Segre's classification of pencils of quadrics is well known, and appears in standard texts such as (1), (2). By contrast, other linear systems of quadrics do not lend them-selves to a similar exhaustive treatment. Here we discuss the simplest case, that of nets of conics. The result is intrinsically interesting, and involves some pleasant geometry. As well as deriving a list of types, and enumerating their properties, we study the elementary geometrical properties of the partition of the space Ω of nets into equivalence classes (or strata). We work over the field ℝ of real numbers, after performing preliminary calculations over ℂ.

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:

Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

In this note, we discuss some questions on the arithmetic properties of recurrence sequences. Our primary purpose is to mention some methods which originated as lemmas in certain transcendence studies, but in order to give context to these ideas, we briefly review other techniques applicable to the problems considered. These problems concern the distribution of zeros in a recurrence, the rate of growth of its terms and the size of the greatest prime factor of the terms.

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We write

If α is a real number, ∥α∥ denotes the distance from α to the nearest integer.

Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).

for all X ≥ 1. We say that θ is singular if

Singular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed that

for almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

This paper considers perturbations in locally convex spaces of semi Fredholm operators T by a sequence of operators {Kn} converging compactly to zero in a sense extending that from (2). It is shown that, under suitable conditions, α(T + Kn) ≤ α(T), β(T + Kn) ≤ β;(T) and κ(T + Kn) = κ(T) for large n.

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.

0. Introduction. Let and be connected cohomology theories on the category of all CW spaces. In this paper we examine the relationship between natural homomorphisms (stable operations) of degree zero from to and natural homomorphisms from to in a number of cases of particular interest in topology – foremost the K-theories and the bundle theories classifying ‘surgery problems’. There are two parts to this. On the one hand we ask under what conditions a homomorphism from to extends to a stable operation and on the other hand when is a stable operation determined by its restriction to .

In (1) Atkinson and Reuter prove the existence of wave solutions with velocities strictly greater than some critical velocity ʗ0 of an equation associated with the deterministic model for the spread of an epidemic. The existence of wave solutions with velocity ʗ0 is left as an open question. In this note we show that Atkinson and Reuter's results and a simple limiting argument imply the existence of such wave solutions.

An infinite genetic population of haploid particles is considered in which selection is controlled by a single locus at which there are an infinite number of possible alleles. These alleles are arranged in an infinite sequence and mutation occurs only to nearest neighbours. This is the ‘ladder model’ of Ohta and Kimura which was put forward as a possible explanation of the distributions of electromorphs in electrophoretic observations. Following an earlier paper, conditions are obtained on the selection coefficients which ensure that a stationary stable state exists. One such model is solved explicitly. The problem, important in evolutionary theory, of the rate of approach to such stationary states starting from some other state, is also discussed briefly.

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.

Alexandrov (1), Zeeman (3) and others after them have investigated at length the problem of characterizing as Lorentz transformations those mappings f of Minkowski space M which ‘preserve cones’, i.e. if C(a) denotes the null-cone in M with vertex a ∈ M, then

for all a ∈ M.

The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.

The possibility is explored that the general static solution of Einstein's vacuum field equations is given in terms of a single harmonic function, i.e. solution of the flat space Laplace equation. It is shown that coordinate conditions can be chosen such that one of Einstein's equations reduces to Laplace's equation. To examine the extent to which the metric is determined by the resulting harmonic function ø, a power series expansion is considered in which in the first non-trivial order one gets a system of eight equations for six unknowns. This is reduced to a system of three equations for three unknowns. This is some indication that the coordinate conditions are consistent and shows the sense in which, to this order, the metric is determined by ø. This system of three equations is then solved explicitly for a particular choice of ø, thereby giving an approximate non-axisymmetric solution of Einstein's vacuum equations. This solution is asymptotically flat and could represent the approximate gravitational field of a non-axisymmetric bounded rigid body. The form of a class of time-dependent metric related to the general static metric is discussed.

A circular disc of radius a, made of homogeneous, isotropic, linearly elastic material, contains a radial edge crack of length b(0 < b < 2a). The disc is in equilibrium in a state of generalized plane stress caused by loading the faces of the crack by a constant internal pressure. The problem of determining the resulting stress field throughout the disc is solved analytically in closed form. The principal results are that the stress concentration factor at the crack tip, the total strain energy W, and the opening U at the mouth of the crack, are given exactly by

where A is a constant whose value correct to 6 significant figures is

and , W0, U0 are normalising factors defined in section 6.

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ < ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν < ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesis

is strengthened to

In fact, Baumgartner has proved the desired conclusion under the weaker hypothesis