In this article our purpose is to prove a differentiation theorem for multiparameter processes which are strongly superadditive with respect to a strongly continuous semigroup of positive L 1 contractions (see Section 1 for definitions).

Recently, the differentiation theorem for superadditive processes with respect to a one-parameter semigroup of positive L 1-contractions has been proved by D. Feyel [9]. Another proof is given by M. A. Akçoğlu [1]. R. Emilion and B. Hachem [7] also proved the same theorem, but with an extra assumption on the process (see also [1]). The proof of this theorem for superadditive processes with respect to a Markovian semigroup of operators on L 1 is given by M. A. Akçoğlu and U. Krengel [4]. Thus [1] and [9] extend the result of [4] to the sub-Markovian setting. Here we will obtain the multiparameter sub-Markovian version of this theorem, namely Theorem 3.17 below

Let M be a σ-finite von Neumann algebra and {σt }t∊T be a σ-weakly continuous representation of the unit circle, T, as *-automorphisms of M. Let H ∞(σ) be the set of all x ∊ M such that

The structure of H ∞(σ) was studied by several authors (see [2-13]).

The main object of this paper is to study the σ-weakly closed subalgebras of M that contain H ∞(σ). In [12] this was done for the special case where H ∞(σ) is a nonselfadjoint crossed product.

Let Mn , for n ∊ Z, be the set of all x ∊ M such that

Let D be a bounded domain in the Euclidean space R N(N ≦ 2) and let where is the closure of D. We assume that the boundary ∂G of G is smooth. Consider the boundary value problem (abbreviated to BVP in the sequel).

(1)

(1)

where A is a nonlinear elliptic differential operator in divergence form of Leray-Lions type, ∇u = grad u, f is a distribution on G and p(x, t, η) is a function defined on G × R × R N . Among other hypotheses we shall, roughly speaking, assume that p has completely unrelated growth rates in the first and the third variables. In this paper we prove the solvability of the BVP (1), (2) under the assumption that it has both an upper solution ψ and a lower solution φ with φ ≧ ψ.

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.

Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G) 4.

A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are

• (i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and

• (ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K 0(X)), construit à l'aide des -modules localement libres sur un schéma X[14]. La somme directe et le produit tensoriel de modules font de K 0(X) un anneau, et les opérations de puissances extérieures lui fournissent une structure supplémentaire, que Grothendieck appelle λ-anneau. Un λ-anneau est muni d'une filtration décroissante, la γ-filtration, et un des principaux résultats de Grothendieck est que, si X est lisse sur un corps, le groupe

est isomorphe, à la torsion près, au groupe de Chow CH 1(X) des cycles de codimension i sur X, modulo l'équivalence linéaire.

The main objective of this paper is to find useful q-extensions of Feldheim's [6] bilinear formula for Jacobi polynomials, namely,

1.1

where the Appel function F 4 is defined by

1.2

α 1, α 2, ρ are arbitrary complex parameters such that the series on both sides of (1.1) are convergent, and

1.3

is the Jacobi polynomial of degree k, (a)k being the usual shifted factorial.