In this paper we obtain some permanence properties of a class of locally convex spaces located between quasi-suprabarrelled spaces and quasi-totally barrelled spaces, for which a closed graph theorem is given.

Recently the authors extended to positive semi-definite matrices converses of matrix inequalities corresponding to ratios and differences of means and of Hölder's inequality. Here, further generalisations are given to the case of several vectors with appropriate weights.

We characterise in terms of free topological groups those Banach-Lie algebras with finite-dimensional centre coming from Lie groups.

It is proved in this paper that if a vector field is both C structurally stable and topologically stable, then it satisfies Axiom A.

We show that a bounded linear operator on a dual Banach space X may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius provided the weak star and norm topologies coincide on the unit sphere of X.

We give sufficient and necessary conditions for simple modules of the quantum group or the quantum enveloping algebra Uq(g) to have weight space decompositions, where g is a semisimple Lie algebra and q is a nonzero complex number. We show that

(i) if q is a root of unity, any simple module of Uq(g) is finite dimensional, and hence is a weight module;

(ii) if q is generic, that is, not a root of unity, then there are simple modules of Uq(g) which do not have weight space decompositions.

Also the group of units of Uq(g) is found.

A description is obtained of the free semifields with both fundamental operations commutative.

We generalise to a rectangular lattice a known result about the maximal width of a planar compact convex set containing no points of the integral lattice. As a corollary we give a new short proof that the planar compact convex set of greatest width which contains just one point of the triangular lattice is an equilateral triangle.

We study the topological entropy of triangular maps of the square. We show that such maps differ from the continuous maps of the interval because there exist triangular maps of the square of “type 2∞” with infinite topological entropy. The set of such maps is dense in the space of triangular maps of “type at most 2∞” and the topological entropy as a function of the triangular maps of the square is not lower semicontinuous. However, we show that for these maps the characterisation of the lower bounds of the topological entropy depending on the set of periods is the same as for the continuous maps of the interval.

Let X and Y be T1 spaces and f: X → Y be a closed and onto mapping. If a fiber of the mapping f is defined to be the inverse image of a singleton in the range, then a bound for the tightness of the domain is the product of the tightness of the range and the supremum of the tightness of the fibers of f. Similar bounds can also be shown for the Lindelöf degree and the extent of X. Examples are provided to demonstrate that such results are not possible for open maps. Cellularity and spread are discussed briefly.

In a recent paper the authors showed that certain set-valued mappings from a Baire space into subsets of a Banach space which have a continuity property defined in terms of Kuratowski's index of non-compactness have inherent single-valued properties. Here we generalise the continuity property to one defined in terms of a weak index of non-compactness and we show that this wider class of set-valued mappings also has significant implications for the differentiability of convex functions on Banach spaces.

A semigroup S with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S,

.

It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive.

We discuss characterisations of Riemannian foliations, totally geodesic submanifolds, and totally umbilical submanifolds by sharp inequalities. These derive from the same linear algebraic set up, characterising a linear endomorphism which is a multiple of the identity.

Let f be a continuous self-map on a complete metric space X and p ∈ X. Let c be a positive real. Equivalent conditions are given for the singleton {p} to be an attractor of a set of c−fixed points of f. We also establish equivalent conditions for the existence of a contractive fixed point of f. These results subsume a body of fixed point theorems.

This paper uses Clarke’s generalised directional derivative to describe several types of invexity, pseudoinvexity and quasiinvexity at a point of a nonlinear function. Direct implications of the relations existing between the various types of invexity and generalised invexity are presented, as well as a block diagram of these implications. In particular, similar results in the class of quasiconvex functions are obtained.

The assumption that a Galerkin equation of the reaction-diffusion system of high order has an asymptotically orbitally stable time-periodic solution implies that the full reaction-diffusion system has a nearby asymptotically orbitally stable time-periodic solution with asymptotic phase.

We introduce two graph-colouring 2-person games, and analyse who has a winning strategy on some specific graphs such as the Petersen graph, and paths and cycles of given length.

An algebra is said to be polynomially n−dense if all equational theories extending the equational theory of the algebra with constants have a relative base consisting of equations in no more than n variables. In this paper, we investigate polynomial density of commutative semigroups. In particular, we prove that, for n > 1, a commutative semigroup is (n − 1)-dense if and only if its subsemigroup consisting of all n−factor-products is either a monoid or a union of groups of a bounded order. Moreover, a commutative semigroup is 0-dense if and only if it is a bounded semilattice. For semilattices, we give a full description of the corresponding lattices of equational theories.

The purpose of the paper is to show that the Harte spectrum and the bicommutant spectrum of an arbitrary n−tuple of normal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. It is also proved that many commonly used joint spectra of an n−tuple of normal m by m matrices are equal. These results are non-commutative variants of some theorems proved by McIntosh, Pryde, and Ricker for commuting sets of operators.