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).

Kurosh-Amitsur radical theories have been developed for various algebraic structures. Whenever the notion of a normal substructure is not transitive, this causes quite some problems in obtaining satisfactory general results. Some of the more important questions concerning the general theory of radicals are whether semisimple classes are hereditary, do radical classes satisfy the ADS-property, can semisimple classes be characterized by closure conditions (e.g., is semisimple=coradical), is Sands' Theorem valid and lastly, does the lower radical construction terminate. For associative and alternative rings, all these questions have positive answers. The method of proof is the same in both cases. In [15], Puczylowski used the results of Terlikowska-Oslowska [18, 19] and hinted at a condition which is crucial in obtaining the positive answers to the above questions.

In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.

We study some weight-homogeneous systems which are not algebraically completely integrable (ACI) in the sense of Adler and van Moerebeke, but whose invariant level surface completes into a semi-abelian variety by adding a set of points (thus ACI in the sense of Mumford).

AMS 2000 Mathematics subject classification: Primary 37J35. Secondary 14H70; 37N05; 70E40

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.