In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

We show that an inverse transversal of a regular semigroup is multiplicative if and only if it is both weakly multiplicative and a quasi-ideal. Examples of quasi-ideal inverse transversals that are not multiplicative are known. Here we give an example of a weakly multiplicative inverse transversal that is not multiplicative. An interesting feature of this example is that it also serves to show that, in an ordered regular semigroup in which every element x has a biggest inverse x0, the mapping x↦x00 is not in general a closure; nor is x↦x** in a principally ordered regular semigroup.

The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—

Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

The purpose of this paper is twofold: first to correct the statement of Theorem 1 in [4], and secondly to consider related problems in the class of ideally finite Lie algebras.

Throughout, L will denote a Lie algebra over a field K, F(L) will be its Frattini subalgebra and φ(L) its Frattini ideal. We will denote by the class of Lie algebras all of whose maximal subalgebras have codimension 1 in L. The Lie algebra with basis {u–1, u0, u1} and multiplication u–1u0 = u–1, u–1u1 = u0, u0u1 = u1 will be labelled L1(0).

This paper extends some familiar theorems concerning the relations between the roots of a polynomial and those of its first derivative to the more general case of the rational function with a pole at a single point.

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))

and the generalised Mehler transform pair (7)

These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

If letters a, b, c, … are used to denote points of a nondegenerate plane cubic curve, other than the singular point if any, and if the product ab is defined as the third point of the curve collinear with a and b, we obtain an algebraic system having nonassociative multiplication (ab . c ≠ a . bc in general). It is in fact a totally symmetric entropic quasigroup (these terms are defined below). This idea, which was put forward at a meeting of the Edinburgh Mathematical Society a few years ago, will be exploited in a forthcoming paper. Such quasigroups have many properties which can be interpreted geometrically. Or, conversely, known properties of cubic curves suggest theorems about totally symmetric entropic quasigroups, which if established will involve a gain in generality, since not every such quasigroup can be “placed” on a cubic.

This paper concludes a series of papers (1) on a group of axisymmetric boundary value problems in potential and diffraction theory by considering some potential problems for a circular annulus. The Dirichlet problem for an annulus has recently been considered by Gubenko and Mossakovskiǐ (2), who, by a somewhat complicated method, show it to be governed by either one of two Fredholm integral equations of the second kind. The purpose of the present paper is to show how the method developed in previous papers, by which certain integral representations of the potentials in problems for circular disks arid spherical caps are used to reduce such problems to the solutions of either single Abel integral equations or Abel and Fredholm equations, can be applied to both the Dirichlet and Neumann problems for the annulus to give reasonably straightforward derivations of the governing Fredholm equations.

Let n be a positive integer. Then we know that, if m>– 1,

Consider the integral

which is equal to

In a recent paper, E. T. Copson (2) proves the following result:

Theorem C.Letki >0 (i = 1, …, m), k1 +…+ km = 1, and the real sequence (an) satisfy the inequality

If (an) is bounded, then it must be convergent.

In (1) I obtained † an asymptotic formula for the number of zeros of an arbitrary canonical product II(z) of integral order but not of mean type, all of whose zeros lie on a single radius, from a knowledge of the asymptotic behaviour of (i)log | П(z)| as | z | = r→ ∞ along another radius l, with certain side conditions. After proving the analogous theorem in which log | П(z)| in (i) is replaced by , I show in this note that, at a cost of replacing l by two radii l1 and l2, both of these theorems may be generalised to include a class of canonical products of integral order whose zeros lie along a whole line. In one of the resulting theorems ‡ (Theorem II) I find the asymptotic number of zeros on each half of the line of zeros; another theorem (Theorem III) includes a previous result of mine.§

The object of this note is to state and prove two theorems of the nature of Montel's Limit Theorem for a function which is regular and bounded in a region G, but involving as hypothesis the limit of a mean value of the function instead of the limit of the function itself. Theorem 2 below (with b = b′ = 1), in which G is a half-strip, was stated several years ago in a letter to A. J. Macintyre and myself from J. M. Whittaker, who added that it could be proved by integrating the inequality in Lemma 3 below (which is due to Dr Whittaker). As far as I can discover, such a proof still has not appeared in print; hence the present one.

In connection with the analysis of mathematical models of real processes undergoing short time perturbations, in the last years the interest in the differential equations with impulses remarkably increased. Going back to the papers of Mil'man and Myshkis [4, 5] the investigations of this subject are now extended to different directions concerning applications in physics, biology, electronics, automatic control etc.

The locus of the focus of a parabola touching three given fixed straight lines is both exceedingly simple and widely known. On the other hand the analogous locus of the focus of a parabola passing through three given fixed points is excessively complicated, and its investigation has, so far as the present writer knows, never appeared in any text-book.

1. A well known theorem in classical dynamics tells us that if Q, P, and are written as functions of q, p and t by means of the equations

where W (q, Q, t) is an arbitrary function, and if two functions F (Q, P, t) and f (q, p, t) are related by

then q and p are conjugate with respect to f(q, p, t) if Q and P are conjugate with respect to F (Q, P, t).