The results developed by Watson [1] are interpreted to indicate how the slow viscous flow due to the rotation of a small circular cylinder in the presence of a stationary cylinder can be calculated. It is shown how the stream function is given as a combination of the force-free representations corresponding to a line rotlet and a line stokeslet outside the stationary body, plus the streaming flow past the body. The coefficients which multiply these representations are calculated by techniques already described by Watson.

A double elastic panel, set in a light compressible fluid, is excited by a time harmonic force applied along a line on one of the plates. The double panel consists of two parallel elastic plates, with different elastic properties, each of width a and separated by distance d≪a, set in rigid plane baffles with acoustically soft adjoining side walls. Each plate is taken to have infinite extent in the z-direction, so that the problem treated is a two-dimensional one. The radiated acoustic power is estimated asymptotically, averaged over a small frequency band and over all line force positions, for frequencies that are sufficiently high to ensure the excitation of many panel modes, with ka≫1 and kd >> 1, where k is the acoustic wave number. Transition formulae are given for frequencies ω that are near to either of the coincidence frequencies, ω1 and ω2, of the individual panels.

In Section 1 of this note we will construct an example of a subset of R × Rn such that the parabolic capacity with respect to the heat equation is zero although its orthogonal projection onto {0} × Rn is the whole space. Such examples were already given by R. Kaufman and J.-M. Wu in [5] and [6]. However, our probabilistic approach seems to be more transparent since it does not depend on explicit formulas for Green functions.

Let r, k, h be positive integers. For any Dirichlet character χ modulo k write

In this short paper, we shall give a new estimate for the exponential sum S(H, M, N), where

e( ξ,) = exp (2πiξ;) for a real number ξ, am and bn are complex numbers with |am| ≤ 1 and |bn| ≤ l, H, M, N ≤1, , x is a large number, ε is a sufficiently small positive number, and Y ≤ x(½)−ε (h ∼ H means 1≤h/H < 2 and so on). In making application of the Rosser-Iwaniec linear sieve of Iwaniec [6] to find almost primes in short intervals of the type (x − y, x], Halberstam, Heath-Brown and Richert [4] first considered an estimate for S(H, M, N) to the effect that

with MN as large as possible. Later, better estimates were given in Iwaniec and Laborde [7], and Fouvry and Iwaniec [3]. Of course, the most interesting case would be finding P2 numbers in a short interval (x − y, x]. The related estimate of [7] implies that (1) holds provided that

Let Δ = {z:|Z|<1}, Γ={z:|z|=1}, and ℳ denote the set of complex-valued Borel measures on Γ. Let Kα(z)=(1−z)−α for α>0 and K0(z)=log 1/(1−z). For α > 0 let ℱα denote the family of functions f on Δ having the property that there exists a measure μ∈ℳ such that

for |z|<1. When α=0, this condition is replaced by

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequality

under the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halász [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halász [Hz 1] and Wirsing [W].

The dynamics of cluster growth can be modelled by the following infinite system of ordinary differential equations, first proposed by Smoluchowski, [8],

where cj=cj(t) represents the physical concentration of j-clusters (aggregates of j identical particles), aj,k=aj,k≥0 are the time-independent coagulation coefficients, measuring the effectiveness of the coagulation process between a j-cluster and a k-cluster, and the first sum in the right-hand side of (1) is defined to be zero if j = 1.

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

Suppose X and Y are spaces of analytic functions in the open unit disk D in the complex plane C. A sequence {λ} is called a coefficient multiplier from X to Y if the function belongs to Y whenever the function belongs to X.

For a sequence of polynomials (Pn) orthonormal on the interval [−1, 1], we consider the sequence of transforms (gn) of the series given by . We establish necessary and sufficient conditions on the matrix (bnk) for the sequence (gn) to converge uniformly on compact subsets of the interior of an appropriate ellipse to a function holomorphic on that interior.

It is shown that in all dimensions n ≥ 11 there exists a lattice which is generated by its minimal vectors but in which no set of n minimal vectors forms a basis.

Every homogeneous convex body in ℝd (d≥2) put to sit on a horizontal hyperplane finds a position of stable equilibrium. A cube has 2d such positions and an ellipsoid with pairwise distinct axis-lengths has 2. How many positions of stable equilibrium have most convex bodies?

The term “most” is understood in the Baire category sense. For various other results on most convex bodies, see [2], [4].

In the paper [2] Hsia noted that the forms x2+xy+y2+9z2 and x2+3y2+3yz+3z2 constitute a genus and that both forms are regular; he asked whether there exist any other genera containing two or more regular forms. In this note it is proved that the forms

are regular. They constitute a genus with discriminant 27 (in the normalization used by Brandt and Intrau in [1]). It is noteworthy that Hsia's genus has the same discriminant.

Let I be an ideal of a Noetherian ring R. The purpose of this paper is to study the relationship between the vanishing of the local cohomology modules , and the comparison of the topologies defined by the I-adic {In}n≥0, the symbolic {I(n)}n≥0 and the integral filtration

Wielandt [4] has shown that a common subnormal subgroup of two permutable subgroups of finite group is subnormal in their product. When G is infinite it seems unlikely that Wielandt's theorem will still be true, but an example illustrating this appears to be difficult, even if G is an FC-group (that is groups in which each element has only finitely many conjugates; see [2]). However, if we replace subnormality by ascendancy we have the following.

For a class of functions containing polynomials over ℤm, we give an inequality relating the cardinality of the value set to the additive order of differences of elements in that set. To do this, we find some inequalities concerning the combinatorics of substrings of sequences on finite sets which are related to an interesting matrix inequality.