We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E. We show under a general condition which is easily checked in practice that the elementary subgroup E(M) of G(M) is normal for all quasi-weak Noetherian objects M in the source category of G. This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

The growth of the laminar compressible boundary layer on a moving flat wall is considered analytically for the case of zero velocity in the free stream outside the boundary layer. The results of this analysis are compared with other published results for the cases in which the free stream has some finite velocity. In all the cases considered in the present paper, the boundary layer is taken to originate at some stationary point on the moving wall. This type of boundary-layer flow occurs behind moving shock waves and it is argued that the case of particular interest in the present paper, that of the stationary gas outside the boundary layer, provides bounding values of such parameters as displacement and momentum thicknesses for shock-induced laminar boundary-layer flows.

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold $M$ and any integer $m\,{>}\,0$, a surgery on a link in $M$ of at most $2m\,{+}\,1$ components will provide a closed, orientable, irreducible 3-manifold containing $m$ disjoint, non-parallel, separating, incompressible surfaces of arbitrarily high genus. (2) There exists a 3-manifold $M$ containing separating incompressible surfaces $S_n$ of genus $g(S_n)$ arbitrarily large, such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along $S_n$ can be stabilized $g(S_n)-3$ times to a minimal Heegaard splitting of $M$.

In the first part of this paper we consider partially ordered sets for which the intrinsic ‘interval topology’ ((1), p. 60) is Hausdorffian. The main result of this section is the following: the interval topology of a lattice is Hausdorffian if and only if convergence with respect to the interval topology is equivalent to strong o*-convergence. This may be regarded as an answer either to Birkhoff's problem 23 ((1), p. 62) or to his problem 25 ((1), p. 64). In the case of a complete lattice, we have an alternative formulation. The interval topology of a complete lattice is Hausdorffian if and only if every filter (or, alternatively, every directed net) in the lattice has an o-convergent refinement. This condition may be regarded as a strong type of compactness.

In recent years much attention has been directed towards the asymptotic solution of diffraction problems. In the present work, consideration is given to the relatively simple problem of the diffraction of an E-polarized plane wave by an infinite slit. The solution takes the form of a series in inverse powers of the ratio of slit-width to wavelength of the incident wave, and is based on the solution by successive substitutions of a pair of integral equations. The current densities induced on both halves of the screen are calculated, from which is deduced the electric field in the slit. The far-field is determined from the aperture distribution, and an asymptotic expression is found for the transmission coefficient as a function of the angle of incidence and the ratio of slit-width to wavelength.

While most previous work has been confined to near-normal incidence, the present theory is uniformly valid for all angles of incidence.

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.

The formulae for current and force which were employed in some of the initial work on magneto-hydrodynamic theory are not valid over the whole range of phenomena which the subject itself embraces. Generalized formulae, suitable for wider application, are derived by an extension of the magneto-ionic approach, and are applied to the simple case of plane-wave propagation in the direction of a primary magnetic field. The dispersion equation which results, as well as the basic current and force formulae, are reduced by approximations in a variety of limiting circumstances. Some earlier results are thereby regained and, in particular, the region of known validity of Alfvén's propagation relation is extended.

It is shown that a necessary and sufficient condition that two regular polynomial matrices T, U have relatively prime determinants is that the equation TX + YU = E, where E is a constant matrix, has a unique solution with the degrees of X, Y less than the degrees of U, T respectively. This is a generalization of a well-known theorem for scalar polynomials. An alternative form for the condition in terms of the non-vanishing of a determinant, corresponding to the resultant of two scalar polynomials, is also obtained together with an equivalent determinant of lower order.

The small transverse oscillations of a string excited by bowing are treated by neglecting all external forces except that due to the bow, taking the ends of the string as fixed, and replacing the bow by a concentrated force acting transversely at the ‘bowing point’, whose magnitude depends on the velocity, v, of the string at this point. Non-linear recurrence equations for v are obtained, from which some of the principal features of the motion can be inferred, and some qualitative conditions about the dependence of the transverse force of the bow on v can be drawn. A detailed examination of the case when the string is bowed at the mid-point shows that, in order to account for the stability of the periodic motion excited, it appears to be necessary to allow for the effect of dissipative forces, such as air friction; to a first approximation this can be done by postulating a small concentrated dissipative force at the bowing point hi addition to the frictional force exerted by the bow. The theory predicts that for small bowing speeds a ‘noise’ is produced instead of a note; this phenomenon does not appear to have been noted in the literature, but can easily be verified in practice.

We introduce the idea of dissociatedness for multiply-subscripted random variables: this is a weaker property than independence in general, coinciding with it in the singly-subscripted case; but it is still strong enough to allow a suitable version of the Strong Law of Large Numbers to hold. This result can be applied to prove a weak convergence result of interest in data analysis.

A profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in ℝ.

It is therefore natural to ask to what extent known results concerning Hausdorff dimension in ℝ carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in ℝ. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in ℝ, where the analogous statement is known to be false.

Observations are described in which a cloud chamber was used to detect a penetrating radiation apparently produced in thunderclouds. Comparisons are made between the results obtained by Schonland and Viljoen in South Africa, there being a similarity in the magnitude of the effect observed. An analysis of the directions of arrival of the particles shows that the most favoured direction is from the north at a large angle to the direction of the magnetic field of the earth. The results obtained, though not conclusive, encourage a continuation of the experiments.

It is usual to define Lebesgue outer measure in ℝ by using economical coverings by a sequence of open intervals. We start by outlining a different definition which gives the same answer for a bounded measurable E ⊂ ℝ. Put

Then λ0 defines a pre-measure, but is not an outer measure because it is not countably sub-additive. However it leads to an outer measure by defining

and it can be proved directly that λ is just another definition of Lebesgue outer measure. We do not give the details of this proof as it can also be deduced as a corollary of the main results in the present paper.

The problem of finding the latent roots and vectors of matrices has been treated in a number of papers ((2)–(7)) mainly from the point of view of desk computers. In this paper the problem is treated from the standpoint of users of high-speed automatic computers. In the first section a number of iterative processes are described and, in the second, the techniques developed for using these processes on the Pilot Model of the Automatic Computing Engine. It is shown that the methods give very high accuracy and can be used to deal with matrices of high orders even on a machine of very limited storage capacity. They have been used on numerous matrices of orders up to 60 mainly on problems arising in the aircraft industry and on eigenvalue problems for systems of ordinary differential equations.

This paper describes a method whereby iteration can be carried out simultaneously with two or more trial vectors in order to obtain the latent roots of largest modulus of a symmetric matrix and their corresponding latent vectors. The convergence of any particular vector is automatically accelerated by making use of the current approximations to those vectors subdominant to it, the dominant latent vectors being obtained quite rapidly. The method is particularly suitable where a few of the dominant latent roots and vectors are required of a large symmetric matrix.

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 main results of this paper show that for any odd primes p and q with q dividing p2 − 1 there exist non-associative Bol loops of order pq. The construction is sufficiently flexible so as to allow the resulting loops to be either Bruck loops or non-Bruck loops. It also permits the determination of isotopy classes among the loops constructed. All Bol loops of order 15 are determined.

A metric is constructed, which represents the gravitational field of two spherical static bodies of masses +b and −b, together with a massless source of angular momentum between them. When the separation of the two masses is allowed to tend to infinity the N.U.T. metric can be obtained from the limiting process. This supports the view that the N.U.T. solution represents a static Schwarzschild body together with a rotating semi-infinite spike of pure angular momentum.

A torsion-free discrete subgroup G of PSL(2, C) acts as a group of isometries of hyperbolic 3-space H3. The resulting quotient manifold M has H3 as its universal covering space with G as the group of cover transformations. We shall give examples where M has finite hyperbolic volume and is a link complement in S3. In these examples, G is a subgroup of the Picard group and in most cases is given as an HNN extension or a free product with amalgamation of kleinian groups with fuchsian groups as amalgamated or conjugated subgroups.

We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.