In 1959 C. A. Rogers gave the following estimate for the density ϑ(k) of lattice-coverings of euclidean d-space Ed with convex bodies . Here, c is a suitable constant which does not depend on d and K. Moreover, Rogers proved that for the unit ball Bd the upper bound can be replaced by , which is, of course a major improvement. In the present paper we show that such an improvement can be obtained for a larger class of convex bodies. In particular, we prove the following theorem. Let K be a convex body in Ed, and let k be an integer satisfying k > log2 loge d + 4. If there exist at least k hyperplanes H1,…, Hk with normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to Hl,…,Hk, respectively, then . Actually, for a bound of this type we do not even need any symmetry assumption. In fact, some weaker properties concerning shadow boundaries will suffice.

A basic notion in the classical theory of differentiation is that of a differentiation base. However, some differentiation type theorems only require the less restricted notion of a contraction. We demonstrate that the classical criteria, such as the covering criteria of de Possel, continue to hold in the new setting.

For d, n positive integers, let α(d, n) be the statement “For every abelian variety A bounded by d there exists a point P of order n”. Let l1, l2, l3, … be the sequence of prime numbers and let α be

When a displacement front meets a heterogeneity in a porous medium, its shape will be altered. The amount of distortion depends on the size and shape of the heterogeneity, the amount of variation of the heterogeneous properties, and the mobility ratio between the displaced and displacing fluids. The solution for a circular permeability discontinuity is known when the capillary pressure is uniform and the mobility ratio M is unity [4]. Here, we extend the theory to the case where there is a small change ε in capillary pressure as well as a nearly unit mobility ratio M = 1 + εγ. Corrections can then be found, in closed form, to first order in ε, to the shape of the front and the pressure field. Computations using these expressions are simpler than the full free-boundary problem, and some analytical estimates are possible in further limits. Finally, the theory is extended to the case of a front passing a number of such heterogeneous patches which are widely spaced.

For (x, t) ∈ Rn+1, we put

This paper considers the flow of a dissipative fluid in a radial bearing. By looking at the equations in the thermal boundary layer it is shown that, if a certain parameter m is less than unity, then the temperature in the boundary layer is bounded for all time.

The asymptotic behaviour of a sequence of polynomials cm = cm(v) satisfying

is established. These polynomials occur in Hawkins' formula for the residues of a Bessel-zeta function at its possible poles in the left half plane. The results imply that cm(v)/cm(0) converges uniformly to cos πV on compact sets. This in turn implies that, for v not a half odd integer, all but finitely many of the possible poles are actual poles.

In our paper [12[ we made extensive use of the details of the proofs given in our earlier paper [11], and, in particular, we claimed that Lemma 3 of [11] holds, not just when Y is a metric space, but also when Y is a Hausdorff space, provided X × Y is a Fréchet space. In a corrigendum to [11], we give a corrected version of this Lemma 3, but it seems to depend, in an essential way, on the assumption that Y is a metric space, or at least a perfectly normal space. In this note we show that a modified version of this Lemma 3 enables us to justify all the theorems in [12] by use of a modified method of selection.

Some interlacing properties of the zeros of the generalized Airy functions A1(z, p) are given for non-positive integral values of p. The result that A1 (z,p) has no real zero for is extended to show that all the zeros of A1(z,p) are real and simple if . It is also shown that all the zeros of the functions Bk(z,p, 1) for k = 1, 2, 3 are simple for non-positive integral p.

Consider a point transformation of the biharmonic equation

namely a coordinate transformation

together with a change of dependent variable given by

for some multiplier F(ξ, η).

In the paper which forms the basis for the modern asymptotic theory of linear differential systems, Levinson [22] considered the system

on an interval [a, ∞), where

(1) A is a constant nÜn matrix with n distinct eigenvalues and either

(2a) R(x) is L(a, ∞)

or

(2b) R(x)→0 as x→∞ and R'(x) is L(a, ∞).

Suppose that f(z) is non-constant and meromorphic in the plane and that, for some k≥= 1, a0(z),…, ak(z) are meromorphic in the plane with

for j' = 0,…, k. Here, using standard notation from [3], S(r,f) denotes any quantity satisfying S(r,f) = o(T(r,f)) as r→ ∞, possibly outside a set of finite linear measure. Then, setting

we have ([3, p. 57])

Theorem A. Suppose that f(z) is non-constant and meromorphic in the plane, and thatψ (z) given by (1.2) and (1.1) and is non-constant. Then

where N0(r, l/ψ') counts only zeros of ψ' which are not zeros of ψ − 1, and thecounting functions count points without regard to multiplicity.

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].