We estimate the number of solutions to $|y^{2}-x^{3}|\leqslant X$ with $N\leqslant y\leqslant 2N$, in terms of both $N$ and $X$.

A solution is obtained of the problem of diffusion from an elevated point source into a turbulent atmospheric flow over a horizontal ground z = 0. The mechanism of the turbulence is the one considered by D. R. Davies [1[ when he obtained a solution of the same problem in the case when the source is at ground level, the specifications for the mean wind velocity V, and for the across-wind diffusivity Ky, and for the vertical diffusivity Kz, being V αzm, Ky αzm, Kz αz1−m, where m is a constant. Predictions of the solution are that the maximum concentration at ground level of the diffusing matter varies inversely as h1·8 in adiabatic conditions, where h is source height, and that the distance downwind from the source to the point where this maximum concentration is attained varies, in these conditions, as h1·3.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.

A steady two-dimensional motion of viscous liquid resulting from a maintained tangential velocity on a part of the bounding surface is considered. It is assumed that the liquid is contained in a long circular cylinder of radius a, and that a constant tangential velocity V is maintained over an arcual length 2b of the boundary. It should be possible to approximate to these conditions experimentally.

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

Hyperplane mean values of non-negative subharmonic functions have been studied in many papers, of which [2] and [3] are examples. Recently, Armitage [1] began a study of hyperplane means of non-negative superharmonic functions. One of his results [1, Theorem 2] shows that, if w is a positive superharmonic function on

and n/(n + 1) p < 1, then

as t → ∞, while another [1, Theorem 1] shows that, if 0 < p ≤ n/(n + l), then the integral in (1) is always infinite. However, he did not present a complete analogue of the result of Flett and Brawn [2, 3], which states that, if Ф: [0, ∞ [ → [0, ∞ [ is a non-decreasing convex function such that Ф(u)/u → 0 as u → 0 +, and w is a nonnegative subharmonic function on Rn × ]0, ∞ [, then under certain conditions on the size of w, the integral mean

tends to zero as t → ∞. In this note we present an analogue for superharmonic functions of the above result, in which the mean M(Ф(w); t) is shown to tend to infinity with t, provided that Ф(u)/u → ∞ as u → 0 +, and which therefore generalizes (1). It might be expected that, in dealing with the superharmonic case, the function Ф would have to be concave, so that Ф(w) would also be superharmonic. It turns out that this condition is unnecessary.

An old theorem of Pólya and Carlson [2] states that, if the power series has rational integer coefficients, positive radius of convergence, and can be continued analytically to a region that contains points outside the closed unit disc, then the function that the power series represents is rational. This result has been extended in a number of ways (cf. e.g. Petersson [4]). The present note gives a new extension based on a recent theorem of Güting [3]. My thanks to Professors Henry Helson and Raphael Robinson for introducing me to this subject.

A PV-number is defined to be an algebraic integer θ, of modulus greater than one, all of whose conjugates (excluding θ itself) lie inside the unit circle. Salem [1] has shown that the set S of PV-numbers forms a closed subset of the real line.

The problem dealt with in this paper was suggested by Dr. E. C. Dade and communicated to me, at Stockholm in August 1962, by Dr. Taussky Todd. It may be stated as follows. Consider the equation

where f is a form (i.e., a homogeneous polynomial) with coefficients in some ring R of algebraic integers. An obviously necessary condition for the solubility of (1) with the xi in R is that the coefficients of f should have no common factor, except for units of R. Now let S be a ring which is an algebraic extension of R, and let us try to solve (1) with the xi in S. The condition just mentioned remains necessary; and Dade has proved in [1] that for some S (depending on R and f) it is sufficient. His theorem is valid for forms of any degree but is merely an existence theorem as far as S is concerned. The problem is to do better for the special case in which f is of degree 2 and R the ring of rational integers; and more precisely, to show if possible that S can be taken to be a quadratic extension R(√q) of R, with an integer q which could be estimated in terms of the coefficients of f.

We study two-weight norm inequalities for a vector-valued operator from a weighted $L^{p}(\unicode[STIX]{x1D70E})$-space to mixed norm $L_{l^{s}}^{q}(\unicode[STIX]{x1D707})$ spaces, $1<p<\infty$, $0<q<p$. We apply these results to the boundedness of Wolff’s potentials.

In a recent book, L. H. Loomis has obtained the “Bohr compactification” of a topological group, in terms of almost periodic functions, by applying the representation theory of commutative B-algebras. It is simpler, and perhaps more natural, to consider this matter from the point of view of comparative topology; we can then obtain a more general result, in that the discussion is no longer restricted to the case of numerically valued (or even vector-valued) functions.

The temporal evolution of nonlinear, incompressible ensembles is examined first for the one-dimensional Burgers' equation and then for the incompressible, unsteady Navier-Stokes equations. It is shown that local closure of the averaged problem can be obtained for finite ensembles of Burgers' equation in the limit as the number of moments tends to infinity. This limit behaviour is verified via direct numerical computations for the onedimensional inviscid and viscous Burgers' equation. Closure is found to occur at reasonably low order. It is shown that this technique can be extended to obtain a local closure of the convective terms of the Navier-Stokes Reynoldsaveraged equations.

Let a−N,…, aN be any complex numbers and let

If y1,y2,…,yM are any real numbers such that

one form of the large sieve inequality (see [1], Chapter 2 and the Appendix) asserts that

where yM+1 ≡ y1 (mod 1).

It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^{n}$ tiles $\mathbb{Z}^{d}$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots ,k\}\subset \mathbb{Z}$ with its middle point removed: they showed that this tiles $\mathbb{Z}^{d}$ for $d=2k^{2}$, and they asked if the dimension needed tends to infinity with $k$. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles $\mathbb{Z}^{4}$.

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.

Littlewood [5, Problem 4.19, originally 4] conjectured that there is an absolute constant C > 0 such that, for every sequence of distinct integers n1, n2, n3, …, if

then

Cohen [2] showed

for some absolute constant C, with b = 1/8. Davenport [3] gave a more explicit version of Cohen's proof and improved the estimate to b = 1/4. Pichorides [6] added another refinement to obtain b = ½, and has, more recently, obtained ‖fN‖1>C(log N)1/2. This seems to be the best estimate so far without restriction on the sequences. We shall show that the methods of Davenport and Pichorides may be extended to obtain better results for certain classes of sequences. Specifically, we prove the following theorems.

Let K be a successor cardinal. Call an regularizing family (for short, a regularizing family) if Card ℱ = k and for each with Card . In [7], Prikry proved, assuming V = L, that if is an ultrafilter on K which is uniform (Card X = K, each ) then is regular (some forms a regularizing family).†

In [5] we have developed part of a theory of K- analytic sets that forms a common generalization of the theory of Lindelöf K- analytic sets developed by Choquet, Sion and Frolik and the theory of metric analytic sets developed by Stone and Hansell. As we explain in [5], this theory parallels the recently developed theory of Frolík and Holický, but has certain advantages. In this paper we take the theory rather further, and, in particular, we prove a number of variants of Lusin's first separation theorem and give some of their applications. We make free use of the definitions, notation and conventions introduced in [5].

For a typical convex body in Ed a typical shadow boundary under parallel illumination has infinite (d - 2)-dimensional Hausdorff measurewhile having Hausdorff dimension d 2.