In 1959 C. A. Rogers gave the following estimate for the density ϑ(k) of lattice-coverings of euclidean d-space Ed with convex bodies . Here, c is a suitable constant which does not depend on d and K. Moreover, Rogers proved that for the unit ball Bd the upper bound can be replaced by , which is, of course a major improvement. In the present paper we show that such an improvement can be obtained for a larger class of convex bodies. In particular, we prove the following theorem. Let K be a convex body in Ed, and let k be an integer satisfying k > log2 loge d + 4. If there exist at least k hyperplanes H1,…, Hk with normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to Hl,…,Hk, respectively, then . Actually, for a bound of this type we do not even need any symmetry assumption. In fact, some weaker properties concerning shadow boundaries will suffice.

The Integral Equation. Consider a periodic wave moving with constant velocity c from right to left on the surface of an inviscid, incompressible fluid which is at rest at infinity. The motion is assumed to be irrotational and two-dimensional. The bottom is horizontal, and the depth of the undisturbed fluid is h.

Summary. This paper is concerned with an infinite plate of homogeneous isotropic elastic material in a state of generalised plane stress and having a circular hole with boundary γ divided into two parts. Over one part of γ the stresses are zero; over the other the shear stress is zero and the normal displacement is specified. The problem corresponds to a smooth loose rigid pin pressed against the edge of a circular hole in an infinite plate.

Let g(n) be a complex-valued multiplicative function which satisfies |g(n)|≤ 1 for all n. The aim of this paper is to establish the following theorem.

Let f(x) be a polynomial of degree d over Fq, the finite field with q = pn elements. Let V(f) denote the number of distinct values of f(x), xєFq. Then, it is easy to see that

where [x] denotes the greatest integer ≤x. A polynomial for which equality is achieved in (1) is called a minimal value set polynomial. Minimal value set polynomials have been studied in [1] and [3].

In this note, we investigate those Hausdorff measures which obey a simple scaling law. Consider a continuous increasing function θ defined on with θ(0)= 0 and let be the corresponding Hausdorff measure. We say that obeys an order α scaling law provided whenever K⊂ and c> 0, then

or, equivalently, if T is a similarity map of with similarity ratio c:

The analysis of the intervals which arise between events occurring randomly in time is a problem which is both interesting and important statistically. Two distinct types of data may arise: either the period of time during which the events are observed may be fixed or the number of intervals may be fixed. It may happen that the intervals between pairs of events, close in time, cannot be accurately measured. It is thus necessary to consider the lengths of intervals ordered according to their magnitudes. We derive here functions of these ordered interval lengths which may be used as a basis for tests for randomness of events occurring in a fixed period of time. The mathematical formulation of this problem is in terms of the classical problem of intervals between points on a line.

Given a topological space X, we denote by Cp(X) the space of real-valued continuous functions on X, equipped with the topology of pointwise convergence.

The sum or Minkowski addition of two subsets, A, B ⊂ En is defined by:

This article studies the non-homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

Schneider posed the problem of determining the maximal value of the affine invariant ∣ΠK∣/∣K∣d−1, where ΠK is the projection body of the d-dimensional convex body K. Some three-dimensional conjectures of Brannen, related to Schneider’s problem, are confirmed. Namely, we determine the maximal value of ∣ΠK∣/∣K∣2 in the class of three-dimensional zonoids, cones and double cones. Equality cases are, also, investigated. Moreover, results related to a conjecture of Petty, concerning the minimal value of the above quantity, are obtained. In particular, we provide a negative answer to a question of Martini and Mustafaev.

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $Re(z)=\pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

For q>1, the set Fq of real numbers which can be expanded in base q with respect to the digit set {0,1,q} is just a self-similar set with overlaps. We consider the subset of Fq whose elements have a unique expansion and calculate its Hausdorff dimension for the case where .

In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties:

1. G. B. Jeffery [1] investigated the axisymmetrical flow of an incompressible viscous fluid caused by two spheres rotating slowly and steadily in the liquid about their line of centres. The numerical results he gave were for spheres of equal radii.

The present problem is to investigate what happens when the spheres touch each other either externally or internally and when they are unequal in size. This problem could be approached by taking a limit of Jeffery's solution, but in fact it will be more convenient to use a co-ordinate system different from Jeffery's and, of course, his results would not yield information about unequal spheres.

Blaschke [1] introduced the notion of maximal tetrahedra inscribed in two and three dimensional convex sets (maximal in the sense of volume). From this notion, he derived an inequality relating the volume of such maximal tetrahedra and the volume of the convex set, and used the inequality to characterize an ellipsoid and to obtain some results concerning isoperimetric inequalities.

An isomorphic factorisation of a digraph D is a partition of its arcs into mutually isomorphic subgraphs. If such a factorisation of D into exactly t parts exists, then t must divide the number of arcs in D. This is called the divisibility condition. It is shown conversely that the divisibility condition ensures the existence of an isomorphic factorisation into t parts in the case of any complete digraph. The sufficiency of the divisibility condition is also investigated for complete m-partite digraphs. It is shown to suffice when m = 2 and t is odd, but counterexamples are provided when m = 2 and t is even, and when m = 3 and either t = 2 or t is odd.

It is well known that every set of at least d + 2 points of ℝd may be decomposed into two disjoint parts whose convex hulls intersect. This result, called Radon's theorem, has been generalized in different ways. The easiest generalization consists in replacing the field of real numbers by any ordered division ring (field or skew-field): here the original proof remains valid. A less immediate generalization is the following one, conjectured by Birch [1] and proved by Tverberg [3]:

Every set of at least r(d + 1) – d points of ℝd may be decomposed into r disjoint parts whose convex hulls intersect.

Previous workers in this field have only considered plane waves. In this paper the Fourier integral method recently devised by Lighthill is used to estimate displacements at large distances from a harmonic point source in an isotropic elastic medium with infinite electrical conductivity subject to a uniform magnetic field. The effect of the applied field is to introduce anisotropy, and the method used gives a complete geometrical description of wave and energy propagation.