Théorie des genres des corps globaux

We establish the fundamental results of genus theory for finite (non necessary Galois) extensions of global fields by using narrow S-class groups, when S is an arbitrary finite set of places. This exposition, which involves both the number fields and the functions fields cases, generalizes most classical results on this subject. Received: 8 February 1999 / Revised version: 17 December 1999     

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Thermostatische Zustandsänderungen des trockenen und des nassen Dampfes

Summary The change of states of a closed thermodynamic system having two degrees of freedom can be expressed analytically by means of the isentropic and the isenthalpic exponents, suitably defined. By imposing certain plausible conditions on these exponents, it can be shown that many well-known formulae, originally derived for an ideal gas with constant specific heat, become applicable more generally. Low pressure wet steam of not too low dryness fraction as occurred in the low-pressure stages of condensing turbines obeys these imposed conditions fairly well.The problem of supersaturation and of the drift toward equilibrium has been briefly touched upon.     

Automorphismengruppen des allgemeinen Kreiskegels und des zugehörigen Halbraumes

Ohne Zusammenfassung     

Groupoïdes quantiques

The aim of this paper is to introduce a notion of quantum groupoid, non-commutative analogue of Lie groupoids. This notion generalizes simultaneously the notion of a Hopf bigebroid [Mal] (corresponding to a quantum groupoid with commutative space of units) and the notion of a braided quantum group of Majid [Maj]. The commutativity hypothesis of the base in [Mal] is no more necessary and a construction of L. Vainerman [V] enters in this frame.     

Approximation des ensembles ω-limites des difféomorphismes par des orbites périodiques

Let M be a compact manifold, D the set of its C¹-diffeomorphisms (possibly symplectic or volume preserving). We prove that there exists a dense G_δ of D such that if , every ω-limit set of f is the limit (for the Hausdorff topology) of a sequence of periodic orbits. This has certain interesting consequences concerning the structure of the ω-limit sets. Moreover, we define a new notion of attractors and describe them precisely in different cases.     

Verallgemeinerung des Verbandsbegriffs durch Abschwächung des Axioms der Idempotenz

Ohne Zusammenfassung     

Propagation des singularités des courants positifs fermés

Given a closed positive currentT on a bounded Runge open subset Ω ofC ⁿ , we study sufficient conditions for the existence of a global extension ofT toC ⁿ . WhenT has a sufficiently low density, we show that the extension is possible and that there is no propagation of singularities, i.e.T may be extended by a closed positiveC ^∞-form outside \(\bar \Omega \) . Conversely, using recent results ofH. Skoda andH. El Mir, we give examples of non extendable currents showing that the above sufficient conditions are optimal in bidegree (1, 1).     

Über die Dimension des singulären Ortes des Modulraumes Mg

The modulus space M^g of compact Riemann surfaces of genus g2 is a normal complex space of dimension 3g–3. In this paper we give necessary and sufficient conditions for the singular locus S^g of M^g to contain points of dimension 0, i. e. isolated singularities, and points of dimension 1. Further we give examples for Riemann surfaces representing those singular points.     

Coût des relations d’équivalence et des groupes

We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra. Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999     

Algébricité des partitions markoviennes des automorphismes hyperboliques du tore

We prove that, to every Markov partition for a hyperbolic automorphism of the torus, one can associate an algebraic sofic coding similar to the one constructed in [6].     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

Anwendung des Galerkinschen Verfahrens auf nichtkonservative Stabilitätsprobleme des elastischen Stabes

Summary The paper deals with the application ofGalerkin's method to non-conservative stability problems of elastic bars. Two results established byBeck andPflüger by means of other methods are verified.     

Invariance des probabilités de retour sur des groupes localement compacts

We construct an asymptotic invariant for locally compact separable, compactly generated unimodular groups. If G is such a group and if F is a symmetric bounded density on it with second order moment (with respect to a word metric), we show that the asymptotic behavior of $n?F^{1(2n)}(e)$ does not depend on the choice of the density F. As an example we show that the asymptotic of the return probabilities on $\mathrm{sol}(K)$ where K is a p-field behaves like $\exp (?t^{(1/3)})$. In the case where $K={\mathbb{Q}}_p$ this answers a question of Pittet and Saloff-Coste published recently by Mustapha. To cite this article: D. Gretete, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Abschwächungen des Adjunktionsbegriffs

Generalizations of right adjointness (i.e. having a left adjoint) of a functor G:AX are studied. G is called weakly right adjoint, if for any X ObX there exists an A_X ObA and an arrow e_X:X GA_X, such that for any f:XGB there is a (not necessarily unique) morphism f:A_XB inA with (Gf)e_x=f. As weakly right adjoint functors do not have so many interesting properties, it is useful to consider weakly right adjoint functors with a certain uniqueness condition. There are three ways for doing this, first by assuming uniqueness only for special f' s, second by assuming uniquness only up to automorphisms, and third by assuming a canonical choice of f. A different way of generalizing right adjointness are the locally adjunctable functors of Kaput [5]. These weaker notions of adjointness are compared, their continuity properties are studied and the problem, when they imply right adjointness is discussed.     

Über eine ausgezeichnete Eigenschaft der Koeffizientenmatrizen des Laguerreschen und des Hermiteschen Polynomsystems

Ohne Zusammenfassung