Let β be a cyclotomic integer. The question of the solvability of the diophantine equation xq = β in a cyclotomic field has been considered by many authors (see [4], [5], [12]). Some of the methods used in these investigations also work in J-fields. (As to the definition, see Section 2.) It is well known that J-fields share some important properties with cyclotomic fields. It is also easy to give interesting examples where the solution belongs to a. J-field but not to a cyclotomic field. It seems therefore to be of some importance to consider in general the solvability of xq = β in a. J-field, or in other words whether β1/q generates a. J-field.

The isometries of the space of convex bodies of Ed with respect to the symmetric-difference metric are precisely the mappings generated by measurepreserving affinities of Ed.

The 24 classes of even-valued, unimodular, positive quadratic forms in 24 variables have been determined by Niemeier [6]. One class is distinguished from the rest by the property that its forms have arithmetic minimum 4, rather than 2. The 24-dimensional lattice Λ corresponding to this form-class can therefore be identified with one found earlier by Leech [4], which has been studied extensively in connection with sporadic simple groups (e.g. [1]).

The Dedekind η-function is defined by

and is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], and the further references cited there.

In this paper we shall consider the well known mean value theorem,

where γ is Euler's constant. The error term E(T) has been estimated by various writers; in particular the bounds E(T) = o(T log T), E(T) ≪ T1/2 log T, E(T) ≪ T5/12(log T)2 and

where ε is any positive quantity, have been given by Hardy and Littlewood [7], Ingham [8], Titchmarsh [10], and Balasubramanian [2], respectively.

§1. With respect to coordinates {z, x1, …, xq+1, …, x2q} a contact transformation is a local diffeomorphism of ℝ2q+1, which preserves the 1-form

up to multiplication by a non-zero real valued function. The family of such maps contains identities, inverses and admits a partial composition law; denote it by the contact pseudogroup. Passing to germs of local diffeomorphisms we obtain a topological groupoid Γ2q+1, ω, to which by any of several constructions, see [1] for example, there corresponds a classifying space BΓ2q+1, ω, By analogy with BΓq this space classifies codimension (2q + l)-foliations, which locally admit a contact structure normal to the leaves. In particular, at least when q is odd, the structural group of the normal bundle reduces to Uq. We shall be most interested in the case, when the foliation is by points, and the underlying manifold M2q+1 admits a global 1-form co such that

A two-dimensional fluid layer of height d is confined laterally by rigid sidewalls distance 2Ld apart, where L, the semi-aspect ratio of the layer, is large. Constant temperatures are maintained at the upper and lower boundaries while at the sidewalls it is assumed that the horizontal heat flux has magnitude λ. If λ = 0 (perfect insulation) a finite amplitude motion sets in when the Rayleigh number R reaches a critical value Rc, but in part I (Daniels 1977) it was shown that if λ = O(L−1) this bifurcation (in a state diagram of amplitude versus Rayleigh number) is displaced into a single stable solution in the region |R – Rc| = O(L−2), representing a smooth increase in amplitude of the cellular motion with Rayleigh number. All other solutions (or “secondary modes”) in this region were shown to be unstable. In the present paper an examination of the two intermediate regimes λ − O(L−5/2) and λ = O(L−2) is carried out, to trace the location of an additional stable solution in the form of a secondary mode, which stems from Rc when λ = 0, and which in the limit as λL2 → ∞ is shown to be removed from the region ߋR − Rc| = 0(L−2), consistent with the results of I.

Suppose that X is a set, and E a Riesz subspace of ℝx. Let f : E → ℝ be an increasing linear functional which is smooth in the sense of [1, 72C]; i.e.

§1. Preliminaries. Let X be a stable subordinator in R, B a subset of R and A a time set. In this paper we shall consider the Hausdorff dimension properties of the random sets

Let S be a compact set in some euclidean space, such that every homo-thetic copy λS of S, with 0 < λ < 1, can be expressed as the intersection of some family of translates of S. It is shown that S has this property precisely when it is star-shaped, and is such that every point in the complement of S is visible from some point (necessarily on the boundary) of the kernel of S. Alternatively, S can be characterized as a compact star-shaped set, whose maximal convex subsets are cap-bodies of its kernel.

I have committed a couple of elementary errors which relate to the proof of Lemma 2.3 of the paper with the above title (Mathematika, 22 (1975), 60–70). These are easily corrected, and the remainder of the paper is not affected by them.

The study of plasma instabilities has led to the question whether a certain third order linear differential equation involving a parameter p has solutions which vanish as x → ± ∞. Assuming existence, it is first easily shown that Rep must be positive and then, after a Fourier transform has changed the equation to one of second order, standard comparison equation techniques are used to obtain a contradiction, valid for large enough p.

This paper is a sequel to [1], but can be read independently. However, we do quote one of the results of [1] in the proof of our theorem here.

As in [1], we define the continuantK(a1, a2, …, an) formed from the positive integers a1 …,an to be the denominator qn of the continued fraction

where pn/qn is in lowest terms.

We consider a Markoff spectrum for the set of indefinite binary quadratic forms with real coefficients which represent zero non-trivially. As was done for the classical Markoff spectrum, we show that 1/3 is the largest accumulation point of the set and explicitly determine the countably infinite number of elements greater than 1/3. Unlike the situation for the classical Markoff spectrum, there is a countably infinite number of limit points greater than 1/3.

From time to time results about partitioning a given set into subsets have been established. (See for example [2], [3].) We consider here the reverse problem of forming the union of three sets in a certain best possible way. For simplicity we work in Euclidean n-space, En. Let mX denote the measure of the set X.

Positive definite quadratic forms are associated with pointlattices in the following way. If x ∊ Zm and H = MT for a real m × m matrix M, then xTHx is the square of the distance from the origin to the point Mx of MZm (equally, to W Mx ∊ W MZm, for orthogonal W).

It is known that if Λ is a self-dual lattice in ℝn, then

.

If equality holds the lattice is called extremal. In this paper we find all the extremal lattices: there are unique lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 23, 24 and no others.

One of the important properties of the Dirichlet L-functions is that they may be expanded as Euler products. Unfortunately there is no corresponding expansion for the Kubota-Leopoldt p-adic L-functions since the infinite products involved do not converge. In this paper we give an explicit formula which shows how any finite number of Euler factors may be factored off in a natural manner. Of course, Euler factors may be factored from any function; but we show that, among p-adic analogues of Dirichlet series with periodic coefficients, the p-adic L-functions are precisely the ones which allow this to be done in a natural way which we call a weak Euler product. This might indicate that weak Euler products are the best that can be hoped for. Finally we make a few remarks concerning other analytic properties of the p-adic L-functions.