Let R be a directly indecomposable finite ring. Let p be a prime, let m be a positive integer and suppose the radical of R has pm elements. Then we show that . As a consequence, we have that, for a given finite nilpotent ring N, there are up to isomorphism only finitely many finite rings not having simple ring direct summands, with radical isomorphic to N. Let R* denote the group of units of R. Then we prove that (1 − 1/p)m+1 ≤ |R*| / |R| ≤ 1 − 1/pm. As a corollary, we obtain that if R is a directly indecomposable non-simple finite 2′-ring then |R| < |R*| |Rad(R)|.

Let K be a 2-dimensional Cantor set. In this note we prove, in two cases, the analytic capacity and the continuous analytic capacity of K are equal.

We introduce a coefficient on general Banach spaces which allows us to derive the weak normal structure for those Banach spaces whose Banach-Mazur distance to James quasi reflexive space is less than .

Sufficient conditions are established for the oscillation of solutions of hyperbolic equations of neutral type of the form

where R+ = {0, ∞), Ω is a bounded domain in Rn with a piecewise smooth boundary ∂Ω.

An associative ring R is called a BT-ring if R is strongly right bounded, but not right duo, and not strongly left bounded. We show that the order of the smallest BT-rings (without unity) is 16, while we prove earlier that the order of the smallest unitary BT-rings is 32.

Some generalised invex conditions are given for a nondifferentiable constrained optimisation problem, generalising those of Hanson and Mond for differentiable problems. Some duality results are obtained.

For f ∈ S, we study the Schwarzian coefficients sn defined by {f, z} = Σ snzn. Sharp bounds on s0, s1 and s2 are given, together with an order of growth estimate as n → ∞. We use the Grunsky Inequalities to estimate combinations of coefficients.

We apply Straßmann's theorem to p–adic power series satisfying linear differential equations with polynomial coefficients and note that our approach leads to our estimating the number of integer zeros of polynomials on a given interval and thence to an investigation of the number of p–adic small values of a function on such an interval, that is, of the number of solutions of a congruence modulo pr.

Let X be a real Banach space with a uniformly convex dual, X*, and let C be a nonempty closed convex and bounded subset of X. Let T: C → C be a strongly accretive and a continuous mapping. For any f ∈ C, let S: C → C be defined by Sx = f + x – Tx for each x ∈ C. Then, the iteration process xo ∈ C,

under suitable conditions on the real sequence converges strongly to a solution of the equation Tx = f in C. Furthermore, if T is strongly accretive and Lipschitz with Lipschitz constant L ≥ 1 then the iteration process x0 ∈ C,

under suitable conditions on the real sequences and converges strongly to a solution of the equation Tx = f in C. Explicit error estimates are obtained.

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.

We study the asymptotic behaviour as t → ∞ of solutions of the initial-boundary value problem vt = G(u, v), ut = uxx + F(u, v), and t > 0, x ∈ ℝ or x ∈ ℝ+ for a wide class of initial and boundary values, where F and G are smooth functions so that the system has three rest points.

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

The Hecke groups

are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).

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.

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 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.