importante. Le projet « télédétection des infrastructures agroécologiques » regroupant spécialistes de la télédétection et utilisateurs s’est proposé d’évaluer des méthodes existantes dans des cas variés et d’en développer de nouvelles. Un site web présentant les résultats du projet guide l’utilisateur vers des grands types d’options techniques en fonction de son projet et lui donne accès à diverses ressources. La pleine appropriation des méthodes et outils implique toutefois un décloisonnement des métiers au delà des considérations purement techniques

Cdc42 and RhoA reveal different spatio-temporal dynamics upon local stimulation with Semaphorin-3A

Small RhoGTPases, such as Cdc42 and RhoA, are key players in integrating external cues and intracellular signaling pathways that regulate growth cone (GC) motility. Indeed, Cdc42 is involved in actin polymerization and filopodia formation, whereas RhoA induces GC collapse and neurite retraction through actomyosin contraction. In this study we employed F\uf6rster Resonance Energy Transfer (FRET) microscopy to study the spatio-temporal dynamics of Cdc42 and RhoA in GCs in response to local Semaphorin-3A (Sema3A) stimulation obtained with lipid vesicles filled with Sema3A and positioned near the selected GC using optical tweezers. We found that Cdc42 and RhoA were activated at the leading edge of NG108-15 neuroblastoma cells during spontaneous cycles of protrusion and retraction, respectively. The release of Sema3A brought to a progressive activation of RhoA within 30 s from the stimulus in the central region of the GC that collapsed and retracted. In contrast, the same stimulation evoked waves of Cdc42 activation propagating away from the stimulated region. A more localized stimulation obtained with Sema3A coated beads placed on the GC, led to Cdc42 active waves that propagated in a retrograde manner with a mean period of 70 s, and followed by GC retraction. Therefore, Sema3A activates both Cdc42 and RhoA with a complex and different spatial-temporal dynamics