1. Introduction. In a principal ideal domain R, any two-sided ideal is of the form Rc = cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

We shall prove the following Theorem. There are exactlyisomorphism classes of residually finite groups with every proper quotient finite.

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).

Let O2 denote the group of isometrics of the plane R2 fixing the origin and let PL2 denote the space of piecewise-linear (PL) homeomorphisms of R2 fixing the origin. Akiba has proved that PL2 is homotopy equivalent to O2(1). The purpose of this note is to point out that this result can be proved by the methods of the author in (4). A complete proof of this result using these methods appeared in the author's Ph.D. thesis(5).

A famous chain of theorems, due originally to de Longchamps (l) and afterwards rediscovered by Pesci(2), Morley(3) and Grace(4), goes as follows:

(1) Given four lines in a plane, the four circumcentres O3 of the triangles formed by omitting each one of the lines in turn lie all on the same circle C4 with centre O4 say.

(2)Given five lines in a plane, the centres O4 of the five circles C4 obtained by omitting each of the five lines in turn lie all on the same circle C6 with centre O5 say.

And in general

(3) Given (n+1) lines in a plane, the (n+1) centres On of the circles Cn+1 formed by omitting each of the lines in turn lie all on the same circle Cn+1 with centre On+1.

0. Introduction: In (14), higher order operations in K-theory, called Massey products, were introduced. These were motivated by the construction, in (7), of a spectral sequence in equivariant K-theory

It is pointed out that many axiomatic systems include a condition of closure under binary composition. In the present development a study is made of the consequences of the weaker requirement of closure under ternary, or three-fold composition. As a preliminary step in the investigation, subsets of groups closed under n-fold multiplication are examined and characterized abstractly. Attention is then focused upon ternary algebras defined as linear subspaces of ℬ (ℋ) (bounded operators on some Hilbert space ℋ) closed under three-fold multiplication. These are shown to satisfy, under suitable conditions, a secondanticommutanttheorem, the analogue of the second commutant theorem which holds for (binary) algebras. It is a result which provides a useful technique for studying such ternary algebras, the structure of which are examined in some detail.

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

The purpose of this paper is to give two characterizations of scalar-type spectral operators.

1. Introduction: Let B be a space and let E be an orthogonal (m + 1)-sphere bundle over B with projection p: E → B. Consider the fibre-preserving map T: E → E which is given by the antipodal transformation in each of the fibres. Following (4) we say that E is homotopy-symmetric if there exists a fibre-preserving homotopy of T into the identity map. Since T maps the fibres with degree (– 1)m this condition requires m to be even.

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined as

and the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).

It has often been assumed in cosmology theory(1) that there exists an average density of matter in space which is everywhere greater than zero. Under this assumption the space-time M will be foliated by curves each of which represents the life history of a particle. In keeping with the postulates of general relativity theory we shall refer to these curves as geodesics. Letting X denote the space of particles one obtains a projection f: M → X which assigns to every P ∈ M the particle found at P. Conversely, given the projection f:M → X, one can recover the geodesics: they are precisely the fibres f−1(x), x∈X.

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:

Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient that

If 0 < κ ≤ 1 then (2) alone is necessary and sufficient.

In this paper we shall consider contractible 2-dimensional cell complexes K with one 0-cell v, one l-cell a, and one 2-cell D, where the boundary of D is attached to a by a word in a and a−1. In order for such a complex K to be contractible, the attaching word in a and a−1 must have total degree ± 1. The simplest example occurs when the attaching word is a; in this case K is a 2-cell, and is of no interest since K is collapsible. If any other word of degree ± 1 is used, the resulting K is contractible but not collapsible.

Consider X and Y to be independent and non-degenerate random variables. If X and Y are non-negative then it follows from (6) that for all given values z of X + Y, the expected values of X and X2 are of the type λz and λ′ z2, respectively, if and only if X and Y have the gamma distributions with the same scale parameter. This is an improvement over Lukacs's result (3). Since the characterization problems for the Poisson, binomial, negative binomial and normal distributions resemble the corresponding problem for the gamma distribution it is quite reasonable to expect the characterizations similar to the above for these distributions. This is established in what follows.

The free field equations for particles with spin are invariant under a group SL(2, c) whose transformations correspond to changes of representation of the twocomponent spinor algebra. The generalization of the equations which extends this invariance to a guage invariance in the Yang–Mills sense necessitates the introduction of auxiliary fields (which are also necessary to maintain Lorentz covariance). These fields can be interpreted as the potentials of a spin-2 field, just as the auxiliary fields for the charge gauge group are the potentials of a spin-l field (electromagnetism); this spin-2 field is then self-interacting. The Bargmann–Wigner formulation of the linear spin-2 field, when modified by the proposed self-interaction, yields a non-linear theory of a spin-2 field which is shown to be identical with Einstein's gravitational theory. With this interpretation the auxiliary fields take on an extra role of Yang–Mills field for the general coordinate transformation group – that is, they are the components of the affine connexion.