The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

In this paper we prove Pesin’s entropy formula for general C2 (or C1+α) (non-invertible) endomorphisms of a compact manifold preserving a smooth measure.

La recherche de la structure des espaces homogènes kählériens d’un groupe de Lie semi-simple fait l’objet de plusieurs travaux récents. L’étude détaillée a été faite au cas compact. Il a été montré en particulier que ces espaces homogènes compacts sont des variétés algébriques (même rationnelles) et simplement connexes.

Le but essentiel de ce travail est de montrer que tout espace homogène kählérien compact est produit direct d’un tore complexe et d’un espace homogène kählérien d’un groupe de Lie compact semi-simple (Théorème 3). Pour ce but nous étudierons au paragraphe 1 la structure des espaces homogènes symplectiques d’un groupe de Lie réductif. La structure et la situation du groupe d’isotropie seront clarifiées dans le théorème 1. Au paragraphe 2 on en déduit un théorème sur la structure des espaces homogènes kählériens d’un groupe de Lie réductif (Théorème 2). Le théorème 3 est une conséquence immédiate du théorème 2.

We give an explicit determinant formula for a class of rational solutions of the Painlevé V equation in terms of the universal characters.

Let k be an algebraically closed field of characteristic p > 0, and let X be a curve defined over k. The aim of this paper is to study the behavior of the Frobenius map F*: H1(X, E) → H1(X, F*E) for a vector bundle E.

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

In [10], P. Montel defined the notion of a quasinormal family of meromorphic functions and obtained several results relating to this. Afterwards, in [13], H. Rutishauser generalized some of them to the case of meromorphic functions of several variables. By definition, a quasi-normal family of meromorphic functions on a domain D in Cn is a family such that any sequence in has a subsequence which converges compactly outside a thin analytic subset of D.

In this paper we are dealing with the following problem: determine all normal (or smooth) projective varieties X over an algebraically closed field k supporting a given variety Y as an ample Cartier divisor.

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

Ce travail est consacré aux groupes d’automorphismes de certaines algebres quantiques de dimension 2 ou 3. Dans la théorie classique des algebres enveloppantes, si désigne l’algèbre de Lie de Heisenberg de dimension 3, U() admet l’algèbre de Weyl A1 comme seul quotient primitif de dimension 2, avec les propriétés suivantes: d’ une part tout automorphisme de A1 se relève en un automor-phisme de U(), d’autre part U() admet des automorphismes non modérés (cf. [A1], [Di1], [ML]). On retrouve la même situation pour les quotients primitifs minimaux de U(sl(2)) paramétrés par C (cf. [Di2], [Jo]). En outre, dans ce cas, on dispose des plongements de Conze de ces quotients dans A1 (cf. [Di2], [Co], [Ro]); bien que les groupes d’automorphismes soient comparables, la restriction correspondant à un tel plongement est seulement définie sur le sous-groupe des automorphismes triangulaires de A1 (cf. annexe).

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.

In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.

A locally strongly convex hypersurface in the affine space Rn + 1 is called an affine hypersphere if the affine normals (§ 1) through each point of the hypersurface either all intersect at one point, called its center, or else are all mutually parallel. It is called elliptic, parabolic or hyperbolic according to whether the center is, respectively, on the concave side of the hypersurface, at infinity or on the convex side. This class of hypersurfaces was first studied systematically by W. Blaschke ([1]) in the frame of affine geometry. In his paper [3] E. Calabi redefined it and proposed a problem of determining all complete hyperbolic affine hyperspheres and raised a conjecture that these hypersurfaces are asymptotic to the boundary of a convex cone and every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature.

The homological dimension of a module MR is often related to the cardinality of a set of generators for M or for right ideals of R. In this note, upper bounds for this homological dimension are obtained in two situations.

Throughout this paper, a space means a T1-space. A space is called fully normal if every open covering of it has a Δ-refinement , that is, an open covering for which the stars (x, ) form a covering which refines . A space is called paracompact if every open covering of it has a locally finite (= neighborhood finite) open covering which refines . It is well known that paracompactness is identical with full normality in a Hausdorff space ([3], [7]).

Nevanlinna’s lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (cf., e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves f: C → M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kähler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves f: C → V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for f and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinna’s lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23].

The present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary then

The common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:

in particular ε(0) is the Fock vacuum.

Eine wichtige Aufgabe der Darstellungstheorie ist die Aufstellung eines vollstandigen Systems von inäquivalenten irreduziblen Darstellungen einer endlichen Gruppe Obwohl man Verfahren kennt, die dies theoretisch leisten (etwa die Reduktion der regularen Darstellung), ist die praktische Durchführung nur in wenigen Fallen möglich. Die Aufgabe vereinfacht sich jedoch wesentlich, wenn sich alle irreduziblen Darstellungen von auf monomiale Gestalt transfor-mieren lassen; denti alle monomialen Darstellungen einer endlichen Gruppe kann man nach einem sehr einfachen Verfahren gewinnen (siehe [4], S. 140).