In § 1 we introduce our system and prove a theorem about its syntax. In § 2 we recall some stock results about the propositional calculus. In § 3 we consider a method of deriving an expression from a given expression and a real number. In § 4 we use this to derive a sequence of expressions from a given expression. In § 5 this sequence is shown to be just all the terms of a conjunctive normal form of the given expression. In § 6 we note that we may not need to produce all of these terms. In § 7 we describe a practical method (suitable for a binary digital electronic computer) which results from all this, and in § 8 we attempt to explain just why this is so.

In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.

1. The object of this note is to prove the restricted Burnside conjecture for exponent 5, that is, to prove, for n = 5, the proposition:

Rn: For each positive integer k there is an integer rn, k such that every finite group of exponent n that can be generated by k elements has order at most rn, k.

The present note is concerned with the proof and applications of the following theorem: Let A be a commutative ring, N be an A-module, and g1, …, gk be elements of A such that (g1,…, gi−1) A:gi = (g1,…, gi−1) N:gi = (g1,…, gi−1) N (i = 1,…, k), so that 0:g1 = 0 in N. Let g denote the ideal (g1 …, gk), let B = A/g and let M be any A-module such that gM = 0. then

where, on the right-hand side of (ii), M, N/gN are considered as B-modules.

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G by

Hi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

1. Introduction. 1·1. Notation. Let

and

be, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

The purpose of the present note is to prove a basis theorem for polynomial modules (Theorem 1 below). As an application, a proof of the Hilbert function theorem for homogeneous modules (Snapper (1)) is obtained. Both these proofs are elementary in the sense that they do not depend on the decomposition theory for modules over a Noether ring.

In the following pages there is developed an abstract theory of infinitely near points for two-dimensional regular local rings which have infinite residue fields. The scope of the theory may be indicated by the fact that it contains generalizations of Noether's formula for intersection multiplicities (Theorem 8), the theory of proximate points (§ 8) and the conditions for a set of integers to be curve multiplicities at a given finite sequence of consecutive points (§ 10). To this extent our account resembles that given by van der Waerden ((6), Chap. 9), but the starting point and (with the exception of the concluding section) the type of reasoning employed are very different. There is also a marked contrast with the methods used by Zariski (7) in an account of related topics.

The present paper is a sequel to a previous study (7) of the completely regular threefolds which possess anticanonical systems, i.e. for which the virtual canonical system, reversed in sign, is effective of positive order. On any such threefold the process of successive adjunction, applied to any linear system of surfaces, must terminate; we have thus to deal with a special case of the adjunction problem for the regular threefolds. By making certain simplifying hypotheses (such as irreducibility and absence of base elements) concerning the anticanonical systems, one can classify the threefolds in broad outline and show that, provided the anticanonical systems are sufficiently ample, the corresponding threefolds are either unirational or birational.

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it an isobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.

1. Let (x1, x2, …, xn) be Cartesian coordinates in Rn. We identify the point X = (x1, x2, …, xn) with these coordinates with the vector X of components x1, x2, …, xn and use the usual vector notation.

In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.

The recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and let

The purpose of this note is to prove that the Jacobian of these systems is given by the equivalence

where Kh is Eger's operator, defined as follows:

General shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

In Schubert's book Abzählende Geometrie(3) there are listed, with no formal verifications, eleven first-order degenerations of the twisted cubic curve in S3. The curve, in this connexion, was seen by Schubert as a union C = (CL, CE, CT) of three simply infinite element systems, the locus CL of its points, the envelope CE of its osculating planes and the system CT of its tangent lines, the complete twisted cubic so envisaged being regarded as a single geometric variable of freedom 12, and any degeneration (projective specialization) of it, C¯ = (C¯L, C¯E, C¯T), being said to be of the first order if it has freedom 11. Schubert used the symbols λ, λ′, k, k′ w, w′, θ, θ′, δ, δ′, η to denote the eleven first-order degenerations in question, the dashed symbol denoting always the dual of that denoted by the undashed. In this paper we exhibit analytically the existence of the above degenerations and indicate further how degenerations of higher order can be systematically investigated. We use for this purpose the analytical theory of complete collineations which has just recently been developed (1,4), and also some of the ideas contained in van der Waerden's earlier development of his general method of degenerate collineations (5).

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric function

where

There are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.