In this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

All subnormal subgroups of hypernormalizing groups have by definition subnormal normalizers. It is shown that finite soluble HN-groups belong to the class of groups of Fitting length three. Finite HN-groups are considered including those with subnormal quotient isomorphic to SL(2,5).

Let ξ be a C*;-bundle over T with fibres {At}t∈A. Suppose that A is the C*-algebra of sections of ξ which vanish at infinity, and that (A, G, α) is a C*-dymanical system that, for each t ∈ T, the ideal It = {f ∈ A|f(t) =; 0} is G-invariant. If in addition, the stabiliser group of each P ∈ Prim(A) is amenable, then A ⋊αG is the section algebra of a C*-bundle with fibres {At ⋊αG}t∈T.

The above theorem may be used to prove a structure theorem for crossed products built from C*-dynamical systems (A, G, α) where the action of G on A is smooth. Assuming that the stabiliser groups are amenable, then A ⋊αG has a composition series such that each quotient is a section algebra of a C*-bundle where the fibres are of the form Aσ ⋊αG; moreover, the Aσ correspond to locally closed subsets of Prim(A), and G acts transitively on Prim(Aσ). In many cases, in particular when (G, A) is separable, the Aσ ⋊αG have been computed explicitly by other authors.

These results are actually proved for twisted C*dynamical systems.

A method is proposed for driving degenerate feasible solutions to linear programming problems away from essential degeneracy and in particular for identifying essentially degenerate optimal solutions. An essentially degenerate cycling example is also given, so answering a question raised earlier.

We prove that in a non-isotropic Euclidean space, homogeneous Lipschitz spaces of distributions, defined in terms of (generalized) Weierstrass integrals, can be characterized by means of higher order difference operators.

Let Q be the space of all rational numbers and (X, τ) be a topological space where X is countably infinite. Here we prove that (1) τ is the join of two topologies on X both homeomorphic to Q if and only if τ is non-compact and metrizable, and (2) τ is the join of topologies on X each homeomorphic to Q if and only if τ is Tychonoff and noncompact.

Given the equations for the n + 1 hyperplanes that bound an n-simplex in Rn, simple formulae are derived for the contents of the n − r simplices (0 ≦ r < n) embedded in it. For example, when n = 3, the formulae include the volume of the tetrahedron, the areas of its faces and the lengths of its edges.

In this note the L2-angle between two concentric rings and between the ring and the exterior of the disc in the complex plane are calculated. In the second part we prove that the L2-angles between domains A and B and between A × C and B × C are equal. We give also some examples of discontinuity of the L2-angle between domains.

A number of sufficient conditions for stability or strong stability, as used in the context of Hamiltonian systems, are found for the differential equation where the continuous function f(t) is periodic ω in t, D = d/dt and p(s), q(s) are real monic polynomials having special properties which allow the differential equation to be transformed into a canonical system of k second order equations.

A.McIntosh and A. Pryde introduced and gave some applications of notion of “spectral set”, γ(T), associated with each finite, commuting family of continuous linear operators T in a Banach space. Unlike most concepts of joint spectrum, the set γ(T) is part of real Euclidean space. It is shown that γ(T) is always non-empty whenver there are at least two operators in T.

All known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.

It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram Γ can be classified by the following data: a partially ordered set U with maximum element 1 and a map λ: U → 2Γ with λ(1) = Γ and with the property that for all J1, J2, J3 ∈ U with J1 > J2 > J3, any connected component of λ(J2) is contained in either λ(J1) or λ(J3). In this paper we show that λ comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a ∧-semilattice; (2) If J1, J2 ∈ U, then λ(J1)∧ λ(J2) λ(J1 ∧ J2); (3) If θ ∈ Γ, J ∈ U, then max{J1 ∈ U|J1 > J, θ ∈ λ (J1)} exists; (4) If J1, J2 ∈ U with J1 > J2 and if X is a two element discrete subset of λ(J1) ∪ λ(J2), then X λ(J) for some J ∈ UJ with J1 > J > J2.

In this note, a brief and accessible proof is given of an extension of the Pták homomorphism theorem to a larger class of spaces—spaces that are not necessarily assumed to be locally convex. This is done by first proving a counterpart of the Bourbaki-Grothendieck homomophism theorem for the non-locally-convex case. Our presentation utilizes the simplifying properties of seminorms.

A group G is said to be conjugacy p-separable if two non-conjugate elements of G remain non-conjugate in some finite p-group endomorphic image of G. We show that the non-cyclic free centre-by-metabelian groups are not conjugacy p-separable for any prime p. On the other hand, we show that every free centre-by-metabelian group has the solvable conjugacy problem