Journal of the London Mathematical Society - current issue

Two and a Half Remarks on the Marica-Schonheim Inequality

Aharoni, R., Holzman, R. — 1993-12-01

The Marica-Schönheim inequality states that the number of distinct differences of the form A\B, with A, B taken from a given finite family A of sets is at least |A|. We prove that equality occurs essentially if and only if A is the product of an ideal and a filter. We also prove an infinite version of the theorem, conjectured (in weaker form) by Daykin and Lovasz. Finally, we note that a generalization (due to Ahlswede and Daykin) of the inequality which considers two families A and B holds under a weaker assumption on the relation between A and B.

Submodules of the Deficiency Module

Migliore, J. C. — 1993-12-01

Proof of the Evans-Root Conjectures for Selberg Character Sums

Van Wamelen, P. B. — 1993-12-01

From a Large Sieve to the Orthogonality of Operators

Elliott, P. D. T. A. — 1993-12-01

On the Number of Characters in a Block and the k(GV) PROBLEM FOR SELF-DUAL V

Gow, R. — 1993-12-01

Sobolev Inequalities and Harmonic Functions of Polynomial Growth

Alexopoulos, G., Lohoue, N. — 1993-12-01

We prove a Sobolev inequality for functions not necessarily with compact support, on a connected Lie group G of polynomial volume growth. To prove this inequality we have to characterise the harmonic functions of polynomial growth on G.

Integral Operators on the Cone of Monotone Functions

Stepanov, V. D. — 1993-12-01

Necessary and sufficient conditions for the boundedness of linear integral operators from $${L}_{U}^{P}$$(R⁺) to $${L}_{Q}^{W}$$(R⁺) restricted to the cones of monotone functions are given. In addition a general approach to a number of classical operators is explicitly described. In particular, we determine when the Hardy-Littlewood maximal operator is bounded in the classical Lorentz space _p(v) consisting of those measurable functions on Rⁿ such that $${\left({\int }_{0}^{\infty }f**{\left(t\right)}^{p}\upsilon \left(t\right)dt\right)}^{1/p}\hbox{ \hspace{0.17em} } < \infty .$$

The Boundary Behaviour of Bloch Functions

Rohde, S. — 1993-12-01

Proof of a Conjecture of Hayman Concerning f and f''

Langley, J. K. — 1993-12-01

We prove the following, which confirms a conjecture of W. K. Hayman from 1959. If f is meromorphic in the plane such that f and f'' have only finitely many zeros, then f(Z) = R(z) exp (P(Z)), where R is rational and P is a polynomial. The theorem is related to earlier results of Frank, Mues and others.

Uniqueness and Extension Theorems for Subharmonic Functions

Gardiner, S. J. — 1993-12-01

Automatic Well-Posedness with the Abstract Cauchy Problem on a Frechet Space

Delaubenfels, R. — 1993-12-01

For an arbitrary closed linear operator, A, on a Fréchet space, we introduce what we shall call its solution space. This is a Fréchet space that contains all initial data for which the corresponding abstract Cauchy problem has a unique global mild solution. We show that A, restricted to this space, generates a locally equicontinuous strongly continuous semigroup.

Corollaries include an almost immediate proof of the fundamental relationship between generating a strongly continuous semigroup and having a unique mild solution, for all initial data. More generally, we show how the solution space may be used to present a simplified and unified approach to different types of semigroups and their relationships to each other and the abstract Cauchy problem.

Quantum Flows with Unbounded Structure Maps and Finite Degrees of Freedom

Fagnola, F., Sinha, K. B. — 1993-12-01

We prove a general existence theorem for quantum flows with finite degrees of freedom and unbounded structure maps satisfying an analyticity assumption.

The Rate of Escape for Pairs of Windings on the Riemann Sphere

Gruet, J-C., Mountford, T. S. — 1993-12-01

At a typical time t, the distribution of the continuous argument of Brownian motion on the sphere about a typical point is Cauchy like. One might expect that a pair of windings would have the same rate of escape as two independent Cauchy processes (as computed in Taylor [8]), we show however that this is not the case. We also give an exact integral test for the rate of escape.

Transplantation Et Isospectralite II

Berard, P. — 1993-12-01