Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if

(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

Let the Nodal cubic have the equation

or,

Three points t1, t2, t3 form an apolar triad to the curve when they satisfy an equation of the form

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]

In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.

The Abel and Cesàro summabilities of two alternating gap series are investigated. We prove that the series is summable at x = 1 (in both senses), but that is not. In 1907, Hardy obtained essentially the same result for the latter series; our proof is shorter and more elementary: we use the Poisson summation formula to derive an explicit estimate for the size of the oscillations as x → 1_. This represents an example of a general method for determining the Abel summability of similar series.

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

A class of finite semigroups is called a genus if it is closed under homomorphic images, subsemigroups and finite direct products. During a talk at the Symposium on Semigroups held at the University of St Andrews, in 1976, M. P. Schützenberger posed the problem of characterising the smallest genus which contains finite groups and finite semigroups, all of whose subgroups are trivial.

Binet shewed that the function

can be expanded as an inverse factorial series. This note furnishes a new and much simpler proof of his result, based on a formula which is an analogue of the Binomial Theorem for factorials.

This formula is that, if we denote by [x]n the ratio

then

where denotes the coefficient of xr in the expansion of (1 + x)m.

In this paper we consider the questions of existence and uniqueness of solutions to a singular, nonlinear boundary value problem arising from a model problem in isothermal autocatalytical chemical kinetics. The boundary value problem occurs in the construction of a small time asymptotic solution to an initial-boundary value problem (King and Needham [14]), and existence and uniqueness for the boundary value problem are required for consistency of this formal asymptotic solution.

The purpose of this note is to solve a problem of Dr A. M. Sinclair. Denote by Aw(I, T) the algebra with identity generated by a bounded linear operator T in the weak operator topology. We prove the following result.

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equation

does not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

When points and lines are not specifically designated in the course of the following pages it will be understood that the notation for them is that recommended in the Proceedings of the Edinburgh Mathematical Society, Vol. I. pp. 6–11 (1894). It may be convenient to repeat all that is necessary for the present purpose.

Our starting point is the differential equation

where A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],

for all , b∈C\{0}, at least outside a set of r of finite measure.

In Vol. XXXV. (Session 1916–17), Part I., of the Proceedings of the Edinburgh Mathematical Society, I discussed in considerable detail the properties of the Apolar Locus of two tetrads of points. I showed there that, subject to certain defined conditions, a unique quartic curve would be obtained, which would be the Apolar Locus of the two given tetrads. I mentioned, however, in §7 of the paper, that in the case when the two tetrads lie on the same conic, the above-mentioned conditions are not independent, and that, in fact, not a unique quartic but a pencil of quartics is obtained.