Some techniques developed by Eilenberg (1) in the theory of automata are used to investigate certain types of near-ring.

A finite presentation F/N of a group G is called efficient if dF(N) = d(H2(G)) + d(F) – r(H1(G)). A finitely presented group is called efficient if it admits an efficient presentation. We show that a finitely presented group embeds into an efficient group.

In this paper we shall develop a new method for the computation of eigenvalues of singular Sturm-Liouville problems of the Bessel type. This new method is based on the interpolation of a boundary function in Paley-Wiener spaces. Numerical results are provided to illustrate the method.

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

This paper shows briefly how, for a singly infinite family of curves on a given surface, the fundamental properties of the family at any point are associated with three central conies determined by the curves, in a manner resembling that in which the curvature properties of a surface at any point are associated with Dupin's indicatrix. The differential invariants employed are the two-parametric invariants for the given surface.

If XYZ is the Simson Line P(ABC); if PM is perpendicular to XYZ and cuts the sides BC, CA, AB in U, V, W: then PU.PV.PW=PA.PB.PC=4R2.PM.

We construct a non-exposed extreme function f of the unit ball of H1, the classical Hardy space on the unit disc of the plane, which has the property: f(z)/(1−q(z))2 ∉ H1 for any nonconstant inner function q(z). This function constitutes a counterexample to a conjecture in D. Sarason [7].

The general linear matrix equation has been investigated by Sylvester in a series of interesting papers published in the Comptes Rendus. His method, however, does not show that the solution can be exhibited as an analytical function of the coefficients. The object of this note is to supply such a solution.

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chain

of type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.

Throughout this paper a near-ring N will satisfy the distributive law α(β + γ)=αβ + αγ for all α, β and γ in N. We shall also assume that 0α = 0 for all α in N

We prove the following theorem.

1. The object of this paper is to correlate the geometrical and analytical aspects of the elements of the theory of points at infinity, etc., in a plane. It is assumed that the reader is acquainted with the method of tracing the graphs of rational functions of x, by using the artifices of change of origin and approximation by Ascending or Descending Division (see Chrystal's Introduction to Algebra, Ch. XXV.).

In his notable memoir on the Bessel Functions, Sonine proves the elegant result that

m > –½, provided a plane triangle can be drawn with sides a, b, c; otherwise the value of the integral is zero.