Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures improved, references update

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Hub loads analysis of the SA349/2 helicopter

The forces and moments at the rotor hub of an Aerospatiale SA349/2 helicopter were investigated. The study included three main topics. First, measured hub forces and moments for a range of level flight conditions (mu = 0.14 to 0.37) were compared with predictions from a comprehensive rotorcraft analysis to examine the influence of the wake model on the correlations. Second, the effect of changing the blade mass distribution and blade chordwise center of gravity location on the 3/rev nonrotating frame hub loads was studied for a high-speed flight condition (mu = 0.37). Third, the use of higher harmonic control to reduce nonrotating frame 3/rev hub shear forces was investigated. The last two topics were theoretical studies only

The level set method for the two-sided eigenproblem

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomia

Réalisation d'un banc de tests de sources d'ions

La physique nucléaire a permis de comprendre de nombreuses propriétés des noyaux stables (masse, forme, structure nucléaire...) et d'élaborer des théorie capables de reproduire ces propriétés. Les recherches des vingt dernières années dans ce domaine ont montré que le comportement des noyaux exotiques est souvent différent de ce que l'on peut extrapoler des théories sur les noyaux stables. Les noyaux exotiques représentent plus de 80% de la carte des noyaux. Les physiciens nucléaires ont vite exprimé le besoin d'utiliser des faisceaux d'ions radioctifs pour étudier leurs propriétés. C'est ainsi qu'a été décidée la construction d'un nouvel accélérateur au GANIL capable de fournir des faisceaux exotiques: c'est le projet SPIRAL ou "Système de Production d'Ions Radioactifs Accélérés en Ligne". Un faisceau d'ions stables de haute énergie provenant des accélérateurs du GANIL est stoppé dans une cible de graphite, provocant des réactions nucléaires. Les atomes radioactifs produits diffusent hors de la cible portée à une température d'environ 2000°C et sont ionisés dans une source ECR (Electronic Cyclotronic Resonance). Le faisceau extrait est enfin accéléré dans le cyclotron CIME (Cyclotron à Ions de Moyenne Energie) avant d'être distribué dans les salles expérimentales. De nombreux domaines de recherche fondamentale sont concernés par les expériences avec ces faisceaux radioactifs: Physique nucléaire (matière nucléaire diluée, cohésion des forces nucléaires, noyaux super-lourds , symétrie dans les noyaux, nombres magiques), théorie du modèle standard, astrophysique nucléaire, physique de la matière condensée, médecine... L'ensemble cible-source (ECS) de SPIRAL est le point crucial du système: il produit des atomes radioactifs, puis les ionise pour former le faisceau d'ions radioactifs. En fonctionnement, la réaction de production des atomes radioactifs génére un flux de neutrons intenses. Ce flux détériore les caractéristiques magnétiques des aimants permanents de la source d'ions et active également tous les matériaux présents dans la casemate de production. Ainsi, la durée de vie d'un ECS est estimée à quinze jours et la casemate est un local où les rayonnements seront toujours présents. La mise en place d'un ensemble neuf est entièrement automatisée pour éviter tout travail sous rayonnement. La préparation de l'ECS neuf doit donc être optimale avant sa mise en place dans la casemate. Ceci nécessite la réalisation d'un banc de tests afin d'assurer les connexions, le fonctionnement de l'ECS et le dégazage des cibles

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur

Reachability problems for products of matrices in semirings

We consider the following matrix reachability problem: given $r$ square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied at left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any $r\geq 2$ is equivalent to the specialization to $r=2$. As an application of this result and of a theorem of Krob, we show that when $r=2$, the vector and matrix reachability problems are undecidable over the max-plus semiring $(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are ``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.Comment: 21 page

Assimilation of IASI partial tropospheric columns with an Ensemble Kalman Filter over Europe

Partial lower tropospheric ozone columns provided by the IASI (Infrared Atmospheric Sounding Interferometer) instrument have been assimilated into a chemistry-transport model at continental scale (CHIMERE) using an Ensemble Square Root Kalman Filter (EnSRF). Analyses are made for the month of July 2007 over the European domain. Launched in 2006, aboard the MetOp-A satellite, IASI shows high sensitivity for ozone in the free troposphere and low sensitivity at the ground; therefore it is important to evaluate if assimilation of these observations can improve free tropospheric ozone, and possibly surface ozone. The analyses are validated against independent ozone observations from sondes, MOZAIC<sup>1</sup> aircraft and ground based stations (AIRBASE – the European Air quality dataBase) and compared with respect to the free run of CHIMERE. These comparisons show a decrease in error of 6 parts-per-billion (ppb) in the free troposphere over the Frankfurt area, and also a reduction of the root mean square error (respectively bias) at the surface of 19% (33%) for more than 90% of existing ground stations. This provides evidence of the potential of data assimilation of tropospheric IASI columns to better describe the tropospheric ozone distribution, including surface ozone, despite the lower sensitivity. <br><br> The changes in concentration resulting from the observational constraints were quantified and several geophysical explanations for the findings of this study were drawn. The corrections were most pronounced over Italy and the Mediterranean region, we noted an average reduction of 8–9 ppb in the free troposphere with respect to the free run, and still a reduction of 5.5 ppb at ground, likely due to a longer residence time of air masses in this part associated to the general circulation pattern (i.e. dominant western circulation) and to persistent anticyclonic conditions over the Mediterranean basin. This is an important geophysical result, since the ozone burden is large over this area, with impact on the radiative balance and air quality. <br><br><br> <sup>1</sup> Measurements of OZone, water vapour, carbon monoxide and nitrogen oxides by in-service AIrbus airCraft (<a href="http://mozaic.aero.obs-mip.fr/web/"target="_blank">http://mozaic.aero.obs-mip.fr/web/</a>)

Allelopathic interactions between the brown algal genus Lobophora (Dictyotales, Phaeophyceae) and scleractinian corals

Allelopathy has been recently suggested as a mechanism by which macroalgae may outcompete corals in damaged reefs. Members of the brown algal genus Lobophora are commonly observed in close contact with scleractinian corals and have been considered responsible for negative effects of macroalgae to scleractinian corals. Recent field assays have suggested the potential role of chemical mediators in this interaction. We performed in situ bioassays testing the allelopathy of crude extracts and isolated compounds of several Lobophora species, naturally associated or not with corals, against four corals in New Caledonia. Our results showed that, regardless of their natural association with corals, organic extracts from species of the genus Lobophora are intrinsically capable of bleaching some coral species upon direct contact. Additionally, three new C-21 polyunsaturated alcohols named lobophorenols A-C (1-3) were isolated and identified. Significant all elopathic effects against Acropora muricata were identified for these compounds. In situ observations in New Caledonia, however, indicated that while allelopathic interactions are likely to occur at the macroalgal-coral interface, Lobophora spp. rarely bleached their coral hosts. These findings are important toward our understanding of the importance of allelopathy versus other processes such as herbivory in the interaction between macroalgae and corals in reef ecosystems