The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.Comment: 29 pages, 8 figure

Impostazione e realizzazione di un Sistema di Gestione Ambientale nell'ambito di un progetto di registrazione EMAS

La presente relazione tratta il lavoro svolto presso la S.p.A. Cartiere Modesto Cardella per l’impostazione e la realizzazione di un Sistema di Gestione Ambientale (SGA) nell’ambito di un progetto di registrazione EMAS. Tale percorso si inserisce all’interno di un progetto più ampio denominato PIONEER, che sperimenta l’applicazione di una metodologia per registrare un distretto omogeneo. Il percorso di registrazione prevede una serie di attività, alcune delle quali propedeutiche all’obiettivo del lavoro ed altre direttamente conseguenti. In particolare con il mio progetto, iniziato con il conoscere la realtà produttiva della cartiera, ho completato l’Analisi Ambientale Iniziale e realizzato tutte le procedure gestionali e operative, necessarie per il rispetto della norma ISO 14001 richiesta dal regolamento EMAS II. Al termine dello sviluppo del SGA, ho anche redatto la prima versione del Manuale del SGA

Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983

Convergence in shape of Steiner symmetrizations

There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.Comment: 11 page

Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional $$J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.$$ We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.Comment: 36 pages, 4 figure

Validation of rotational thromboelastometry during cardiopulmonary bypass : a prospective, observational in-vivo study

Le ROTEM est un test de coagulation réalisable au près du malade qui permet d'objectiver la coagulopathie, de distinguer la contribution des différents éléments du système de coagulation et de cibler les produits procoagulants comme le plasma frais congelé (PFC), les plaquettes, le fibrinogène et les facteurs de coagulation purifiés ou les antifibrinolytiques. 3 des tests disponibles pour le ROTEM sont: EXTEM, INTEM, HEPTEM. Le premier test est stable sous hautes doses d'héparine alors que le deuxième est très sensible à sa présence. Dans le dernier test on rajoute de l'héparinase pour mettre en évidence l'éventuel effet résiduel de l'héparine en le comparant à l'INTEM. Idéalement, le ROTEM devrait être effectué avant la fin du bypass cardiopulmonaire (CEC), donc sous anticoagulation maximale pas héparine, afin de pouvoir administrer des produits pro¬coagulants dans les délais les plus brefs et ainsi limiter au maximum les pertes sanguines. En effet la commande et la préparation de certains produits procoagulants peut prendre plus d'une heure. Le but de cette étude est de valider l'utilisation du ROTEM en présence de hautes concentrations d'héparine. Il s'agit d'une étude observationnelle prospective sur 20 patients opérés électivement de pontages aorto-coronariens sous CEC. Méthode : l'analyse ROTEM a été réalisée avant l'administration d'héparine (TO), 10 minutes après l'administration d'héparine (Tl), à la fin de la CEC (T2) et 10 minutes après la neutralisation de l'anticoagulation avec la protamine (T3). L'état.d'héparinisation a été évalué par l'activité anti-Xa à T1,T2,T3. Résultats : Comparé à TO, la phase de polymérisation de la cascade de coagulation et l'interaction fibrine-plaquettes sont significativement détériorées par rapport à Tl pour les canaux EXTEM et HEPTEM. A T2 l'analyse EXTEM et INTEM sont comparables à celles de EXTEM et HEPTEM à T3. Conclusion: les hautes doses d'héparine utilisées induisent une coagulopathie qui reste stable durant toute la durée de la CEC et qui persiste même après la neutralisation de l'anticoagulation. Les mesures EXTEM et HEPTEM sont donc valides en présence de hautes concentrations d'héparine et peuvent être réalisés pendant la CEC avant l'administration de protamine