We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

The purpose of this paper is to define and study a natural rank function which associates to each differentiable function (say on the interval [0,1]) a countable ordinal number, which measures the complexity of its derivative. Functions with continuous derivatives have the smallest possible rank 1, a function like x2 sin (x−1) has rank 2, etc., and we show that functions of any given countable ordinal rank exist. This exhibits an underlying hierarchical structure of the class of differentiable functions, consisting of ω1, distinct levels. The definition of rank is invariant under addition of constants, and so it naturally assigns also to every derivative a unique rank, and an associated hierarchy for the class of all derivatives.

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

The Brianchon-Gram and Sommerville theorems on angle-sums for convex polytopes and polyhedral cones are here shown to be particular cases of an angle-sum relation for general polyhedral sets. The new relation is proved on the level of an equidissectability theorem, and this approach yields yet other angle-sum relations, including a different generalization of the Brianchon-Gram theorem. Further results extend, again to equidissections, earlier angle-sum relations of the author and others.

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

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 $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$. We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.

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.

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|.

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.

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

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

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.

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).

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.

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.

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 .

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.

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 an inverse theorem for the Gowers $U^{2}$-norm for maps $G\rightarrow {\mathcal{M}}$ from a countable, discrete, amenable group $G$ into a von Neumann algebra ${\mathcal{M}}$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert \cdot \Vert$. We use this result to prove a stability result for unitary-valued $\unicode[STIX]{x1D700}$-representations $G\rightarrow {\mathcal{U}}({\mathcal{M}})$ with respect to $\Vert \cdot \Vert$.