Let (X, Σ, μ) be a probability space, let f1, f2, ..., Fk be k σ-subalgebras of Σ, and let p ∊ R be such that 1 < p < + ∞. Let Pi :LP(X, Σ, μ) → LP(X, Σ, μ) be the conditional expectation operator corresponding to fi for every i = 1,2,…, k, and set T = P1 . . . Pk. Our goal in the note is to give a new and simpler proof of the fact that for every f ∊ LP(X, Σ, μ), the sequence (Tnf)n∊N converges in the norm topology of LP(X, Σ, μ), and that its limit is the conditional expectation of f with respect to f1 ∩ f2 ∩ … ∩ Fk.

The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.

Let d(n; l, k) be the number of positive divisors of n which lie in the arithmetic progression l mod k. Using the complex integration technique the formula

is proved. This formula holds uniformly in l, k and x satisfying 1 ≦ l ≦ k, (lx)1/2 ≦ k ≦ x1-∊; the exponent α ≦ 1/3.

Unique global strong solutions of a Cauchy problem arising in electrophoretic separation are constructed with arbitrary initial data in L1, thus generalizing an earlier global existence result. For small diffusion coefficients, the solutions can be viewed as approximate solutions for the corresponding zero-diffusion Riemann problem.

Nous montrons qu'il existe deux contractions T1 et T2 de classe C0 telles que T1 — T2 soit compact et une fonction intérieure h telle que h(T1) — h(T2) soit non compact.

For h € ℝ and ϕ : ℝ2 → ℝ define Lhϕ: ℝ2 → ℝ by (Lhϕ)(x, y) = ϕ (x + h, y) + ϕ (x — h,y) — ϕ (x,y + h) — ϕ (x,y — h) for all (x,y) € ℝ2. The aim of the paper is to establish the following "stability" theorem concerning the functional equation

if δ > 0, f : ℝ2 → ℝ and

then there exists ɛ > 0 ami ϕ : ℝ2 → ℝ such that (Lhϕ)(x, y ) = 0 for all x, y,h ∊ ℝ and

For each pair (m, n) of integers such that m+1 ≦ n ≦ 2m+1, we give, by original and straightforward methods, some examples of rings A' such that dim A' = m and dim A'[X] = n.

The class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

A characterization is given of those families ℱ of subsets of the positive integers for which σ(ℓ1, ) is sequentially complete, where denotes the linear span of the characteristic functions of ℱ. Consequences include some known matrix results, such as Hahn's Theorem and conditions required for a matrix to "sum" the characteristic functions of the lacunary sets.

We show that for a discrete ICC group G, which is not inner amenable and is such that has a weakly outer automorphism, is not isomorphic to .

A linear map A : C(T) → C(S) is called separating if f • g ≡ 0 implies Af • Ag = 0. We describe the general form of such maps and prove that any separating isomorphism is continuous.

We compare, under some assumptions on mass density, the eigenvalues of the Sturm-Liouville problems satisfying homogeneous Dirichlet and Neumann boundary condition.

Generalizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,

In this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.

Let M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

It is shown that if a trigonometric series is (R, 3), respectively (R, 4), summable then its (R, 3) sum, respectively (R, 4) sum, is James P3—, respectively P4—, integrable and that such series are Fourier series with respect to these integrals.

W. Deeb, R. Khalil and M. Marzuq have studied some properties of H(ϕ), the Hardy-Orlicz spaces. They introduced the functions class Np (0 < p ≦ 1 ) and discussed some properties of Np. In the present short note we prove that Np = N+ for 0 < p ≦ 1. We also give a condition of H(ϕ) = H(ψ).

We use the general factorisation theorems of Grothendieck, Nikishin and Maurey to characterise coefficient multipliers between Bergman spaces and into the Nevanlinna class.

Enochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.

The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.