A unified method is used to improve the order of the lower bound sieve.

Let A = {ala2,…, an} be a finite set of (not necessarily distinct) positive integers and

be the corresponding set of multiples. My primary object here is to show that in fairly general circumstances there are significant irregularities in B(A), regarded as an ordered sequence.

In this paper some new Opial-type integrodifferential inequalities in one variable are established. These generalize the existing ones which have a wide range of applications in the study of differential and integral equations.

Let K be an algebraic number field of degree n and discriminant d. Let K(1),…, K(n) be the embeddings of the field. Then n = r1 + 2r2 where K(1), …, K(rl) are real and the remainder complex, satisfying . The conjugates of the number μ in K(i) are denoted by μ(i.

Let be the sum of the m-th powers of the zeros of the Bessel polynomial yn(x). It is known that for m = 0, 1, 2, …,

where cm(v) is the Hawkins polynomial. In this paper we find rational functions wm(v) such that for m = 0, 1, 2, …

In recent papers on fractals attention has shifted from sets to measures [1, 5, 10]. Thus it seems interesting to know whether results for the dimension of sets remain valid for the dimension of measures. In the present paper we derive estimates for the dimension of product measures. Falconer [3] summarizes known results for sets and Tricot [8] gives a complete description in terms of Hausdorff and packing dimension. Let dim and Dim denote Hausdorff and packing dimension. If then

There exists a family of pairwise disjoint congruent infinite circular cylinders such that no two cylinders of are parallel and the density of is greater than zero.