Comodules over weak multiplier bialgebras

This is a sequel paper of arXiv:1306.1466 in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.Comment: LaTeX source, 47 page

Canonical coordinates for the planetary problem

In 1963, V. I. Arnold stated his celebrated Thorem on the Stability of Planetary Motions. The general proof of his wonderful statement (that he provided completely only for the particular case of three bodies constrained in a plane) turned out to be more difficult than expected and was next completed by J. Laskar, P. Robutel, M. Herman, J. F\'ejoz, L. Chierchia and the author. We refer the reader to the technical papers \cite{arnold63, laskarR95, rob95, maligeRL02, herman09, fej04, pinzari-th09, ChierchiaPi11b} for detailed information; to \cite{fejoz13, chierchia13, chierchiaPi14}, or the introduction of \cite{pinzari13} for reviews. The complete understanding of Arnold's Theorem relied on an analytic part and a geometric one, both highly non trivial. Of such two aspects, the analytic part was basically settled out since \cite{arnold63} (notwithstanding refinements next given in \cite{fejoz04, chierchiaPi10}). The geometrical aspects were instead mostly unexplored after his 1963's paper and have been only recently clarified \cite{pinzari-th09, chierchiaPi11b}. In fact, switching from the three--body case to the many--body one needed to develop new constructions, because of a dramatic degeneracy due to its invariance by rotations, which, if not suitably treated, prevents the application of Arnold's 1963's strategy. The purpose of this note is to provide a historical survey of this latter part. We shall describe previous classical approaches going back to Delaunay, Poincar\'e, Jacobi and point out more recent progresses, based on the papers \cite{pinzari-th09, chierchiaPi11a, chierchiaPi11b, chierchiaPi11c, pinzari13, pinzari14}. In the final part of the paper we discuss a set of coordinates recently found by the author which reduces completely its integrals, puts the unperturbed part in Keplerian form, preserves the symmetry by reflections and is regular for zero inclinations.Comment: 27 pages. Final version. Introduction slightly modified and misprints correcte

René Géronimo Favaloro : pioneer of Cardiac Surgery

Dr. René G. Favaloro moved to the Cleveland Clinic in 1962 and proceeded to reshape the face of cardiac surgery as we knew it. Together with his colleagues at the Cleveland Clinic, Drs. Effler, Sones, Proudfit, Groves, Sheldon and countless others, he contributed to the double internal mammary arterymyocardial implantation by the Vineberg method, and by May 1967, he reconstructed the right coronary artery by the saphenous vein graft interposition. These landmark procedures paved the way for the aorto-coronary saphenous vein bypass graft in October 1967. Many similar breakthroughs ensued, with the application of the bypass technique to the left coronary artery, the combination of coronary artery bypass graft with left ventricular reconstruction and valve repair/replacement and finally, by December, a double bypass to the right coronary artery and anterior descending branch of the left coronary artery. In June, 1971, Dr. Favaloro decided to leave the Cleveland Clinic and return to Argentina where he created a medical centre, a teaching unit, a research department and finally an Institute of Cardiology and Cardiovascular Surgery. This was his greatest personal ambition. Over and above his brilliant mind and craft, Dr. Favaloro was a man of integrity, courage, honesty and humility, whose name will never cease to reverberate throughout the history of medicine.peer-reviewe

Search for SM and MSSM Higgs Bosons at LEP

Latest results from the LEP Collaboration on searches for neutral Higgs bosons predicted by the Standard Model and its minimal supersymmetric extension, the MSSM, are summarized.Comment: Talk presented at SUSY 2003: Supersymmetry in the Desert, University of Arizona, Tucson, AZ, June 5-10, 200