The structure of various classes of annihilator algebras has been known for some time. Bonsall and Goldie (1) considered semi-simple Banach algebras with the properties

(i)r(L)≡{x:xε,yx=0(yεL)} ≠(0) for each proper closed left ideal L of ,

(ii)l(K)≡{x:xε,xy=0(yεK)}≠(0) for each proper closed left ideal K of ,

Let

where the {lν} are numbers near to an arithmetic progression of common difference unity. Let

b being a constant. Write

so that k(z) is an integral function, and

is meromorphic with simple poles at {lν}.

A ring R is said to be a P-ancestral ring if all proper non-zero sub-rings of R have property P. If p is the property that every proper non-zero sub-ring of R is a (two-sided) ideal then the ring Z of rational integers furnishes an example of a P-ancestral ring.

In this paper we first obtain elementary solutions of the integral equations

and

Using these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.

With every graph G (finite and undirected with no loops or multiple lines) there is associated a graph L(G), called the line-graph of G, whose points correspond in a one-to-one manner with the lines of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of Gare adjacent. This concept was originated by Whitney (3). In a similar way one can associate with G another graph which we call its total graph and denote by T(G). This new graph has the property that a one-to-one correspondence can be established between its points and the elements (the set of points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent (if both elements are points or both are lines) or theyare incident

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλby

where m is planar measure.

The Klein–Gordon equation

Ω being a constant of dimensions [time]-1 and c being a constant velocity, appears in nuclear physics (1) and, when the Laplacian operator is twodimensional, in the theory of long gravity waves on a rotating earth (2). If Ω is zero it reduces to the wave equation

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factor

which is extremely important to workers in fracture mechanics.

A q-analogue of the integral ∣f(t)dt is defined by means of

which is an inverse of the q–derivative

The present author (2) has recently obtained a q–nalogue of a formula of Cauchy, namely,

where, for real or complex α and N a positive integer,

Perhaps the simplest elementary proof of the prime number theorem, see Erdös (2) and Selberg (5), is Wright's modification (8), (3, p. 362) of Selberg's original proof (5). Another variant is due to V. Nevanlinna (4). Wright's proof uses Selberg's idea of smoothing the weighting process which occurs in the Selberg inequality, (1.2) below, by iterating this inequality. Here it will be shown that the proof requires less ingenuity if use is made of a further smoothing operation, namely first integrating the Selberg inequality itself. Integration has been used on a related inequality by Breusch (1 to obtain a remainder term. This method also makes proof by contradiction unnecessary.

A well-known theorem in group theory [(8), p. 11, Satz 3] asserts that, when H is a subgroup of finite index in a group G, there exists a system of common representatives of the right cosets and the left cosets of H in G. Various proofs and generalisations, mainly involving combinatorial rather than grouptheoretical ideas, are known, and an excellent account of the subject is to be found in Chapter 5 of Ryser's book (6), where references to the literature are given. The purpose of the present paper is to use group-theoretical ideas to prove theorems of a similar nature. The motivation for this work comes from the theory of Hecke operators, and one of the main objects is to provide a simple proof of a result given by Petersson (4), which is needed in order to prove the normality of these operators.