The Poincaré problem for the normal modes of oscillations of an inviscid, incompressible fluid contained in an infinitely long cylinder rotating about a direction perpendicular to its axis is investigated.

In this paper, we consider a class of spaces for which the convolutions with any set of regularizes converge in the topology of the space. We have already dealt with this matter in [2], but the conditions on the topology were unnecessarily restrictive and the proof somewhat unnatural. The present theorem is not only substantially more general, but is also more satisfying in that the argument reveals an unexpected connection between two topics; namely, the approximation of Lebesgue integrals by means of Riemann sums, and the uniqueness of certain types of locally convex topologies in vector lattices.

A graph G (finite, undirected, and without loops or multiple lines) is n-connected if the removal of fewer than n points from G neither disconnects it nor reduces it to the trivial graph consisting of a single point. We present in this note a sufficient set of conditions on the degrees (valences) of the points of a graph G so that G is n-connected.

Let S be a set of points in n-dimensional space, and suppose that an open sphere of unit radius is centred at each point of S. Suppose that no point of space is an inner point of more than two spheres. We say that S provides a double packing for spheres of unit radius. We define δ2(S), the density of this double packing, to be

where Jn is the volume of a sphere of unit radius, and Nt(S) is the number of points of S inside a cube of side 2t, centred at the origin O. We define δ2, the density of closest double packing, to be

where the supremum is taken over all sets S with the property described above.

The purpose of this paper is to give a short proof of an important recent theorem of Bombieri [2] on the mean value of the remainder term in the prime number theorem for arithmetic progressions. Applications of the theorem have been made by Bombieri and Davenport [3], Rodriques [9], and Elliott and Halberstam [5]. For earlier versions of the theorem and a survey of other applications, see Barban [1], and Halberstam and Roth [7, Chapter 4].

Helly's theorem asserts that if W is a family of compact convex sets in a j-dimensional linear space, and if any j + 1 members of ℱ have a non-empty intersection, then there is a point common to all members of ℱ. If one attempts to generalise this result to the case when ℱ consists of sets which are expressible as the union of at most n disjoint compact convex sets then, in general, one finds that there is no number h(n, j) such that if any h(n, j) members of ℱ have a non-empty intersection, then there is a point common to all members of ℱ. The difficulty lies in the fact that, in general, the intersections of members of ℱ are more complicated in structure than are the members of ℱ.

Let A be a subset of a compact metric space Ω, and suppose that A has non-σ-finite h-measure, where h is some Hausdorff function. The following problem was suggested to me by Professor C. A. Rogers:

If A is analytic, is it possible to construct 2ℵodisjoint closed subsets of A which also have non-σ-finite h-measure?

At this level of generality the problem, like others which involve selection of subsets, appears to offer some difficulty. Here we prove two results which were motivated by it.

Truesdell and Noll [1; sections 22, 27, 34] have discussed the concepts of material uniformity and homogeneity in continuum mechanics. A body is said to be materially uniform if, roughly speaking, all the particles composing the body are of the same material and homogeneous if there exists a global reference configuration which can be taken as a natural state for the whole body. To make the ideas precise for elastic materials, consider a small neighbourhood of each particle X and suppose that a reference configuration κ is chosen for each . Then during the motion, the deformation gradients may be calculated at each point X relative to the local reference configurations k. The stress at X is a function of these deformation gradients and if the stress relation does not depend explicitly on X the body is said to be materially uniform. If each local reference configuration κ can be taken as the configuration of its associated set of particles in some global reference configuration for the whole body, the body is said to be homogeneous. In general, however, the configurations κ need not fit together to form a global reference configuration. The body is then said to contain a distribution of dislocations.

Let Pn (n ≥ 0) be an n-polytope, that is, a convex polytope in n-dimensional euclidean space (Grünbaum [5], 3.1), and for 0 ≤ j ≤ n − 1 let be its j-faces. If Pn itself and Ø (the empty set) are also allowed to be faces of Pn, of dimensions n and − 1 respectively, then the set of faces of Pn forms a lattice partially ordered by inclusion ([5], 3.2). Two polytopes P1n and P2n are said to be combinatorially isomorphic, or of the same combinatorial type if their respective lattices of faces are isomorphic; that is, if there is a one–to–one correspondence between the set of faces of P1n and the set of faces of P2n which preserves the relation of inclusion ([5], 3.2). Similarly, any permutation of the set of faces of Pn which preserves inclusion will be called a (combinatorial) automorphism; it is clear that the set of automorphisms of Pn forms a group Γ(Pn), called the automorphism group of Pn.

There are many convenient ways in which a plane triangle can be defined and given projective coordinates. It can most simply be treated as an ordered triad of points (A, B, C) or dually as an ordered triad of lines (a, b, c), but it may seem more natural to regard it as a triad of points and an associated triad of lines which together satisfy the familiar incidence conditions. Again, the triangle for which Schubert [1] developed a calculus was a septuple, but Semple [2] has shown the advantages of a calculus for a triangle defined as an octuple.

Let θ1, …, θk be k real numbers. Suppose ψ(t) is a positive decreasing function of the positive variable t. Define λ(N), for all positive integers N, to be the number of solutions in integers p1 …, pk, q of the inequalities

and

The vertex-connectivity and the edge-connectivity of a graph involve minimum sets of vertices and edges, respectively, whose removal results in a disconnected graph. However, the mixed case of separating sets consisting of both vertices and edges appears to have been overlooked. Such considerations might apply to vulnerability problems, such as that of disrupting a railway network with both tracks and depots being destroyed. Depending on the relative costs, a particular combination of tracks and depots might be optimal for the purpose.

In the present paper the researches initiated in the two earlier papers of this series are continued, and, by suitable generalizations of the techniques employed therein, solutions are obtained to some further well known problems from the theory of transcendental numbers. It will be proved, for example, that a non-vanishing linear form, with algebraic coefficients, in the logarithms of algebraic numbers, cannot be algebraic. This implies, in particular, that π + log α is transcendental for any algebraic number α ≠ 0, and also eα π+ß is transcendental for all algebraic numbers α, β with β ≠ 0.

Let be the class of normalised polynomials of the form:

of degree n which are univalent in |z| < 1, and let be the class of “polynomials” of the form

of degree n which are univalent in 0 < |z| < 1. In this note we discuss the coefficients of polynomials in (k = 2, 3, 4) and (k = 2, 3).

Let p be a prime number, Qp the field of p-adic numbers and Ωp the completion of the algebraic closure of Qp with its valuation normed by setting |p| = 1/p. We shall designate by log the p-adic logarithm defined by the usual series

Let Λ be a lattice in 3-dimensional space which provides a double covering for spheres of unit radius. By this we mean that if X is any point of space, there are at least two distinct lattice points P, Q such that XP ≤ 1, XQ ≤ 1. Let d(Λ) be the determinant of Λ. We shall prove that

where

In the author's paper [1] in which an exact solution is given to the Stokes equations for the steady motion without rotation of a rigid sphere through a viscous fluid in a direction parallel to and at an arbitrary distance d from a fixed plane, expressions were obtained for the forces and couples which are exerted on the sphere and the plane by the fluid. The forces were shown to be equal and opposite. The couple acting on the sphere was found to have Cartesian components (0, Gy, 0), when moments of the surface stress are taken about the centre of the sphere, where

and the couple acting on the plane was found to be (0, Gy′, 0) where

The notation of (1) and (2) follows that as explained in [1] but a misplaced minus sign is corrected.