The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. We shall assume throughout that we are dealing with Banach algebras over the field of complex numbers, C.

In this paper we study a property which we call residual -commutativity. This idea was kindled by a paper of Ayoub (1) and one of Stanley (6). Durbin has defined a similar property in (2) which has also been studied by Slotterbeck in (4). Durbin's property implies residual -commutativity but we have not been able to decide if they are equivalent. However, we have shown that they coincide in certain circumstances. The notation, unless otherwise stated, is that of Robinson (3).

The generalised Stieltjes transform of a function f(t) ∈ L(0, ∞) is defined by

The integral converges for complex s in the region Ω, where Ω is the s-plane cut from the origin along the negative real axis.

In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:

Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Then

where

and

Harnack inequalities are known to be of great importance in the theory of quasilinear elliptic partial differential equations. In the case of such equations defined over a domain Ω in Rn, inequalities of this type have been proved for solutions of second-order equations in divergence form which are of either elliptic or degenerate elliptic structure. More recently Bombieri and Giusti (2) have proved a Harnack inequality for solutions of linear elliptic equations on a minimal surface in Rn+1. The equations are of the form

where summation over i, j = 1, …, n+l is understood, and δ = (δ1, …, δn+1) is the tangential derivative on S. In (2), the inequality is used to give much simplified proofs of some classical results on minimal surfaces, and to generalise some more recent ones.

The purpose of this paper is to expose a method which will match a function f(z) existing in a domain D to a formal series whose radius of convergence may be zero. This matching process has to be done in a “natural way”, and has to be “regular”, which means that if a power series converges absolutely in the circle E = {z | |z|<r} then the summability function f(z) produced by our method in the domain D and matched to will coincide with in the domain E∩D. Euler, in his time, matched the function to the power series .

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.

In (1) a measure λ∈M(ℝ) is constructed and shown to satisfy the following:

(i) is absolutely continuous, where denotes the measure with for all Borel sets E,

(ii) λ*λ = λ2 is singular,

(iii) λk = λk−1*λ is absolutely continuous for K ≧6.

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:

(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,

(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.

Let G be a non-discrete LCA group with dual group Γ. Denote by M(G) the usual convolution algebra of bounded Borel measures on G and Ma(G) those μ ∈ M(G) which are absolutely continuous with respect to mG—the Haar measure on G.

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.