§1. Introduction and notation. In [1] and [2], Besicovitch demonstrated that there exist plane sets of measure zero containing line segments (and indeed entire lines) in all directions in the plane. It is natural to ask about the existence of analogous sets in Euclidean spaces of higher dimensions, and in [3] we defined an (n, k)-Besicovitch set to be a subset A of Rn, of n-dimensional Lebesgue measure zero, such that for each k-dimensional subspace Π of Rn, some translate of Π intersects A in a set of positive k-dimensional measure. (Thus Besicovitch's original constructions were for (2,1)-Besicovitch sets.) Recently, Marstrand [5] has shown (by approximating to sets by unions of cubes) that no (3, 2)-Besicovitch sets exist, and simultaneously the author [3] proved using Fourier transform methods that (n, k)- Bsicovitch sets cannot exist if k > ½n.

In my recent paper [1], the corollary on p. 241 is not correct. It should be

Corollary. There is a sequence S, tending to infinity, such that for each t in S we have where γ is any zero of

Let FN be the set of real numbers x whose continued fraction expansion x = [a0; a1, a2,…, an,…] contains only elements ai = 1,2,…, N. Here N ≥ 2. Considerable effort, [1,3], has centred on metrical properties of FN and certain measures carried by FN. A classification of sets of measure zero, as venerable as the dimensional theory, depends on Fourier-Stieltjes transforms: a closed set E of real numbers is called an M0-set, if E carries a probability measure λ whose transform vanishes at infinity. Aside from the Riemann-Lebesgue Lemma, no purely metrical property of E can ensure that E is an M0-set. For the sets FN, however, metrical properties can be used to construct the measure λ.

The purpose of this paper is to prove two integralgeometric formulae for convex bodies. Our results are expressed in terms of integrals with respect to the rigid-motion-invariant measure μd, r on the space ℰ(d, r) of all r-dimensional affine flats in d-dimensional Euclidean space Ed. Rolf Schneider, in an unpublished note [6], has shown that for a convex polytope P in Ed and 1 ≤ r ≤ d – 1 one has

where ηr(P) is the sum of the contents of the r-dimensional faces of P, ηo(Ed–r ∩ P) is the number of vertices of the (d – r)-dimensional section Ed–r ∩ P, and α(r) is the content of the r-dimensional unit ball.

Consider a homogeneous linear viscoelastic filament, one of whose ends is held fixed, while the other is subjected to a given traction τ(t) (t ≤ 0). I shall prove first that the total energy E(t), which is the sum of the kinetic and free energies, its derivative E′(t), and the rate of working W(t) satisfy

and

An exact solution for the velocity field induced by the slow rolling motion of a sphere in contact with a permeable surface is derived. The porous solid and the viscous fluid each occupies a semi-infinite space.

It is shown that, by the use of conformal mapping, the problem is reduced to a fourth order ordinary linear boundary value problem. As the permeability of the porous body diminishes the important functionals, e.g. the force and torque resisting the rolling motion of the sphere, are divergent. The nature of this divergence is found and the dependence on the permeability of the porous solid is explicitly evaluated.

The quasigeostrophic equations describe large scale motion in the atmosphere and oceans at middle latitudes. Being considerably simpler than the primitive equations, they have been widely used for modelling atmospheric and oceanic circulation, and for studies of stability, frontogenesis and turbulence. A number of assertions have been made about these equations; first, that finite element approximate solutions converge in an open flow region [7]; secondly, that solutions depend discontinuously or nondeterministically on initial data [13]; and thirdly, that the energy wave number spectra of the solutions asymptote to the statistical equilibrium spectra of the spectrally truncated equations [12].

It is shown that a convex body K tiles Ed by translation if, and only if, K is a centrally symmetric d-polytope with centrally symmetric facets, such that every belt of K (consisting of those of its facets which contain a translate of a given (d – 2)-face) has four or six facets. One consequence of the proof of this result is that, if K tiles Ed by translation, then K admits a face-to-face, and hence a lattice tiling.

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grössencharakter Xi in Ki and consider the L-function

associated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined by

where

and a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.

Let K be a convex body in Ed and let skelsK denote the s-skeleton of K. Let ηs(K) denote the Hausdorff s-measure of skelsK and

It is the purpose of this note to draw attention to some connections between the topics mentioned in the title. These allow familiar results from one area of mathematics to be usefully exploited in another, to the mutual benefit of both. Moreover, the basic ideas suggest possible generalizations whose examination should prove worthwhile. In this introduction we give a brief account of the results to be derived and discussed in more detail in the later sections.