We give a construction of 2s, s = n(n – 1)/2, many natural almost complex structures on the orthogonal twistor bundle over a 2n-dimensional Riemannian manifold. The usual almost complex structures are then characterised by the condition that they correspond to integrable invariant complex structures on the standard fibre which is identified with the hermitian symmetric space SO(2n)/U(n).

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

We consider the following non-linear nonautonomous second order differential equation

where h(x) is continuous, f, p are continuous and periodic with respect to t of period w. Using the Leray-Schauder fixed point technique we prove that the above equation possesses at least one non-trivial periodic solution of period w.

Let R be a ring with centre Z. In this note, we prove the following: If the additive group Z+ of Z has finite group-theoretic index in R+, then R has an ideal I contained in Z such that R/I is a finite ring. This is a solution of a problem posed by F.A. Szász.

Neumann's totally ordered power series fields and division rings may be tipped over to form archimedean lattice-ordered fields and division rings. This process is described and then generalised to produce non-archimedean lattice-ordered fields and division rings in which 1 > 0 and, as well, ones in which 1 ≯ 0.

Section 1 addresses the problem of covers in Sub D, the lattice of subalgebras of a Boolean algebra; we describe those BA's in whose subalgebra lattice every element has a cover, and show that every small and separable subalgebra of P(ω) has 2ω covers in SubP(ω). Section 2 is concerned with complements and quasicomplements. As a general result it is shown that Sub D is relatively complemented if and only if D is a finite– cofinite BA. Turning to Sub P(ω), we show that no small and separable D ≤ P(ω) can be a quasicomplement. In the final section, generalisations of packed algebras are discussed, and some properties of these classes are exhibited.

In this paper, we shall characterise the B(k)-property in generalised ordered (GO) spaces as follows.

For every uncountable regular cardinal K, every GO space has the B(K)-property if and only if it has no closed subspace which is homeomorphic to a stationary set in K (with the subspace topology in K).

We prove that the classic interpolation spaces of Calderón can be defined using spaces of functions that satisfy weaket conditions. For Calderón's second space we use a space of functions defined by Cwickel and Janson; we then modify their definition to find another space of functions which defines Calderón's first space.

For α ≥ 0 and β ≥ 0, let K(β) consist of those functions f(z), analytic and non-zero in |z| < 1, such that for θ1 < θ2 < θ1 + 2π and . It is conjectured that for 1 ≤ α ≤ β and α an integer, the weighted convolution of polynomials having their zeros on |z| = 1 and belonging to K(α, β), also belong to K(α, β). This conjecture is known to be true for the case α = 1, which leads to an alternative proof for the generalised Polya-Schoenberg conjecture. The case α = 2 is also known to be true for cubic polynomials. We prove the conjecture for certain quartic polynomials when 2 ≤ α ≤ 4.

A self-contained elementary account is given of the theorem of S. Agou that classifies all composite irreducible polynomials of the form over a finite field of characteristic p. Written to appeal to a wide readership, it is intended to complement the original rather technical proof and other contributions by the author and by Moreno.

Let S = {x1, x2, …, xn} be a set of distinct positive integers. The n × n matrix [S] = (Sij), where Sij, = (xi, xj), the greatest common divisor of xi, and xj, is called the greatest common divisor (GCD) matrix on S. H.J.S. Smith showed that the determinant of the matrix [E(n)], E(n) = { 1,2, …, n}, is ø(1)ø(2) … ø(n), where ø(x) is Euler's totient function. We extend Smith's result by considering sets S = {x1, x2, … xn} with the property that for all i and j, (xi, xj) is in S.

A topological space is said to be locally dyadic if every neighbourhood of a point contains a dyadic neighbourhood of that point. It is proved here that every locally compact Hausdorff topological group is locally dyadic.

Consider the first order differential equation (1) , where pi, and τi, for i = 1,…,n, are positive constants. To find necessary and sufficient conditions, in terms of the coefficients and the delays only, under which all solutions of (1) oscillate, is a problem of great importence. In a recent paper, Bowcock and Yu claimed that is a necessary and sufficient condition for all solutions of (1) to be oscillatory. In this paper a counterexample shows that the above result is not valid and the error in this paper is indicated.

It is well-known that the approximation to f(x) ∈ C2π, by nth trigonometric Lagrange interpolating polynomials with equally spaced nodes in C2π, has an upper bound In(n)En(f), where En(f) is the nth best approximation of f(x). For various natural reasons, one can ask what might happen in Lp space? The present paper indicates that the result about the trigonometric Lagrange interoplating approximation in Lp space for 1 < p < ∞ may be “bad” to an arbitrary degree.

Let G be a locally compact abelian group, with character group Ĝ. Let ψ be an arbitrary continuous real-valued homomorphism defined on Ĝ. For f in LP(G), 1 < p ≤ 2, let

where 1[−ν, ν] is the indicator function of the interval [ − ν, ν ], and I is an unbounded increasing sequence of positive real numbers. Then there is a constant Mp, independent of f, such that ‖M#f‖p ≤ Mp ‖f‖p. Consequently, the pointwise limit of the function exists, almost everywhere on G, as ν tends to infinity. Using this result and a generalised version of Riesz's theorem on conjugate functions, we obtain a pointwise inversion for Fourier transforms of functions on Ra × Tb, where a and b are nonnegative integers, and on various other locally compact abelian groups.

Let f(z) be a n-valued algebroid function of order λ(0 ≤ λ ≤ 1), Sato obtained an elliptic theorem for f(z) with a condition. In this paper, we prove that Sato's theorem is true without conditions and give a generalisation.

Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

We prove the Legendre Necessary Condition of the Calculus of Variations in Mean an arbitrary finite dimension. When the Lagrangian is convex, we establish that if the Euler-Lagrange equation possesses an almost periodic solution then it possesses periodic and constant solutions. We deduce from this fact various consequences on the structure of the set of almost periodic solutions.

In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument.

Huq presented a general study of semi-homomorphisms of rings, following, amongst others, Kaplansky's study of semi-automorphisnis of rings and Herstein's study of semi-homomorphisms of groups. Huq gave several “sufficient” conditions for a semi-homomorphism and a semi-monomorphism of rings to be a homomorphism and a monomorphism respectively. In this note we introduce semi-subgroups of groups, provide counterexamples to four of Huq's assertions and show how a minor, albeit forced, change to one of the conditions of the fourth assertion turns it into a special case of another theorem of Huq's.