Let be a (finite) non-empty sequence of integers, and let K be a positive integer. The aim of the Brun-Selberg sieve is to obtain bounds for the “sifting function”

where z ≥ 2 is a real number and

For any fixed integer a and real variables x, y with y < x let Na(x, y) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y. As an elementary application of the following deep theorem of Bombieri on arithmetic progressions,

Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that ifwith l = log x, then for x > 1

where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m;

we shall prove the following theorem:

Theorem 1. Let a be any fixed integer and let x > e. We then have

where the double sum is taken over primes p and q.

For a 3-connected graph G imbedded in the plane let νk(G) denote the number of k-valent vertices, and let pk(G) denote the number of k-gonal faces (countries) of the map defined by G. (We consider only graphs without loops or multiple edges; hence, νk(G) and pk(G) are defined only for k ≥ 3.) It is a wellknown consequence of Euler's formula that

The uniform approach of a solid sphere of radius a towards a stationary solid sphere through a viscous fluid is considered when the fluid motion may be regarded as essentially steady with Reynolds number sufficiently small to permit the linearization of the Navier–Stokes equations by neglect of the inertia terms. Discussion is given when the minimum clearance εa between the spheres is arbitrary and a detailed analysis is made of the asymptotic behaviour of the solution as the minimum clearance tends to zero by constructing matched asymptotic expansions. The forces on the spheres are shown to be of the form as a0ε−1 + b0logε + c0 + O(εlogε) when ε is small where a0, b0 and c0 are constants. The values of a0 and b0 are determined explicitly in the general case of arbitrary sized spheres and c0 is also determined explicitly when the stationary sphere degenerates into a plane. For this case, it is shown that the asymptotic expression for the force is useful in practice even when ε is not very small, the error being less than 3% when ε ∼ 0·5.

Recently there has been much interest in theories describing the behaviour of a mixture of two or more continua, which are allowed to diffuse through each other and which interact mechanically and thermally. In the present note we consider such a mixture consisting of two isotropic incompressible non-linear elastic solids; physically this may describe a composite of two rubber-like materials. Such a mixture possesses internal friction between the constituents when deformed (called diffusive resistance or diffusive force), which acts as a damping mechanism. This damping mechanism has a static part and a dynamic part, and we here show that the exact static solutions which exist for an isotropic incompressible non-linear elastic solid are such that the damping mechanism is expressible as a gradient, and hence are also controllable for such a mixture. We also discuss the corresponding dynamic solutions.

From a practical viewpoint, some experiments carried out on vulcanized synthetic rubber-like materials may involve a mixture rather than a pure solid.

Let D be a compact set in the plane, t a real number, and Dt the linear set {x + ty|x + iy ε D}. We are interested in the Hausdorff dimensions of D and Dt, and assume that dim D = d ≤ 1. A number t is “exceptional” if dim Dt < d; the exceptional numbers form a Borel set of d-dimensional measure zero [3]. (Marstrand [4, p. 268] proves a similar conclusion for the Lebesgue measure of the exceptional set.) In this note we exhibit a planar set D of Hausdorff dimension d and a linear set E for which dim Dt ≤ r < d for every t in E, but dim E > 0. The method does not come very close to the probable truth, that the set E can have dimension r. But perhaps the result can be improved by more subtle calculation, for example, as in Jarník [2].

Necessary and sufficient conditions for Abel's integral equation to have a solution have been given by Tamarkin [18]. The form of the solution was obtained by Abel [1]. The corresponding integral equation for an infinite range of integration was introduced by Liouville [12], who found a solution in a restricted class of cases. In the present paper, we find necessary and sufficient conditions for Liouville's equation to have a solution, and also give the form of the solution.

Let be the class of non-constant functions f(z), holomorphic in |z| < 1, which have asymptotic values at a dense set of points on |z| = 1. MacLane [4; p. 18] asked whether the sum and product of two functions in had to be either constant or in . Recently Barth and Ryan [2] have shown that this was not necessarily so; in this note we will demonstrate, in a totally elementary way, why not. Our principal result is:

Theorem 1. Any non-constant function R(z), holomorphic in |z| < 1, may be represented both as a sum and as a product of pairs of functions in.

It is well known [1; pp. 333–339, 2; pp. 181–183] that the quadrics of S4 through a normal elliptic quintic curve 1C5 form a homaloidal system, and that this gives rise to an interesting Cremona transformation T2, 3 of S4 into S4′ for which the reverse system consists of the (Segre) cubic primals of S4′ that contain an elliptic quintic scroll 1R5. Less well known, though the fact was noted by one of us many years ago, is that the quadrics of S6 through the most general type of normal elliptic septimic scroll 1R7 likewise form a homaloidal system giving rise to a T2, 4 of S6 into S6′. We show in this paper that the latter transformation has properties no less remarkable than those of the above mentioned T2, 3. We propose also to show in a subsequent note that T2, 3 and T2, 4 are the initial members of a sequence of Cremona transformations T2, k+2 of S2k+2 (k = 1,2, …) of which the general member is generated by the homaloidal system of quadrics of S2k+2 that contain an elliptic ruled variety 1Rk2k+3, locus of an elliptic pencil of [k − 1]'s. Since the T2, 4 of S6, however, has certain cyclic properties peculiar to itself, we have thought it best to describe it separately in the present paper.

The reviewer [Math. Reviews, 32 (1966), 7526] has stated that he could not follow case (iii) of the proof of Lemma 7 and upon examination we find that the argument is incomplete (on p. 183, the degree of d1 may not be smaller than that of g). The following version of the proof (which is given with full details), circumvents the difficulty and has been designed to show, further, that the constant implied in the O-symbol is independent of the number of variables. This minor refinement may be of some interest as the corresponding estimates of Lang and Weil [Amer. J. Math., 76 (1954), 819–827; cf., Lemmas 1, 2 and Theorem 1] for general varieties over a finite field lack this feature. Throughout, we use the symbol deg F to denote the total degree of an element F of a polynomial domain , where k = [p] is the field of residue classes mod p and . Nr(†) will denote the number of r-tuples (xn−r+1, …, xn) ε kr with some specific property †. We prove that, if n ≥ 1 and f, g are elements of with no non-constant common factor in , i.e. (f, g)p = 1, then

where the constant in the O-symbol depends only upon d = max (deg f, deg g). The argument is entirely elementary and uses a method of descent in which the number of variables n and the minimum degree, defined by

are simultaneously diminished at each step of the descent. It makes use of a corresponding (trivial) estimate for the case of one polynomial, i.e. if and ∂ = deg F, then Nn(F = 0) = O∂(pn−1). Thus, if f, g are disjoint in the sense that f ε k[x1, …, xr], g ε k[xr+1, …, xn] after a suitable permutation of x1, …, xn, then (f, g)p = 1 implies that

Littlewood posed the following problem: for each pair of real numbers, θ and φ, and each ε > 0, is there a positive integer n such that

where ‖α‖ denotes the difference between a and the nearest integer.

Consider an incompressible fluid of density p and kinematic viscosity v in an infinite two-dimensional domain. We assume that the fluid has a uniform velocity U∞, in the direction of the positive x*-axis of a rectangular Cartesian coordinate system Ox* y*, at large distances from a fixed flat plate of zero thickness which occupies the interval −l < x* < 0 of the line Ox*. Of special interest here is the structure of the flow when ε ≪ 1 where

Re being the Reynolds number of the flow. A first examination was made by Blasius (1908), using Prandtl's theory of the boundary layer, who found inter alia that the leading term of the drag D on one side of the plate is given by

the numerical factor being determined by Goldstein (1930).

This paper presents an analysis of two canonical problems for the steady supersonic inviscid flow past a wing-body combination with the body nonlifting and the wings at a very small angle of attack. Pressure distributions in the vicinity of the lines separating the interaction regions from the remainder of the wing and body are determined from asymptotic expansions by modifying a method due to Friedlander, who considered the diffraction of sound pulses by a cylinder. In a manner similar to Keller's geometrical theory of diffraction the results of the canonical problem are directly applicable to the case where the cross section of the body is any closed convex curve which meets the wings at right angles. In the first canonical problem, the leading edge of the wings has no sweepback, that is, it is normal to the body surface. Here results are compared with those of an approach using boundary layer methods by Stewartson and of a Green's function method by Jones. In the second problem which is even more important in practice and has been neglected in the literature, the leading edge of the wings has positive sweepback; our results are valid provided that the leading edge of the wings is supersonic.