Let E be a continuum and n ≥ 4 a given positive integer. A system of points z1,…, zn єE that maximizes

is called a system of Fekete points. Such a system may be not unique.

When the motion of a viscous fluid around a gas bubble is discussed, it is frequently assumed, especially for flows at low Reynolds numbers, that the bubble takes on a spherical shape in three dimensions or a circular cross-section in a two-dimensional flow. If this assumption is made, arid the gas within the bubble is assumed to have negligible density and viscosity, then the problem of finding the exterior flow is mathematically overdetermined and it is not obvious that a solution to the problem exists. Moreover, if such a solution does exist, then the over-determination of the system should, in general, give rise to relationships between the flow parameters, that is, certain conditions must be satisfied to ensure the existence of a solution. It is the purpose of this paper to derive these conditions in the case of a two-dimensional Stokes flow. The problem is generalised to the extent that part of the circular boundary is taken to be rigid, on which the no-slip condition is to be satisfied and part is to be a free streamline, on which stress conditions are to be satisfied. The conditions for the existence of a solution to this problem are derived and the solution is found in closed form. The method of solution is that of reducing the problem to one of a mixed boundary-value problem in analytic function theory. The classical solutions for the Stokes flow around a circular bubble and around a rigid circle are then easily derived as limiting cases.

The problem of estimating accurately the order of magnitude of the least primitive root g(p) to a large prime modulus p is as yet unsolved. The first non-trivial estimate was obtained by I. M. Vinogradov (see [5]) who in about 1919 showed that

The integral equation

occurs in the theory of the motion of an electrically conducting fluid, occupying the space between two electrodes at different potentials, in the presence of a strong magnetic fluid [Hunt and Malcolm 1]. In this theory a solution is required in which

and one of the principal properties of the solution to be determined is

Christoffel's problem, in its classical form, asks for the determination of necessary and sufficient conditions on a function φ, defined over the unit spherical surface Ω, in order that there exist a convex body K for which φ (u) is the sum of the principal radii of curvature at that boundary point of K where the outer unit normal is u. The figures Ω and K are in Euclidean n-dimensional space (n ≥ 3). It is assumed that φ is continuously differentiable and that K is of sufficient smoothness. A solution of Christoffe's problem was given in [6]. Yet that treatment is rather unsatisfactory in that the smoothness restrictions are set by the method rather than the problem, cf. [5; p. 60]. The present paper overcomes this defect. To do this it is first necessary to generalize the original problem so as to seek conditions on a measure M, defined over the Borel sets of Ω, in order that M be a first order area function for a convex body K. When K has sufficient smoothness, then φ is the Radon-Nikodym derivative of M with respect to surface area measure on Ω. It is this generalized Christoffel problem which is solved in what follows.

If θ is a real algebraic number of degree r ≥ 2, there is a computable number c = c(θ) > 0 such that

for all rational numbers p/q (q > 0). This follows directly from the definition of an algebraic number, as was shown by Liouville in 1843; and if r = 2 there is no more to be said. Axel Thue was the first to prove a stronger result when r ≥ 3; he showed that if

there are at most finitely many rational numbers p/q that satisfy

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.