Let denote the contracted semigroup ring of the ompletely 0-simple semigroup D over the ring R. The Rees structure theory of completely 0-simple semigroups is used to obtain necessary and sufficient conditions that have zero radical (Theorem 3.8). By using Amitsur's construction of the upper π-radical [1], we are able to treat the Jacobson, Baer (prime), Levitzki (locally nilpotent) and possibly the nil radicals simultaneously. Our results generalize a theorem of Munn [6] on semigroup algebras of finite 0-simple semigroups.

The following result is found quite widely. Suppose f(z) is a non-constant entire function such that |f(z)| = 1 along |z| = 1. Then, f (z) has form czm, |c| = 1, m ≧ 1. See Ahlfors [1, p. 172, exercise 3], Dienes [4, p. 172, exercise 23], Hille [6, p. 317, exercise 2]. It is natural to inquire about a generalization of this result.

One of the concepts introduced in [2] is that of a hyperbornological space, an idea which effectively replaces that of a bornological space when semiconvex spaces are being considered. In Section 2 of the present paper, it is shown how the topology of such a space may be described in terms of bounded pseudometrices. This is used in Section 3 to tackle the problem of when a product of separated hyperbornological spaces has the same property. It is shown that, as in the classical case of bornological spaces, this problem is equivalent to one in measure theory.

Let Г(1) denote the homogeneous modular group of 2 × 2 matrices with integral entries and determinant 1. Let (1) be the inhomogeneous modular group of 2 × 2 integral matrices of determinant 1 in which a matrix is identified with its negative. (N), the principal congruence subgroup of level N, is the subgroup of (1) consisting of all T ∈ (1) for which T ≡ ± I (mod N), where N is a positive integer and I is the identity matrix. A subgroup of (1) is said to be a congruence group of level N if contains (N) and N is the least such integer. Similarly, we denote by Г(N) the principal congruence subgroup of level N of Г(1), consisting of those T∈(1) for which T ≡ I (mod N), and we say that a sub group of Г(1) is a congruence group of level N if contains Г (N) and N is minimal with respect to this property. In a recent paper [9] Rankin considered lattice subgroups of a free congruence subgroup of rank n of (1). By a lattice subgroup of we mean a subgroup of which contains the commutator group . In particular, he showed that, if is a congruence group of level N and if is a lattice congruence subgroup of of level qr, where r is the largest divisor of qr prime to N, then N divides q and r divides 12. He then posed the problem of finding an upper bound for the factor q. It is the purpose of this paper to find such an upper bound for q. We also consider bounds for the factor r.

We are here concerned with following result of Trench:

Theorem. (Trench [5]). Let v1 and v2 be two linearly independent solutions of the differential equation

where a(t) is continuous on [0, ∞), and let b(t) be a continuous function of t for t ≧ 0 satisfying

where m(t) = max {|v1(t)|2, |v2(t)|2}. Then, if α1and α2are two arbitrary constants, there exists a solution u of

which can be written in the form

,

with

fori = 1, 2.

This paper considers the determination of the coefficients in two sets of triple trigonometrical series and shows that these can be obtained in closed form. The series considered are special cases of some triple series in Jacobi polynomials studied by K. N. Srivastava [1]. Srivastava, however, shows that the problem for the more general series can be reduced to the solution of a Fredholm integral equation of the second kind and he does not discuss special cases which may lead to closed form solutions.

Let s = s(a1, a2,...., ar) denote the number of integer solutions of the equation

subject to the conditions

the ai being given positive integers, and square brackets denoting the integral part. Clearly s (a1,..., ar) is also the number s = s(m) of divisors of which contain exactly λ prime factors counted according to multiplicity, and is therefore, as is proved in [1], the cardinality of the largest possible set of divisors of m, no one of which divides another.

A group G is said to be an FC-group if each element of G has only a finite number of conjugates in G. We are concerned with the class of periodic locally soluble. FC-groups. Clearly subgroups and factor groups of -groups are also -groups.

Every finite soluble group is a -group, and we consider here the generalization of a concept from the theory of finite soluble groups.

R. A. Rankin [3] considered the problem of finding, for each integer n ≧ 3, a sequence of positive integers containing no n−term geometric progression. He constructed such sets Bn having asymptotic density

For example A3 ≑ 0·71975, A4 ≑ 0·8626, and An→1 as n → ∞.

Riemann's method for solving the Cauchy problem for hyperbolic differential equations in two independent variables has been extended in a number of papers [4], [5], [2] to the wave equation in space of higher dimensions. The method, which consists in the determination of a so-called Riemann function, hinges on the solution of a characteristic value problem. Accordingly, if Riemann's method is to be used in solving a characteristic value problem, one will have to consider another characteristic value problem and thus the process becomes circular. This difficulty was first overcome by Protter [7] in solving the characteristic value problem for the wave equation in three variables. There he employed a variation of Riemann's method developed by Martin [5]. Martin's result was later extended by Diaz and Martin [2] to the wave equation in an arbitrary number of variables. This made it possible to extend Protter's result to the wave equation in space of higher dimensions [8].

Let T denote the class of functions

f(z) = z+a2z2+…

that are analytic in U = {|z| <1}, and satisfy the condition

Imf(z). Imz≧ 0 (zεU).

Thus T denotes the class of typically real functions introduced by W. Rogosinski [5].

One of the most striking results in the theory of functions

g(z) = z + b2z2…

that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <¼} ⊂ g(U). In this note we point out that the same result holds for functions in the class T, a fact which seems to have been overlooked previously. We also determine the largest subdomain of U in which every f(z) in T is univalent, extending previous results in [1] and [2].

In teaching the elements of transform theory to students of physics and engineering it is very useful to have available, as early as possible, the inversion theorem for the Hankel transform

The difficulty is that a valid proof for general values of v (cf. [1], p. 456) is complicated and involves a greater familiarity with the processes of analysis and the properties of Bessel functions than is possessed by most science students.

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.

1. The problem of determining the state of stress in the vicinity of a penny-shaped crack which is opened by thermal means has been considered by Olesiak and Sneddon [1]. In that paper no simple closed expressions were given either for the stress-intensity factor at the tip of the crack or for the normal component of the surface displacement. The purpose of this note is to show how such expressions may be derived.