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

Following A. S. Besicovitch [1] and L. J. Mordell [3], we say that a polygon is rational if the lengths of all its sides and diagonals are rational. Besicovitch proved that the set of all rational right-angled triangles is dense in the set of all right-angled triangles and that the set of all rational parallelograms is dense in the set of all parallelograms. Then Mordell showed that the set of all rational quadrilaterajs is dense in the set of all quadrilaterals.

Since my article McMullen [1980] has appeared, Professors S. S. Ryškov and B. B. Venkov have drawn my attention to two previously published papers. B. A. Venkov [1954] proves my main Theorem 1 (and its corollary Theorem 2) by methods apparently very similar to mine (1 have not checked all the details), while A. D. Aleksandrov [1954] generalizes Venkov's result to tilings of spaces of constant curvature by polytopes (not necessarily convex) congruent to ones in some finite collection. I am happy to acknowledge their priority.

It has been shown by Mirsky and Perfect [1] that the theorem of Rado [2], linking matroid theory and transversal theory, has important applications in combinatorial theory. In this note I use it to obtain necessary and sufficient conditions for two families of sets to have a common transversal containing a given set, and then I show how it may be used to obtain a variant of a well-known theorem that was obtained by Hoffman and Kuhn [3] using linear programming methods.

This paper uses systems of image sources to construct suitable generalised Green's functions for considering small amplitude short surface waves due to an oscillating immersed sphere. A sphere of radius a is half-immersed in a fluid under gravity and is making vertical oscillations of small constant amplitude and period 2π/σ about this position. It is required to find the fluid motion, and in particular the virtual mass and wavemaking coefficients. For sufficiently small amplitudes the motion depends non-trivially on only a single dimensionless parameter

where g is the gravitational acceleration. This was shown by Ursell in an unpublished U. S. Navy report in which the methods of an earlier paper (Ursell, 1949) were adapted from cylindrical to spherical symmetry and resolved the conflict between Havelock (1955) and Barakat (1962) in favour of the former. Ursell showed that the virtual mass coefficient is

i.e. infinitely increasing initially so that Barakat's results must be incorrect near N = 0. However, although existence is proved for all N, the same difficulty arises as with the heaving cylinder, namely computation is only practical for values of N up to about 1. In the cylindrical case, Ursell (1953) developed a method for finding the asymptotic solution for large N and here it will be adapted, surprisingly perhaps, to deal with the spherical case. Ursell (1954) then published a formal solution which gave the same virtual mass as the rigorous treatment and if this formal method is applied to the heaving sphere, the virtual mass coefficient obtained is

Given a probability density, we estimate the rate of decay of the measure of the level sets of its evolutes by the Ornstein–Uhlenbeck semigroup. The rate is faster than what follows from the preservation of mass and Markov’s inequality.

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

The purpose of this article is to give the proof of a theorem announced in [1]. We refer the reader to [1] for the terminology used here.

Some of the elastic properties of liquids in shear can be detected and measured by observing suitable types of oscillatory motion. Oscillating systems have proved to be fairly simple to design and to control in practice, and can lead, in the case of purely viscous liquids, to accurate measurements of the viscosity [see, for example, 1, 2, 3]. One such system, used for the purpose of investigating the rheological properties of dilute polymer solutions, is the coaxial-cylinder elastoviscometer of Oldroyd, Strawbridge and Toms [4]; in this experimental arrangement, the liquid is contained between two cylinders with a common, vertical axis, the inner cylinder being suspended by a vertical torsion wire. The theory of the motion of such an instrument in the case when the outer cylinder is forced to oscillate about its axis, which is fixed, with a known constant frequency, and the resulting motion of the inner cylinder is constrained by the torsion wire, has been considered by Oldroyd [5], who shows that the experimental results available for some typical polymer solutions can be interpreted in terms of an idealised elastico-viscous liquid characterised by three constants (a viscosity and two relaxation times). A new representation of the relaxation spectrum of a liquid has subsequently been used by Walters [6] in order to develop the theory of oscillatory flow of the most general (linear) visco-elastic liquid. Walters has shown that the experimental results for dilute polymer solutions (previously interpreted in terms of a discrete relaxation spectrum by Oldroyd [5]) can, equally, be interpreted in terms of a simple continuous relaxation spectrum characterised by three constants.

Let ℳ be the collection of all intersections of balls, considered as a subset of the hyperspace ℳ of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. It is proved that ℳ is uniformly very porous if and only if the space fails the Mazur intersection property.

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.

We say a motion g brings a mobile convex body K into inner contact with a fixed body K0 if the image gK lies in K0 and shares a boundary point with K0; we speak of the inner contact being at the common boundary point. The mobile body K is said to roll freely in K0 if, corresponding to each boundary point x of K0 and each rotation R, there is a translation t such that RK + t = gK has inner contact with K0 at x.

§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehen.

For any subsets E, F of a group G, written additively, we write

and

§1. Introduction and Summary. Throughout X is a complete separable metric space. We write K1 for the family of non-empty compact subsets of X. K1 may be endowed with a metric (first introduced by Hausdorff) under which K1 is complete and separable. We shall make use of the subbase for this metrizable topology of K1 given by sets of the two forms

for U open in X (see Kuratowski [4] or E. Michael [9] for a discussion of topologies on the space of subsets of X). if we shall be concerned with sets in [0, 1] × X which are universal for ℋ. To define these let us make the convention that, for D ⊆ [0, 1] × X, we write

D is said to be universal for ℋ if

Let p be an odd prime, let q be a divisor of p – 1, and let α be a primitive q-th root modulo p. For each natural number r the metacyclic group Gr is defined by

Let m and n be integers with 0<m<n and let μ be a Radon measure on ℝn with compact support. For the Hausdorff dimension, dimH, of sections of measures we have the following equality: for almost all (n − m)-dimensional linear subspaces V

provided that dimH μ > m. Here μv,a is the sliced measure and V⊥ is the orthogonal complement of V. If the (m + d)-energy of the measure μ is finite for some d>0, then for almost all (n − m)-dimensional linear subspaces V we have

Here we give necessary and sufficient conditions foi a prime ι to divide the class number of the Galois closure of a pure field of degree ι over the rationals. The work extends that of Honda in [4] and that of the first author in [8].

We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.

We study the Goldbach problem for primes represented by the polynomial $x^{2}+y^{2}+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^{2}+y^{2}+1$. This improves a result of Matomäki, which tells us that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^{2}+y^{2}+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^{2}+y^{2}+1$ contain infinitely many three-term arithmetic progressions, and that the numbers $\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$, with $\unicode[STIX]{x1D6FC}$ irrational and $p$ running through primes of the form $x^{2}+y^{2}+1$, are distributed rather uniformly.