In 1984 Heath-Brown [5] proved the following conjecture of Erdős and Mirsky [2] (which seemed at one time as hard as the twin prime problem):

“There exist infinitely many integers n for which d(n) = d(n + 1).”

Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K = r (mod 2 π), on [-1, 1 ] and on ℝ. The key result is the establishment of the inequality

for every rational function f=pn/qn, where pn is a polynomial of degree at most n with complex coefficients and

with | aj | ≠ 1 for each j and for every zo∈ δ D, where δ D,= {z∈ ℂ: |z| = l}. The above inequality is sharp at every z0∈δD.

This paper is a contribution to the general problem of differentiability of Lipschitz functions between Banach spaces. We establish here a result concerning the existence of derivatives which are in some sense between the notions of Gâteaux and Frechet differentiability.

In this paper we prove that a maximal common summand of two compact convex sets in R2 is unique up to translation.

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

The equation

where ℱ is a certain complex-valued function of the given real periodic function λ, is studied analytically and numerically. The equation is motivated physically by a boundary-layer stability problem in which λ represents the skin-friction of the undisturbed basic flow profile. It is proved that no periodic neutral solutions exist for any attached basic flow and the implications of this result for certain vortex-wave interactions are discussed.

In [3] the authors introduced the notion of a completely 0-simple semigroup of quotients. This definition has since been extended to the class of all semigroups giving a definition of semigroups of quotients which may be regarded as an analogue of the classical ring of quotients. When Q is a semigroup of quotients of a semigroup S, we also say that S is an order in Q.

The prototype of isoperimetric problems is to minimize the surface area of a convex body with given volume. The minimal body is naturally the suitable ball. The solution to this problem in the planar case was already known to the ancient Greeks. In the higher dimensional cases, the first proofs were provided with the help of Steiner's symmetrization method towards the end of the last century. Important later contributors are, among others, Minkowski, Blaschke, Hadwiger. By their work, the optimality of the ball has been also verified for a much wider class of sets (see [14]).

We recall that if S is a d - simplex then each facet and each vertex figure of S is a (d − 1)-simplex and S is a self-dual. We introduce a d-polytope P, called a d-multiplex, with the property that each facet and each vertex figure of P is a (d − 1)-multiplex and P is self-dual.