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Ce mémoire porte sur quelques notions appropriées d'actions de groupe sur les variétés symplectiques, à savoir en ordre décroissant de généralité : les actions symplectiques, les actions faiblement hamiltoniennes et les actions hamiltoniennes. Une connaissance des actions de groupes et de la géométrie symplectique étant prérequise, deux chapitres sont consacrés à des présentations élémentaires de ces sujets. Le cas des actions hamiltoniennes est étudié en détail au quatrième chapitre : l'importante application moment y est définie et plusieurs résultats concernant les orbites de la représentation coadjointe, tels que les théorèmes de Kirillov et de Kostant-Souriau, y sont démontrés. Le dernier chapitre se concentre sur les actions hamiltoniennes des tores, l'objectif étant de démontrer le théorème de convexité d'Atiyha-Guillemin-Sternberg. Une discussion d'un théorème de classification de Delzant-Laudenbach est aussi donnée. La présentation se voulant une introduction assez exhaustive à la théorie des actions hamiltoniennes, presque tous les résultats énoncés sont accompagnés de preuves complètes. Divers exemples sont étudiés afin d'aider à bien comprendre les aspects plus subtils qui sont considérés. Plusieurs sujets connexes sont abordés, dont la préquantification géométrique et la réduction de Marsden-Weinstein., This Master thesis is concerned with some natural notions of group actions on symplectic manifolds, which are in decreasing order of generality : symplectic actions, weakly hamiltonian actions and hamiltonian actions. A knowledge of group actions and of symplectic geometry is a prerequisite ; two chapters are devoted to a coverage of the basics of these subjects. The case of hamiltonian actions is studied in detail in the fourth chapter : the important moment map is introduced and several results on the orbits of the coadjoint representation are proved, such as Kirillov's and Kostant-Souriau's theorems. The last chapter concentrates on hamiltonian actions by tori, the main result being a proof of Atiyah-Guillemin-Sternberg's convexity theorem. A classification theorem by Delzant and Laudenbach is also discussed. The presentation is intended to be a rather exhaustive introduction to the theory of hamiltonian actions, with complete proofs to almost all the results. Many examples help for a better understanding of the most tricky concepts. Several connected topics are mentioned, for instance geometric prequantization and Marsden-Weinstein reduction.

A sharp upper bound is given for the degree of nonconvexity of the partition range of a finite-dimensional vector measure, in terms of the maximum (one-dimensional) mass of the atoms of that measure. This upper bound improves on a bound of Hill and Tong (1989, Anal. Stat.17, 1325–1334) by an order of magnitude n. Its proof uses several ideas from graph theory, combinatorics, and convex geometry. Applications are given to optimal-partitioning and fair division problems.

Minimax risk inequalities are obtained for the location-parameter classification problem. For the classical single observation case with continuous distributions, best possible bounds are given in terms of their Lévy concentration, establishing a conjecture of Hill and Tong (1989). In addition, sharp bounds for the minimax risk are derived for the multiple (i.i.d.) observations case, based on the tail concentration and the Lévy concentration. Some fairly sharp bounds for discontinuous distributions are also obtained.

Let M =G/H be a semisimple symmetric space,τ the corresponding involution and D =G/K the Riemannian symmetric space. Then we show that the followingare equivalent: M is of Hermitian type; τ induces a conjugation on D; thereexists an open regular H-invariant cone Ω in q =h[bottom] such that k ∩ Ω ≠ 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under τ and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory.

\noindent We consider eigenvalue problems for the vibrating string \vspace{-0.1cm} \[ u^{\prime \prime }(x)+\lambda \rho (x)u(x)=0,\;\;\;\;u(0)=u(a)=0 \] \vspace{-0.6cm} \noindent where the density $\rho (x)\;$is a positive continuous function on $% [0,a]$. Let $\lambda _{n}(t)$ be the $n$th eigenvalue of the string with $\rho =\rho (x,t)$. A classical convexity theorem of P\'{o}lya and Schiffer states that for any $k\geq 1$, the sum $% \sum_{n=1}^{k}\frac{1}{\lambda _{n}(t)}\;$is a convex function of $t$ if $\rho (x,t)$ is convex with respect to $t$. In this paper, we shall give a different approach to this result based on variational analysis. The ideas used also lead to applications in the case of symmetric densities and in the case of concave densities.

Optimal-partitioning and minimax risk inequalities are obtained for the classification and multi-hypotheses testing problems. Best possible bounds are derived for the minimax risk for location parameter families, based on the tail concentrations and Levy concentrations of the distributions. Special attention is given to continuous distributions with the maximum likelihood ratio property and to symmetric unimodal continuous distributions. Bounds for general (including discontinuous) distributions are also obtained.