If λ = (λ1,…, λm) where λ1,…,λm are nonnegative integers with λ1 ≥…≥ λm, then λ is a partition of |λ| = λ1 + …+λm, and we write λ⊢ |λ|. The non-zero λi's are the parts of λ, so λ1 is the largest part, and ℓ(λ) is the number of parts of λ. Two partitions with the same parts, so they differ only in number of zeros, are the same. The set of all partitions, including the partition of 0 (with 0 parts) is denoted by The conjugate of λ, denoted by , is the partition (μ1,…, μk), in which μi is the number of λ's that are ≥i , for i = 1,…, k, where k=λ1.

Let X be a compact Hausdorff space, let C(X) be the algebra of complex-valued continuous functions on X, and let A be a uniform algebra on X. Fix a nonzero complex homomorphism τ on A and a representing measure m for τ on X. The abstract Hardy space Hp = Hp(m), 1 ≤ p ≤ ∞, determined by A is defined to the closure of Lp = Lp(m) when p is finite and to be the weak*-closure of A in L∞ = L∞(m) p = ∞.

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

Let p be a prime number. Let k be a field of characteristic different from p and containing the p-th roots of unity. Let be a finite group. Let L/k be a finite normal extension with Galois group and let c be a 2-cocycle on with coefficients in , where acts trivially on By Emb(L/k, c) we denote the question of the existence of a finite normal extension M of k, such that M contains L, such that [M: L] = p, and such that, denoting by the Galois group of M/k, the extension is given by the class of c.

Varadarajan [13] named a map f: A → X a cyclic map when there exists a map F: X × A → X such that for the folding map ∇X: X ∨ X → X. He defined the generalized Gottlieb set G(A, X) of the homotopy classes of the cyclic maps F: A → X and studied the fundamental properties of G(A, X) If A is a co-Hopf space, then the Varadarajan set G(A, X) has a group structure [13]. The group G(A,X) is a generalization of G(X) and Gn(X) of Gottlieb [2,3]. Some authors studied the properties of the Varadarajan set, its dual and related topics [4, 5, 6, 7,12,15,16,17].

Recently MacCluer and Shapiro [6] have characterized the compact composition operators in Hardy and weighted Bergman spaces of the disc, and MacCluer [5] has made an extensive study of these opertors in the unit ball of Cn. Angular derivatives and Carleson measures have played an essential role in these studies. In this article we study composition operators in poly discs and characterize those operators which are bounded or compact in Hardy and weighted Bergman spaces. In addition to Carleson measure theorems resembling those of [5], [6], we give necessary and sufficient conditions satisfied by the maps inducing bounded or compact composition operators. We conclude by considering the role of angular derivatives on the compactness question explicitly.

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.

Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

By P(ω) we denote the Boolean algebra of all subsets of the set ω of natural numbers. We identify each natural number with the set of its predecessors and define: the ideal of sets of density zero, and the ideal of sets of logarithmic density zero.

Let H be a finite set; we can suppose H = { 1, 2, …,d } . Consider Hn, the set of length n words over the alphabet H. For every A ∈ Hn define the waiting time for A as the number of experiments needed till A appears as a connected sub-sequence of random elements of H. Formally, let X1, X2,... be i. i. d. random variables, P(X1 = i) = d−1, 1 ≤ i ≤ d then

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).