The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.

Let S be a finite semigroup, A be a given subset of S and L, R, H, D and J be Green's equivalence relations. The problem of determining whether there exists a supersemigroup T of S from the class of all semigroups or from the class of finite semigroups, such that A lies in an L or R-class of T has a simple and well known solution (see for example [7], [8] or [3]). The analogous problem for J or D relations is trivial if T is of arbitrary size, but undecidable if T is required to be finite [4] (even if we restrict ourselves to the case |A| = 2 [6]). We show that for the relation H, the corresponding problem is undecidable in both the class of finite semigroups (answering Problem 1 of [9]) and in the class of all semigroups, extending related results obtained by M. V. Sapir in [9]. An infinite semigroup with a subset never lying in a H-class of any embedding semigroup is known and, in [9], the existence of a finite semigroup with this property is established. We present two eight element examples of such semigroups as well as other examples satisfying related properties.

In an earlier paper (3), polycyclic groups in which every subgroup can be generated by d, or fewer, elements were studied. In this paper we investigate the structure of those polycyclic groups G such that every abelian normal subgroup of F(G), the Fitting subgroup of G, can begenerated by at most d elements.

In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to be

and the spectrum of T is the complement of ρ(T) in Z.

The present note contains generalisations and new proofs of certain theorems in the theory of Young Tableaux and Invariant Matrices. For an account of Young Tableaux and their applications, and an introduction to the method of Clebsch-Aronhold symbols, reference should be made to Rutherford [1], and Turnbull [1], respectively. An invariant matrix T(A) of a given square matrix A is, as appears from the context in § 4 below, a matrix of polynomials in the elements of A, regarded as independent variables, such that T(AB) = T(A) T(B). Further details, and references to original sources, are given in Wallace [1].

In the quotient ring of differential polynomials modulo cubic terms the usual odd order hierarchy of Korteweg–de Vries equations can be supplemented by an even order hierarchy. The full (even and odd) sequence is generated by an Olver recursion operator of order one and any pair has zero bracket in the quotient ring. The even order equations do not possess a Hamiltonian structure and thus their related Rosencrans densities are considered.

The notion of transversality has proved of immense value in differential topology. The Thom transversality lemma and its many variants show that transversality is a dense,and often open, property. In one parameter families the occurrence of non-transversality is inevitable; for example one cannot pull two linked curves in ℝ3 apart without a non transverse intersection. The aim of this note is to prove the following. In any generic family of mappings each map in the family fails to satisfy some fixed transversality conditions at worst at isolated points, and even at these points in rather special sorts of way. So, returning to the above example, given two space curves C1 and C2 without a (necessarily non-transverse) intersection we expect, in any genericisotopy of C2, that it will meet C1 if at all, at isolated points In particular generically we do not expect C1 and C2, any time, to have an arc in common

Consider the equation

where is the generalised derivative of x defined as follows: for every t ≧ T

.

In this paper are introduced what we shall term “successive oscillation functions.” These functions are derived from functions of a real variable. The word “function” as here used has its widest meaning. We say y is a function of x in an interval of the the x-axis, if given any value of x, in the interval one or more values of y are thereby determined. The values of the function may be determined by any arbitrary law whatsoever. We shall deal with discontinuous functions; the theorems will be true for continuous functions, but will be trivial, except in the case of functions which are discontinuous and whose points of discontinuity are infinite in number. We shall assume in what follows that the values of the function lie between finite limits.

If the physical state of a substance is completely defined when the simultaneous values of three of its properties are given, then, by measuring off along three rectangular axes, from any point chosen as origin, lengths proportional to these values, we determine a point which represents completely the physical state of the substance. And, evidently, each point lies on a surface, the equation to which is determined by the three co-ordinate properties. If, in the equation to the surface, we give one of the variables a definite value, we get the equation to a contour-line of the surface which represents the necessary relation subsisting between the remaining two properties when the other is constant.

We give an abstract description of the kernel of a proper primitive inverse congruence on a categorical semigroup. More specifically, we show that it is a *-reflexive, *-unitary, *-dense subsemigroup, and that on a given categorical semigroup there is a one–one correspondence between such subsemigroups and the proper primitive inverse congruences. Our results allow us to give a description of the minimum proper primitive inverse semigroup congruence on a strongly E*-dense categorical semigroup.

Suppose that we have c candidates, e electors, s seats, v votes.

There are at least three different kinds of voting to consider: Simple, Combinational, and Cumulative.

I. Simple voting. By simple voting I mean any case in which an elector has only one vote. Denote the candidates by A, B, C, D.

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we consider

where Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.