Compact convex sets of the plane and probability theory

The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of $n$ random variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to a CCS associated with $\mu$ at speed $\sqrt{n}$, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations

Dietary live yeast alters metabolic profiles, protein biosynthesis and thermal stress tolerance of Drosophila melanogaster

International audienceThe impact of nutritional factors on insect's life-history traits such as reproduction and lifespan has been excessively examined; however, nutritional determinant of insect's thermal tolerance has not received a lot of attention. Dietary live yeast represents a prominent source of proteins and amino acids for laboratory-reared drosophilids. In this study, Drosophila melanogaster adults were fed on diets supplemented or not with live yeast. We hypothesized that manipulating nutritional conditions through live yeast supplementation would translate into altered physiology and stress tolerance. We verified how live yeast supplementation affected body mass characteristics, total lipids and proteins, metabolic profiles and cold tolerance (acute and chronic stress). Females fed with live yeast had increased body mass and contained more lipids and proteins. Using GC/MS profiling, we found distinct metabolic fingerprints according to nutritional conditions. Metabolite pathway enrichment analysis corroborated that live yeast supplementation was associated with amino acid and protein biosyntheses. The cold assays revealed that the presence of dietary live yeast greatly promoted cold tolerance. Hence, this study conclusively demonstrates a significant interaction between nutritional conditions and thermal tolerance

Indirect inference : which moments to match?

The standard approach to indirect inference estimation considers that the auxiliary parameters, which carry the identifying information about the structural parameters of interest, are obtained from some recently identified vector of estimating equations. In contrast to this standard interpretation, we demonstrate that the case of overidentified auxiliary parameters is both possible, and, indeed, more commonly encountered than one may initially realize. We then revisit the “moment matching” and “parameter matching” versions of indirect inference in this context and devise efficient estimation strategies in this more general framework. Perhaps surprisingly, we demonstrate that if one were to consider the naive choice of an efficient Generalized Method of Moments (GMM)-based estimator for the auxiliary parameters, the resulting indirect inference estimators would be inefficient. In this general context, we demonstrate that efficient indirect inference estimation actually requires a two-step estimation procedure, whereby the goal of the first step is to obtain an efficient version of the auxiliary model. These two-step estimators are presented both within the context of moment matching and parameter matching. View Full-Tex

Magnitude bias of microlensed sources towards the Large Magellanic Cloud

There are lines of evidence suggesting that some of the observed microlensing events in the direction of the Large Magellanic Cloud (LMC) are caused by ordinary star lenses as opposed to dark Machos in the Galactic halo. Efficient lensing by ordinary stars generally requires the presence of one or more additional concentrations of stars along the line of sight to the LMC disk. If such a population behind the LMC disk exists, then the source stars (for lensing by LMC disk objects) will be drawn preferentially from the background population and will show systematic differences from LMC field stars. One such difference is that the (lensed) source stars will be farther away than the average LMC field stars, and this should be reflected in their apparent baseline magnitudes. We focus on red clump stars: these should appear in the color-magnitude diagram at a few tenths of a magnitude fainter than the field red clump. Suggestively, one of the two near-clump confirmed events, MACHO-LMC-1, is a few tenths of magnitude fainter than the clump.Comment: To appear in ApJ Letters. Shortened to match the accepted version, 8 pages plus 1 ps figur

Chemical Abundance Constraints on White Dwarfs as Halo Dark Matter

We examine the chemical abundance constraints on a population of white dwarfs in the Halo of our Galaxy. We are motivated by microlensing evidence for massive compact halo objects (Machos) in the Galactic Halo, but our work constrains white dwarfs in the Halo regardless of what the Machos are. We focus on the composition of the material that would be ejected as the white dwarfs are formed; abundance patterns in the ejecta strongly constrain white dwarf production scenarios. Using both analytical and numerical chemical evolution models, we confirm that very strong constraints come from Galactic Pop II and extragalactic carbon abundances. We also point out that depending on the stellar model, significant nitrogen is produced rather than carbon. The combined constraints from C and N give $\Omega_{WD} h < 2 \times 10^{-4}$ from comparison with the low C and N abundances in the Ly$\alpha$ forest. We note, however, that these results are subject to uncertainties regarding the nucleosynthesis of low-metallicity stars. We thus investigate additional constraints from D and $^4$He, finding that these light elements can be kept within observational limits only for \Omega_{WD} \la 0.003 and for a white dwarf progenitor initial mass function sharply peaked at low mass (2$M_\odot$). Finally, we consider a Galactic wind, which is required to remove the ejecta accompanying white dwarf production from the galaxy. We show that such a wind can be driven by Type Ia supernovae arising from the white dwarfs themselves, but these supernovae also lead to unacceptably large abundances of iron. We conclude that abundance constraints exclude white dwarfs as Machos. (abridged)Comment: Written in AASTeX, 26 pages plus 4 ps figure

Etude des graphes planaires cofinis selons leurs groupes de symétries

Les graphes cofinis constituent une famille de graphes possédant un groupe de symétries non trivial, comme les graphes de Cayley ou les graphes sommet-transitifs. Lorsque ces graphes sont en plus planaires, ces symétries peuvent se traduire de manière simple grâce à des symétries du plan dans lequel les graphes sont dessinés. L’ensemble de ces symétries ou automorphismes permet alors de décrire globalement le graphe à l’aide de données géométriques locales, par des structures appelées schémas d’étiquetage. Dans cette thèse, nous étudions les groupes de symétries et décrivons les schémas d’étiquetage des graphes planaires cofinis possédant une représentation topologique simple : les graphes planaires localement finis. Nous montrons comment ces schémas permettent de caractériser le graphe et ses plongements. Cette analyse permet d’énumérer cette famille des graphes planaires cofinis, en particulier lorsqu’ils sont de Cayley ou sommet-transitifs. A partir de ces résultats, nous nous intéressons à la structure des groupes d’automorphismes de cette famille de graphes. Des problèmes de la théorie combinatoire des groupes usuellement indécidables se trouvent devenir décidables dans notre cadre : c’est le cas en particulier des problèmes du mot, simple et généralisé. Les problèmes de décidabilité de la logique permettent de classifier ces graphes en deux grandes familles, selon leur largeur arborescente et la géométrie de leur plongement. Enfin, la question de l’extension de cette description à une famille de graphes plus généraux est étudiée. La classification de ces graphes en terme de bouts et de points d’accumulation dans les plongements permet d’obtenir des informations sur la forme que peuvent prendre les plongements des graphes planaires cofinis non localement finis. Nous discutons alors des difficultés d’extension de la méthode “localement finie” au cas général.The cofinite graphs represent a family of graphs possessing a non-trivial group of symmetries, such as the Cayley graphs and the vertex-transitive graphs. When such graphs are planar, these symmetries correspond merely to symmetries of the plan in which the graphs are embedded. This set of symmetries – or, more precisely, automorphisms – can provide a global description of the graph from local data, by means of structures called labeling schemes. In this thesis, we study the groups of symmetries and describe the labeling schemes of the planar cofinite graphs possessing a simple topological representation : the planar locally finite graphs. We prove how a labeling scheme allows to characterize the graph and its embeddings. This analysis allows the enumeration of this family of the planar cofinite graphs, in particular when they are vertex-transitive or Cayley graphs. With these results, it is possible to analyze the structure of the groups of automorphisms of this family of graphs. There exist problems of the combinatorial group theory unsolvable in general that become solvable within this framework. That is the case in particular of the simple and generalized word problems. Problems of decidability of logics allow for the classification of these graphs into two families, depending on their treewidth and the geometry of ther embedding. Finally, we raise the question of the extension to the more general family of the cofinite planar graphs. The classification of these graphs in terms of number of ends and of accumulation points in their embeddings provides information on the structure of the embeddings of these more general graphs. We discuss the problems raised by the extension of the “locally finite” method to the general case