Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x3 = x. (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x3 = x. (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If RRR is quasi-projective, then R is a finite direct sum of copies of GF(2) and/or GF(3).

In this paper we give another version of the generalised dominated convergence theorem, which is better than other convergence theorems for Perron integrals in the sense that it can be applied more easily.

We find conditions for which a manifold endowed with an integrable almost transversal structure is the total space of a vector bundle isomorphic to the transversal bundle of a foliation.

In this paper we deal with nonlinear second order boundary value problems for ordinary differential equations including the case of jumping nonlinearities. The set of generalised eigenvalues in the case of nonconstant coefficients is described. It is proved that these generalised eigenvalues are simultaneously bifurcation points of the problem with coefficients also depending on the solution u = u(x).

In this paper, we prove that if h is a generator of a Z2n, action on S1 × S1 × S1, and Fix(hn) consists of two disjoint tori, one torus, four simple closed curves, or two simple closed curves, then h is equivalent to the obvious actions.

Suppose the edges of the complete graph on 17 vertices are coloured in three colours. It is shown that at least five monochromatic triangles must aris.

The commutator has the following order theoretic properties: [α, β] ≦ α ∧ β, [α, β] = [β α],[α1 ∨ α2,β] = [α1, β] ∨ [α2, β] for congruences α, β ∈ Con A of an algebra A in a congruence modular variety generalising the original concept in group theory. A tolerance of a lattice L is a reflexive and symmetric sublattice of L2. We show that to every commutator [ , ] of Con A corresponds a ∧-subsemilattice of the lattice of tolerances of Con A. It can be shown that A in a congruence modular variety is nilpotent if |con A| > 2 and Con A is simple.

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.

Boyle and Goodearl proved that if R is a left V-ring then R has left Krull dimension if and only if R is left Neotherian. In this paper we extend this result to arbitrary V-modules.

We prove a characterisation of locally s-regular manifolds using the theory of homogeneous structures.

For a very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

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.

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 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.

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.