Analysis of the Brylinski-Kostant model for spherical minimal representations

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra \goth g, and furthermore a real form {\goth g}_{\bboard R}. The connected and simply connected Lie group G_{\bboard R} with {\rm Lie}(G_{\bboard R})={\goth g}_{\bboard R} acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$Comment: 42 page

Self-adjoint extensions of operators and the teaching of quantum mechanics

For the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different physical situations. Some consequences are worked out, which could lead to experimental checks.Comment: 25 pages, Latex file, extended version of Am. J. Phys. 69 (2001) 32

A Discrete event model for multiple inhabitants location tracking

6 pagesInternational audienceSmart Home technologies are aiming to improve the comfort and safety of the inhabitants into their houses. To achieve this goal, online indoor location tracking of the inhabitants is often used to monitor the air conditioning, to detect dangerous situations and for many other applications. In this paper, it is proposed an approach to build a model allowing dynamic tracking of several persons in their house. A method to construct such a model by using finite automata and Discrete Event System (DES) paradigms is presented. An approach to reduce the size of the model is also introduced. Finally, an efficient algorithm for location tracking is proposed. For the sake of better understanding, an illustrative example is used throughout the paper

Analyse harmonique et fonctions d'ondes sphéroïdales

Notre travail est motivé par le problème de l'évaluation du déterminant de Fredholm d'un opérateur intégral. Cet opérateur apparait dans l'expression de la probabilité pour qu'un intervalle [?s, s] (s > 0) ne contienne aucune valeur propre d'une matrice aléatoire hermitienne gaussienne. Cet opérateur commute avec un opérateur différentiel de second ordre dont les fonctions propres sont les fonctions d'ondes sphéroïdales de l'ellipsoïde alongé. Plus généralement nous considérons l'opérateur de Legendre perturbé. Nous montrons qu'il existe un opérateur de translation généralisée associé à cet opérateur. En?n, par une méthode d'approximation des solutions de certaines équations différentielles, dite méthode WKB, nous avons obtenu le comportement asymptotique des fonctions d'ondes sphéroïdales de l'ellipsoïde alongé Il s'exprime à l'aide des fonctions de Bessel et d'Airy. Par la même méthode nous avons obtenu le comportement asymptotique des fonctions propres de l'opérateur dfférentiel d'Airy.Our work is motivated by the problem of evaluating the Fredholm determinant of an integral operator. This operator appears in the expression of the probability, for a random matrix in the Gaussien Unitary Ensemble, to have no eigenvalue in an interval [?s, s]. This operator commutes with a differential operator wich have the spheroidal wave functions as eingenfunctions. More generally, we consider the perturbated Legendre differential operator. We show that there exists a generalized translation operator associated to the perturbated Legendre dfferential operator. Finaly, by using the WKB method, we have determined the asymptotic behavior of the prolate spheroidal wave functions. This asymptotic behavior involves Bessel and Airy functions. By using the same method, we have obtained similar results for asymptotic behavior of the eigenfunctions of the Airy differential operator.PARIS-JUSSIEU-Bib.électronique (751059901) / SudocSudocFranceF