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.