Guided by an observation of Hausdorff ([4]; reproduced by Whittaker and Robinson [6, pp. 177–178]), I pointed out a long time ago [7] that his “Fourier“ treatment of certain products can be systematized so as to apply to an inclusive class of infinite convolutions. Recently I noticed [8] that an appropriate application of this method supplies the following curious result on gamma-quotients:

Corresponding to every index θ on the range 0 < θ < 1, there exists on the line - ∞ < t < ∞ a monotone function μ = μθ = μθ(t) in terms of which the identity

holds on the half-plane Re z > -1 and so, in particular, on the half-line z ≧0.

In this paper, which is a continuation of [4], the necessary theoretical background is given to enable the calculation of the irreducible Brauer projective characters of a given finite group to be carried out. As an example, this calculation is done for the alternating group A (7) in §3. In a future paper the calculations for the Mathieu groups will be presented.

In § 2 a number of integrals in which the integrand contains a product of a hypergeometric function and an E-function will be evaluated. The following formulae will be employed in the proofs.

If ρ +σ = α + β + γ + 1, and if α, β or γ is zero or a negative integer,

this is Sallschütz's theorem [1].

If R(γ - ½α - ½β)< - ½,

This theorem was given by Wastson [2] for negative integral values of α and later by Whipple [3] for general values α.

R(γ) > 0,

This formula was given by Whipple [3]

l is a positive integer,

In a number of recent papers, we have developed an abstract approach to dual orthogonal series (see [1], [2], [3], and [4]). Such series arise in crack theory, heat transfer, etc. In this paper, we generalize these results to triple orthogonal series. We also show, via a counterexample, that, surprisingly, the results in the dual case are not generalizable as completely as expected.

In this note we prove that the equation x2 + 1 = yn, x, y, n ɛ ℕ, n>2, has no solutions (x, y, n)with 2 × y. Moreover, all solutions (x, y, n)of the equation with 2| y satisfy n < 5. 106 and y < exp exp exp 30.

Let f(x) and g(x) be real functions defined on the interval [a, b], with f(x) at least twice continuously differentiable, f′(x) monotone increasing, and f(x) of bounded variation. We consider the exponential integral

where e(t) denotes exp 2πit. The purpose of this note is to prove sharp forms of the well-known estimates:

A: If f′(x) is nonzero on [a, b], then I has order of magnitude

The constant of proportionality depends on the function g(x).

B: If f′(x) changes sign at x = c with a < c < b, then

Dr V. O. Ennola has pointed out that there is a mistake in this paper. The inequality for dox at the foot of page 76 does not imply the stated inequality between the second and third lines on page 77. This error vitiates the entire argument, and I have not so far been able to put it right.

Let G be a finite group with neutral element e which operates trivially on the multiplicative group R* of a commutative ring with identity 1. Let H2(G, R*) = Z2(G, R*)/B2(G, R*) denote the second cohomology group of G with respect to the trivial G-module R*. With every factor system (2-cocycle) f ∈ Z2(G, R*) we associate the so called (central) twisted group algebra (R, G, f) of G over R (see [4, Chapter V, 23.7] or [13, §4] for a definition). If f is cohomologous to f', then the R-algebras (R, G, f) and (R, G, f′) are isomorphic. Hence, up to R-algebra isomorphism, (R, G, f) is determined by the cohomology class f∈H2(G, R*) determined by f. If R = k is a field of characteristic not dividing the order |G| of G, then a computation of the discriminant of (k, G, f) shows that (k, G, f) is semisimple (see [13, 4.2]).

In [1], Pall proved an interesting result on a certain class of 2 × 2 integral matrices. He showed that the semigroup of 2 × 2 matrices of determinant 1 and non-negative entries contains exactly 2 primes , and every other non-unit is expressible uniquely as products of these primes. Before formally stating this result, we need some notation. Let Gn denote the semigroup of n × n matrices with determinant 1 and nonnegative integral entries, In the n × n identity matrix, the n × n matrix with a 1 as its (i, j) element and zeros elsewhere, and let . When the dimension is clear, we shall drop the superscripts.

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.

It is well known that when the characteristic p(≠ 0) of a field divides the order of a finite group, the group algebra possesses a non-trivial radical and that, if p does not divide the order of the group, the group algebra is semi-simple. A group algebra has a centre, a basis for which consists of the class-sums. The radical may be contained in this centre; we obtain necessary and sufficient conditions for this to happen.

The paper is related to the area which was recently called topological homology [3, 6, 12, 16, 4]. We consider questions associated with the central Hochschild cohomology of C*-algebras. The study of the latter was begun by J. Phillips and I. Raeburn in [9, 10], when they were investigating some problems of the theory of perturbations of C*-algebras. In [8] we obtained a description of the structure of C*-algebras with central bidimension zero: it was proved that these C*-algebras are unital and have continuous trace. In the special case of separable and a priori unital C*-algebras this statement was proved by J. Phillips and I. Raeburn in [11] with the help of a different approach. The question was raised. Which values can the central bidimension of C*-algebras take? In the present paper it is shown that, for any CCR-algebra A having at least one infinite-dimensional irreducible representation, the central bidimension and the global central homological dimension of A are greater than one. At the same time it is proved that there exist CCR-algebras which are centrally biprojective, but which have both dimensions equal to one. This situation contrasts with the state of affairs in the “traditional” theory of the Banach Hochschild cohomology. Recall [3, Ch. 5] that the bidimension and the global homological dimension of any infinite-dimensional biprojective C*-algebra are equal to two. Besides, there is no CCR-algebra of bidimension one (respectively, global homological dimension one). See [7].

In this paper we evaluate some integrals involving U-functions by the methods of the Operational Calculus. The results obtained are quite general and many of them include, as particular cases, some known results.

A function ψ (p) is operationally related with another function f(t), if they satisfy the integral equation

2. Theorem. If

and

As usual, we shall denote (1) by the symbolic expression

provided that the integral is convergent. HereR(α) > 0, R(p) > 0, n = 2,3,4, …, andmeans that in the expression following it, i is to bee replaced by – i and the two expressions are to be added.

Banach spaces whose duals possess the Radon-Nikodym property have been studied extensively in the past (cf. [5]). It has been shown recently in [4] that a C*-algebra is scattered if and only if its Banach dual possesses the Radon-Nikodym property. This result extends the well-known result of Pełczynski and Semandini [8] that a compact Hausdorff space Ωis dispersed if and only if C(Ω)* has the Radon-Nikodym property. The purpose of this note is to give a transparent proof of a more general result for Jordan algebras which unifies the aforementioned results. We prove that the dual of a JB-algebra A possesses the Radon-Nikodym property if and only if the state space of A is the cr-convex hull of its pure states. We also consider the projective tensor products of the duals of JB-algebras in this context.

In [1], we showed how a collection of physical operations or experiments could be represented by a nonempty set of nonempty sets satisfying certain conditions (irredundancy and coherence) and we called such sets . We also introduced “complete stochastic models” for the empirical universe of discourse represented by such a manual , namely, the so-called weight functions for . These weight functions form a convex set the extreme points of which are called pure weights. We also showed that there is a so-called logic ∏() affiliated with a manual and that each weight function for induces a state on this logic.

In accordance with customary notation, Jv(t) denotes the Bessel function of the first kind and order v, jvk its kth positive zero and jv, 0 = 0.

The object of this note is to prove that

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū(K) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).