1. Introductory. In this paper certain infinite integrals involving products of four Bessel functions of different arguments are evaluated in terms of Appell's function F4 by the methods of the operational calculus. The results obtained are believed to be new.

As usual, the conventional notation will be used to denote the classical Laplace integral relation

In the proofs of the formulae the following results will be required [1, pp. 281, 284], [3, pp. 78, 79].

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . For undefined terminologies used in the paper, we refer to [3] and [6].

Finite soluble groups in which all the Sylow subgroups are abelian were first investigated by Taunt [8] who referred to them as A-groups. Locally finite groups with the same property have been considered by Graddon [2]. By the use of Sylow theorems it is clear that every section (homomorphic image of a subgroup) of an A-group is also an A-group and hence every nilpotent section of an A-group is abelian. This is the characterization that we use here in considering groups which are not, in general, periodic.

The notion of a group G having periodic cohomology after k steps was introduced by Talelli in [10], and is equivalent to having the functors Hm(G, —) and Hm+q(G, —) naturally isomorphic for some q ≥ 1 and all m ≥k + 1. It extends to infinite groups the long-understood phenomenon of cohomological periodicity for finite groups (for which k = 0).

A fundamental problem in the theory of ordinals is the assignation of principal sequences to limit numbers of the second number class.

It is our main object here to show that a certain class of methods, which are a natural generalisation of those used in the solution of the corresponding problem for the real numbers (the description of which we omit), must fail to solve the problem. The methods are those which rest on the following assumption: the principal sequence assigned to any limit number of the second number class is determined once the first i terms of that sequence are known.

In recent years, certain varieties of semigroups with unary operations (of “inversion”) have received considerable attention. Generally speaking, these have been contained in one or other of the two classes of completely regular semigroups (that is, semigroups that are unions of groups) and inverse semigroups. For instances of the former see [1], [2], [3], [6], [10], [14] and [15], and for instances of the latter see [7], [8], [12] and [13].

In [3] P. E. Conner showed that no abelian group with rank greater than 2 can act freely on Sn×Sn, the product of two spheres. G. Lewis [6] studied free actions of p-groups on Sn×Sn, when n is odd, n≢−l(p) and p is an odd prime. He showed that any p-group which has such an action must be abelian.

Let Y be a subspace of a topological space X. Then S(X, Y) denotes the semigroup, under composition, of all continuous selfmaps of X which also carry Y into Y. In the special case Y = X, the simpler notation S(X)is used. We have devoted several recent papers ([4], [7] and [8]) to the problem of determining when S(Z) and S(X, Y)are isomorphic and, more generally, when S(Z) is a homomorphic image of S(X, Y). In this paper, we investigate the analogous problem for certain semigroups of functions on spaces which were introduced in [5]. These include semigroups of closed functions which are treated in further detail.

The integral operator which we will consider in this paper is the operator T denned for suitably restricted functions f on (0, ∞) by

where x >0 and the integral is taken in the Cauchy principal value sense at t = x. This operator plays a considerable role in Wiener–Hopf theory; see [2; Chapter 5].

Since T is clearly the restriction to (0, ∞) of minus the Hilbert transformation applied to functions which vanish on (−∞, 0), it follows easily from the theory of the Hilbert transformation, as given in say [6; Theorem 101], that T is a bounded operator from Lp(0, ∞) to itself for 1 < p < ∞.

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that

(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative.

This paper describes a simple method of evaluating integrals of a type some cases of which have been discussed by Dr. Ragab.

The following formulae are required in the proofs

If p≧q + 1, z ≠ 0,

If. p = q + 1, | z |>1 in this formula.

Let G be a locally compact topological group, let H be a closed subgroup and let G/H be the space of left cosets = xH with the natural topology. We denote by μ a non-negative measure in G/Hdefined on the ring of Baire sets. G acts by left multiplication as a transitive group of homeomorphisms on G/H: Every t ∈ G defines the homeomorphism We write, for E ⊂ G/H, tE = . The measure μ is called stable (cf. [3], [4]) if from t ∈ G, E ⊂ G/H and μ(E) = 0 follows μ(tE) = 0. We say that μ is locally finite [3], [5] if every set of positive measure contains a subset of positive finite measure.

Applying Hopkins's Theorem asserting that each unitary right Artinian ring is right Noetherian, G. Köthe and K. Shoda proved the following theorem (cf. Köthe [7], p. 360, Theorem 1 and p. 363, Theorem 5): If R is a unitary right Artinian ring, then the following statements hold:

(i) Each nilpotent subring of R is contained in a maximal nilpotent subring of R.

(ii) The intersection of all maximal nilpotent subrings of R is the maximal nilpotent twosided ideal of R.

(iii) All maximal nilpotent subrings of R are conjugate.

The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

Given the series ,the n-th Casáro sum of order k is defined by the relation

where is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s if

It is assumed that k and p are positive.