We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

In this paper we study the oscillatory behaviour of solutions of equations of the forms , where pi, and τi, for i = 1, 2,…, n, are positive constants and p is an arbitary constant. Three necessary and sufficient conditions for all solutions of (A) and (B) respectively to be oscillatory are established. These results extend and generalise previous results in a new approach.

A class of problems in forced convection give rise to quadratic eigenvalue problems. In this paper it is shown that the eigenvalues are necessarily real. The extension of this result to a wider class of problems is also discussed.

We apply Newton-like methods to operator equations where the operator has Hölder continuous derivatives. Our results reduce to the ones obtained by Rockne when the ordinary Newton method is applied to find solutions of nonlinear operator equations.

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(M))mn, where Mmn is the matrix Γnm-ring. τ is shown to be a special radical in the variety of Γ-rings. τ1 is the upper radical determined by the class of usp Γ-rings of bound 1. τ ⊆ τ1, but the reverse inclusion does not hold in general. The place of τ and τ1 in the hierarchy of radicals for Γ-rings is shown.

Maximum and minimum values are obtained for some sums invovling the divisor functions of number theory.

We find necessary and sufficient conditions on a topological space X so that S(X), the semigroup of all continuous selfmaps of X, is isomorphic to the multiplicative semigroup of a near-ring. The analogous problem is also considered for the semigroup of all continuous selfmaps which fix some point of X.

We show that if a three dimensional normed space X has two linearly independent smooth points e and f such that every two-dimensional subspace containing e or f is the range of a nonexpansive projection then X is isometrically isomorphic to ℓp(3) for some p, 1 < p ≤ ∞. This leads to a characterisation of the Banach spaces c0 and ℓp, 1 < p ≤ ∞, and a characterisation of real Hilbert spaces.

In this article we give an explicit representation of p-adic Dedekind sums and their reciprocity laws by using p-adic measure theory. We then study the consequences of the p-adic reciprocity law for particular positive integer values in which case we can recover a reciprocity law for Dedekind sums attached to particular Dirichlet characters. This gives a proof different from that of Nagasaka.

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.