The algebraic classification problem with which I shall be concerned in this paper is suggested by the algebraic theory of homomorphisms of free crossed modules whose groups of operators are free groups. This theory arises in the study of the homotopy theory of two-dimensional C.W. complexes*. It is shown in (3) that the homotopy theory of such complexes, including the homotopy classification of mappings, is equivalent to this purely algebraic theory.

The well-known Steinhaus theorem (2) with respect to the set of distances of linear sets of positive Lebesgue measure has been generalized to the case of linearly measurable subsets of rectifiable curves in the Euclidean plane by Besicovitch and Miller (1). An extension of the theorem to rectifiable curves in Euclidean n-space is immediate. Prof. A. P. Morse has suggested the problem as to whether or not the theorem still remains true in a general metric space. By defining a particular curve ℒ in a metric space we prove that the answer to this question is in the negative.

It was shown by Sheffer that all functions of the two-valued propositional calculus can be denned in terms of a single primitive. This result has been extended by Post (3) and Webb (5) to the functionally complete m-valued propositional calculus. We shall consider here similar extensions for the extended propositional calculus and for the (functionally incomplete) systems of Łukasiewicz (1).

It is known that the refinement theorems for direct decompositions, now classical for group theory, are true, under suitable chain conditions, for two large classes of algebras. These are (1) the algebras whose congruences commute and (2) the algebras with a binary operation of a particular type generated by what we shall call a decomposition operator. On the other hand, there exist finite algebras for which the refinement theorems are not true. The problem of characterizing algebras for which the theorems are true arises at once. It is difficult, because the structures of algebras of classes (1) and (2) can be, to a large extent, incompatible. To illustrate this, we mention that any algebra of (2) has a naturally defined centre which is an Abelian group under the binary operation.

This paper is concerned with topologies for vector lattices (in the sense of (1)) or spaces that satisfy the axioms K 1, 2, 3′ and 4 of (5) that are given by neighbourhoods. The notions of convergence introduced by Kantorovitch(5) and Birkhoff(1) do not in general lead to topologies which can be denned in terms of open sets, as, even for directed systems, the notion of closure derived from them is not such that the closure of every set is closed. In order to ensure this Kantorovitch introduces an axiom (his K6) which excludes even some Banach lattices. These notions of convergence are based more on the lattice aspect than the vector-space aspect of a vector lattice. In this paper the reverse is true, and it deals essentially with the application of the theory of topological vector spaces, as developed by von Neumann(9), Mackey (6, 7) and others, to vector lattices.

If A is a subset of a given metric space , and F is a function which correlates with each point x of A a family of non-vacuous subsets of , having the property that there exist members of the family of arbitrarily small diameter, shrinking down upon x, then F is said to be a blanket with domain A. It is well known that if a blanket is endowed with suitable covering properties, then it may be used for the purpose of differentiating one set function with respect to another. It is our intention to consider, in § 3, a type of blanket whose covering families are permitted to overlap in a controlled fashion, and to prove that these may be used for the differentiation of certain set functions with respect to a given one. The paper concludes with the construction, in § 4, of an example which shows that the results achieved are essentially the best possible under the hypotheses.

1. The paper is in three parts, of which the first is devoted to the proof of certain lemmas, which form the basis for the results proved in Parts II and III, and which are summed up in Lemma 6, §3. In Part II we consider questions relating to linear accessibility: a member of a set of points in the plane is said to be (linearly) accessible if through it there exists a straight line (infinite in both directions) containing no other point of the set. The main results are Theorems 3 and 5. Theorem 5 extends the result of Nikodym (2) that a set can be constructed which is of full measure in a square, and each point of which is accessible. Theorem 3 was conjectured by Besicovitch.

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.

In (5), Whitehead introduced a generalization of the Hopfinvariant in the form of a homomorphism H: πr(Sn) → πr(S2n−1), valid† if r < 3n − 3. In (2), the author extended Whitehead's definition to cover the cases r ≤ 4n − 4; r = 5, n = 2; r = 13, n = 4. He also defined, without restriction, a homomorphism

which was related to H, whenever H was defined, by the equation H* = FH, F being the Freudenthal suspension

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

In this note we are concerned with the semi-group Snr, generated by n of its elements, in which each element x satisfies the equation xr = x, the semi-group being otherwise free. The main result proved is that the proposition

(Ar) Snris finite for all finite n, is equivalent to the proposition

(Br) the group Bn, r−1, generated by n of its elements, in which each element x satisfies the equation xr−1 = 1, the group being otherwise free, is finite for all finite n.

A well-known theorem, due to Castelnuovo and Enriques(1), states that a surface on which the process of successive adjunction, applied to any curve system, terminates, must be rational or scrollar; actually the result shows that, if the property in question holds for any one system of sufficiently general type, then it must hold for all systems; but this has not been established directly.

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such that

where, subject to the conditions

the integers αk(r, n) are uniquely determined, and, in fact, clearly

αk(r, n) = [n/rk] − r[n/rk+1]

(square brackets denoting integral parts, according to the usual convention).

1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite series

The series may be either convergent or asymptotic and divergent.

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

1. Let f(x, y) = ax3 + bx2y + cxy2 + dy3 be a binary cubic form with real coefficients. According as the discriminant D = b2c2+ 18abcd − 4ac3 − 4db3 − 27a2d2 satisfies D ≥ 0 or D < 0, the form f(x, y) can be factorized into either three real, or one real and two conjugate complex, linear factors. A form f1(x, y) is said to be equivalent to f2(x, y) if there is an integral unimodular substitution on the variables x, y transforming f1(x, y) into f2(x, y). Thus f2(x, y) is equivalent to f1(x, y) and, in fact, the set of all forms with given discriminant can be divided into classes of equivalent forms. Davenport (3) has specified a member of each class which satisfies

This paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.

1. In this paper I investigate the tangential properties of parametric surfaces, leading to the main result that the Lebesgue area is the Hausdorff two-dimensional measure of the set of points of the surface where an approximate tangential plane exists.

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled

‘Properties of p(n) and τ(n) defined by the relations

and was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties of p(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems

The object of this paper is to extend the range of validity of Ingham's Tauberian theorem for partitions (2). Ingham's theorem deals with many types of partitions, but it cannot be used in some cases of importance. With a slight modification of Ingham's notation we define

where P(u) is the number of solutions of

in integers ni ≤ 0, and the λi are a given set of numbers, not necessarily integers, such that 0 < λ1 < λ2 <.… If N(u), the number of λi not exceeding u, is given by

where B > 0, β > 0, and as u → ∞

then, according to Ingham's theorem, when u → ∞

where