There are at least two indices used to measure the size of bounded sets of ℝn of zero measure—Hausdorff dimension (see [4] for a definition), and the density index [7].

We study the question of whether every centred convex body K of volume 1 in ℝn has “supergaussian directions”, which means θ∈Sn−1 such that for all , where c>0 is an absolute constant. We verify that a “random” direction is indeed supergaussian for isotropic convex bodies that satisfy the hyperplane conjecture. On the other hand, we show that if, for all isotropic convex bodies, a random direction is supergaussian then the hyperplane conjecture follows.

Atkinson, Young and Brezovich [1: 1983] gave a formula for the potential distribution due to a circular disc condenser with arbitrary spacing parameter к (the ratio of separation of the discs to their radius). This was simpler to calculate than the formulation which I gave in [8: 1949]; but unfortunately it fails to satisfy two requirements, as the present paper shows. Together with [8], this paper shows that the potential formulated in [8] satisfies all requirements.

Let a 3-connected graph G be embedded in an orientable surface of genus g in such a way that the connected components of its complement in the surface are topological discs. We denote by vk(G) the number of k-valent vertices of G and by pk(G) the number of k-gonal faces of the map defined by the embedded graph. For the numbers vk(G) and pk(G), it follows from Euler's formula that

Our object in this note is to construct rings such that

More generally we prove the following embedding theorem (Theorem 5.1) from which rings satisfying (I) are easily constructed:

Any algebra over a field F which contains no zero-divisors or unit-element may be embedded in an algebra over F satisfying (I).

In this paper is proved a formula for the number of simplicial neighbourly d-polytopes with d + 3 vertices, when d is odd.

Stewartson and Rickard [1] have shown that Rossby waves in thin spherical shells of an inviscid fluid contain singularities at the critical circles. These may be removed by introducing another wave containing square-root singularities in the velocity on the characteristics which by reflection touch the inner boundary at the critical circles. Stewartson and Walton [2] continued this wave all round the shell and showed it to be an inertial wave. Here we consider the effect of a weak kinematic viscosity v on these waves.

This paper is a sequel to a previous paper [1] on axially symmetric torsion-free stress distributions in isotropic elastic solids and applies the methods used to investigate these distributions to distributions in solids under torsion. The basis of these methods is that in a solid of revolution containing a symmetrically located crack the stresses set up in the neighbourhood of the crack by forces applied over the crack can be found by perturbing on their values in an infinite solid containing a crack of the same radius and under the same applied forces, provided the radius of the crack is small compared with a typical length of the solid of revolution. The problem of determining the stresses in the solid of revolution is shown to be governed by a Fredholm integral equation of the second kind, which holds whatever the ratio of the crack radius to the typical length, but which, when this ratio is small, is readily solved by iteration to give stresses perturbing on those in an infinite solid. A similar method can be applied to an infinite solid containing two or more cracks when the crack radii are small compared with a typical length of the crack array.

If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

We extend Kemperman's structure theorem by completely characterizing those finite subsets A and B of an arbitrary abelian group with |A + B| = |A| + |B|.

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.

Let ν be a rank 1 henselian valuation of a field K having unique extension ῡ to an algebraic closure of K. For any subextension L/K of /K, let G (L), Res (L) denote respectively the value group and the residue field of the valuation obtained by restricting ῡ to L. If a∈\K define

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

We prove that, if (fn)n∈ω is a sequence of continuous functions on some recursively presentable Polish space, such that any pointwise cluster point of (fn)n∈ω is a Borel function, then there exists a -subsequence of (fn)n∈ω which is pointwise convergent. This is an effective version of a well known result of Bourgain, Fremlin and Talagrand.

Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.

In [1], the author considered the distribution of rational integers which are norms of elements of a given algebraic number field K, and in particular obtained the result

where A(K), B(K) and C(K) are positive parameters depending only on K, as does the implied constant in the O-term. In fact 0 < B(K) < 1, and B(K) is a rational number related to the Galois groups Gal and Gal, where is the normal hull of .

Consider a random simplex in a d-dimensional convex body which is the convex hull of d+1 random points from the body. We study the following question: as a function of the convex body, is the expected volume of such a random simplex monotone non-decreasing under inclusion? We show that this is true when d is 1 or 2, but does not hold for d≥4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.

We prove fractional Sobolev–Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

Let T(n, k) denote the number of trees n with n labelled nodes of which exactly k have degree two. We shall derive a formula for T(n, k) and then determine the asymptotic behaviour of T(n,0); this will enable us to calculate the limiting distribution of Xn the number of nodes of degree two in a random tree n. Rényi [5] has treated the corresponding problem for nodes of degree one in random trees.

The system of equations governing the stress in a shear-strained prismatical body are examined and it is shown that, provided the stress-strain law adopts certain multi-parameter forms, the system may be reduced to the Cauchy-Riemann equations. Integration of the system is then immediate and the analysis of the stress concentration round certain notches thereby facilitated.