In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

In this note, we state general conditions which imply that the numerical range of a function mapping a set S into a normed linear space Y is the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

An unbounded two-phase medium is transmitting radiation from a group of uniformly moving, two-dimensional forcing effects. The emission process is governed by two matrix differential equations for both interior regions, accompanied by a common set of boundary conditions along the communicating interface. Anticipating possible hyperbolic modes, in co-existence with elliptic modes during the eventual steady state, a radiation condition is initially applied. This radiation condition, together with a particular boundedness criterion, decides the precise number of boundary equations required. An exact solution is established for moving forcing effects having arbitrary spatial distributions. Those aligned along the interface correspond to sheet sources. Superimposed are additional source sheets virtually induced by the action of internal forcing effects, converting their combined incident field into reflection and refraction. A peculiar feature is the invariant propagation along characteristics of certain hyperbolic modes, under the essential presence of elliptic modes, as real Hilbert transforms (in the sense of principal values) of a net interface source density.

A more detailed analysis has been carried out of earlier results obtained in the scattering of X-radiation by liquid He II. When certain necessary corrections have been applied to these results and a comparison made with theoretical intensity distribution functions to be expected for certain ordered arrangements of atoms in a liquid, it is found that the arrangement of atoms in liquid He II most probably follows either a diamond or a simple cubic configuration. Considerations of energy and atomic volume render the former the more likely arrangement.

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].

However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

A n.c. JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky for C*-algebras. Different characterizations of associativity are given.

Experiments on the emission of secondary electrons from hot nickel surfaces bombarded by Cs+ ions at normal incidence show that the emission is a function of bombarding voltage and target temperature. The emission is negligible below 300 volts and increases steadily with further increase of the bombarding voltage, rising to 4% at 4000 volts with the target at 950° C. Increase of temperature causes the secondary emission to diminish. If positive ions are allowed to evaporate from the hot target the number of Cs+ ions leaving the surface may, under certain conditions, be equal (with an accuracy greater than 1%) to the number arriving in the incident beam.

I much appreciate the kindness shown me in many ways by Lord Rutherford. The experiments were commenced at the instigation of Dr M. L. E. Oliphant and I gratefully acknowledge that this paper represents to a great extent the results of his advice and assistance. The work was made possible by a grant from the Department of Scientific and Industrial Research.

Let h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

If D is an acyclic digraph, define the height h = h(D) to be the length of the longest directed path in D. We prove that the values of h(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Cn and variance C′n, where C ≈ 0·764334 and C′ ≈ 0·145210. Furthermore, define V0(D) to be the set of sinks (vertices of out-degree 0) and, for r ≥ 1, define Vr(D) to be the set of vertices ν such that the longest directed path from ν to V0(D) has length r. For each k ≥ 1, let nk(D) be the number of sets Vt(D) which have size k. We prove that, for fixed k, the values of nk(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Ckn and variance C′kn, for positive constants Ck and C′k. Results of Bender and Robinson imply that our claim holds also for unlabelled acyclic digraphs.

We investigate the best order of smoothness of Lp(Lq). We prove in particular that there exists a [Cscr ]∞-smooth bump function on Lp(Lq) if and only if p and q are even integers and p is a multiple of q.

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

Three generalizations of Neumann's integral connecting the two kinds of Legendre function are obtained. They are not subject to restrictions that some parameter be integral, as the original formula and various known extensions of it all appear to be. These known extensions are exhibited as quite particular cases of the chief generalization obtained; and even when this generalization is specialized to the ‘unassociated’ Legendre functions of non-integral order it still seems to be new. Generalizations of Rodrigues's formula and of the orthogonality property are auxiliary results involved.

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups.

Let K be a compact Hausdorif space, X a Banach space and C(K, X) the Banach space of all continuous functions : KX equipped with the supremum norm. A subset H of C(K, X) is pointwise weakly precompact if, for each t in K, the set Ht) = {(t):H} is weakly precompact. In this note we study the images of a bounded pointwise weakly precompact subset H of C(K, X) under several classes of linear operators on C(K, X).

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.

We compute the subring of H*(CP∞ × CP∞; ) annihilated by the Steenrod algebra, , p being an odd prime. By calculating the subring's structure as a GL(2, )-space we may obtain information about the modular representations of that group.

It is assumed initially that only the increase of entropy defines locally the sense of advance of time. This assumption, together with the feature of global coherence of statistical processes, which is provided by the cosmic line-element, imply that there is a unique way of associating the standards with the cosmic statistical processes. There is hence a connexion between the sense of divergence of geodesics of fundamental observers and the cosmic time-sense. This is in keeping with the unmodified conservation laws of gravitation and the analogous adiabatic conservation law quite generally without dependence on the explicit form of the line element. These remarks correspond exactly to consequences of the ‘elliptic’ model in the case of de Sitter space, obtained by entirely separate geometrical considerations. A remark is made with regard to the irreversibility associated with quantum measurement.

It is a familiar fact that if two quadric forms, in (n + 1) homogeneous variables, be each expressible as a sum of squares, of (n + 1) independent linear functions of the variables, then they are polar reciprocals of one another, in regard to any one of 2n quadrics. The question arises whether this is true for any two quadrics. Segre states that this is an unsettled question. A solution is given, however, by Terracini for any two non-degenerate quadrics, supposed to have been reduced to the Weierstrass canonical form. The present note has the purpose of pointing out that a solution is derivable from a remark made by Frobenius; this requires a knowledge of the roots of the equation satisfied by the matrix of the two quadrics.

This paper aims to provide a complete resolution of the general norm form equation over function fields of positive characteristic. In a previous paper [4] we studied norm forms in the simpler case of zero characteristic; that study forms the starting point for the present investigations. Diophantine problems over function fields of positive characteristic were first investigated by Armitage in 1968 [1], who clamied to have established an analogue of the Thue–Siegel–Roth–Uchiyama theorem for such fields. This claim was refuted by Osgood in 1975 [6], who also derived a correct analogue of Thue's approximation theorem. in 1983 a different attack was made on Diophantine problems over function fields, the principal weapon being a bound [2] for the heights of the solutions of the unit equation