We generalise Chuang's inequality to vector meromorphic functions, which is originally a sort of extension of Nevanlinna's second fundamental theorem.

A number of Szegö-type prediction error formulas are given for two-parameter stationary random fields. These give rise to an array of elementary inequalities and illustrate a general duality relation.

In this paper we study the variational-like inequalities, which generalise some results of Parida, Sahoo and Kumar, and we also investigate the quasivariational-like inequalities. We establish some existence theorems of a solution for the above problem.

It is shown that for the finite Hilbert transform Tp on the Banach space Lp(]–1, 1[), 1 < p < ∞, the linear operator is not strictly singular whenever n is a positive integer.

Let p be a prime and d, e positive integers. We prove that a regular d-generator metabelian p-group whose commutator subgroup has exponent pe has nilpotency class at most e(p – 2) + 1 unless e = 1, d > 2, p > 2 when the class can be p and these bounds are best possible.

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

A simple new proof is given of a result of Vaughan-Lee which implies that if G is a relatively free nilpotent group of finite rank k and nilpotency class c with c < k then the characteristic subgroups of G are all fully invariant. It is proved that the condition c < k can be weakened to c < k + p − 2 when G has p–power exponent for some prime p. On the other hand it is shown that for each prime p there is a 2-generator relatively free p-group G which is nilpotent of class 2p such that the centre of G is not fully invariant.

The one-sided Laplace transform is defined on a space of generalised functions called transformable Boehmians. The space of one-sided Laplace transformable distributions is shown to be a proper subspace of transformable Boehmians. Some basic properties of the Laplace transform are investigated. An inversion formula and an Abelian theorem of the final type are obtained.

The necessary and sufficient conditions for atomic orthomodular lattices to have the MacNeille completion modular, or (o)-continuous or order topological, orthomodular lattices are proved. Moreover we show that if in an orthomodular lattice the (o)-convergence of filters is topological then the (o)-convergence of nets need not be topological. Finally we show that even in the case when the MacNeille completion of an orthomodular lattice L is order-topological, then in general the (o)-convergence of nets in does not imply their (o)-convergence in L. (This disproves, also for the orthomodular and order-topological case, one statement in G.Birkhoff's book.)

We examine the rings in which every sequence has either a subsequence, or a rearrangement, such that for some n the product of the first n terms in zero.

We show that a right duo ring R is strongly regular if and only if for each ideal I of R, the coset product of I in the factor ring R/I is the same as their set product.