This paper discusses rational edged tetrahedra, in 3, 4 and n dimensions, with rational volume. The main results are (i) a proof of the existence of infinitely many tetrahedra with rational edge-lengths, face-areas and volume and (ii) a proof that there exist dimensions for which all regular hypertetrahedra with rational edge-lengths have rational hypervolume.

We study a concept of stability under the gap of isomorphic properties of Banach spaces and apply it to obtain some results of stability under compact or small norm perturbation for non-semi-Fredholm operators with closed range.

We consider twists of matrix algebras by some continuous characters. There are some subgroups of Brauer groups arising from these twists. Some results regarding these subgroups are proved.

In this paper, we shall prove a generalisation of Himmelberg's fixed point theorem and as applications, the existence of equilibrium points for abstract economies given by preference correspondences and utility functions have been established.

In this paper we have evaluated the integral . A new integral representation of the Euler constant is shown. Some special cases of the result are discussed and an open problem is posed.

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

We provide sufficient convergence conditions as well as sharp error bounds for Newton-like iterations which generalise a wide class of known methods for solving nonlinear equations in Banach space.

Suppose that f, g ∈ L∞[0,1] have discontinuities of the first kind only. Using the measure, max{‖f − h‖p, ‖g − h‖p}, of simultaneous Lp approximation, we show that the best simultaneous approximations, hp, to f and g by nondecreasing functions converge uniformly as p → 1. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. We discuss the inheritance of properties of f and g by hp, and of hp by h1.

We study the existence of positive solutions of the periodic, Neumann or Dirichlet problem for the semilinear equation u″ + f(t, u) = 0, 0 ≤ t ≤ T, where f is a Carathéodory function. Our assumptions in each case are such that the problem possesses a lower solution or an upper solution.

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

An example of a Banach space is given which does not contain isomorphically l1 and does not admit the Kadec-Klee property.

Let W be an n−dimensional vector space over a field F. It is shown that the expected dimension of a vector subspace of W is n/2. If F is infinite, the expected dimension of a sum of a pair of subspaces of W is (n + 1)/2 if n > 1; and 3/4 if n = 1. If F is finite, with q elements, the expected dimension of a sum of subspaces of W depends on q and n. For fixed n, the limiting value of this expectation as q → ∞ is n if n is even; and n − 1/4 if n is odd. Moreover, if F is finite and n > 1, the expected dimension of a sum of three (not necessarily distinct) subspaces of W has limit n as q → ∞.

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

In this paper we introduce the concept of ℙ-regular, normal subset of a ℙ-regular semigroup, give an alternate characterisation of the maximum idempotent separating congruence in a ℙ-regular semigroup and finally describe the lattice of the idempotent separating congruences in a ℙ-regular semigroup.

This paper examines conditions for the monotone convergence of general Newton-like methods generated by point-to-point mappings. The speed of convergence of such mappings is also examined.

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

Popov has recently introduced a class of subspaces of Lp(μ) (μ nonatomic) which generalise the finite codimensional ones, and proved that for p ≠ 2 any projection onto such a subspace has a norm strictly greater than one. In this paper we give the quantitative version of Popov's result computing the best possible lower bound for the norms of the considered projections.

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

The loxodromic mapping conjecture of J. Harrison is affirmed for diffeomorphisms of the 2-sphere that embed in flows.