Given an n × n matrix A, an n-dimensional vector q, and a closed, convex cone S of Rn, the generalized linear complementarity problem considered here is the following: find a z ∈ Rn such that

where s* is the polar cone of S. The existence of a solution to this problem for arbitrary vector q has been established both analytically and constructively for several classes of matrices A. In this note, a new class of matrices, denoted by J, is introduced. A is a J-matrix if

The new class can be seen to be broader than previously studied classes. We analytically show that for any A in this class, a solution to the above problem exists for arbitrary vector q. This is achieved by using a result on variational inequalities.

A characterization has previously been given for linear transformations in Hilbert space whose first N + 1 powers are partial isometries. An analogous characterization is now obtained for transformations whose first N+ 1 powers have closed ranges. A hypothesis (that transformations have no isometric part) is found to be unnecessary in previous work.

It is shown that in a variety of (not necessarily associative) algebras which satisfies a variant of Andrunakievich's Lemma, a class C containing no solvable algebras is the semi-simple class corresponding to some supernilpotent radical class if and only ifCis hereditary and is closed under extensions and sub-direct products. Semi-simple classes in general are not characterized by these properties. If the variety satisfies the further condition that some proper power of every ideal is an ideal, then analogous results hold for the semi-simple classes corresponding to radical classes containing no solvable algebras. In particular, for algebras over a field in the latter situation, all semi-simple classes are characterized by the three closure properties mentioned.

In this note we give external characterizations of absolutely topologicai functors U: A →X (where A may be a large category) as functors which are injective with respect to all functors based on the category X, or equivalently, which do not admit essential extensions over X. From this we deduce several characterizations and properties of the MacNeille completion of faithful functors. These results generalize the work of Horst Herriich [Math. Z. 150 (1976), 101–110] in case of small categories, where the crucial point of this generalization lies in the fact that we cannot make use of the existence of MacNeille completions and we therefore have to introduce a local-initial-completion-construction.

Finite O-simple semigroups with a Rees representation over an elementary abelian group are considered. The isomorphism classes of such semigroups with class graph γ and coefficient group of order pd are shown to correspond to the orbits under the automorphism group of γ of those subspaces of the circuit space of γ over Zp which have codimension at most d. A semigroup interpretation of the inclusion relation between subspaces is given and results are obtained on the structure of the automorphism group and the enumeration of the semigroups under consideration.

This paper, in conjunction with its algebraic sequel, aims to provide a foundation, long outstanding, to the theory of quasigroups obeying the law x(xy) = yx , otherwise known as Stein quasigroups.

A few results on uniform and uniform ultimate boundedness properties of solutions of impulsively perturbed systems are presented in this paper, assuming the corresponding properties of solutions of the unperturbed nonlinear system.

A general process for the desingularization of smooth actions of compact Lie groups is described. If G is a compact Lie group, it is shown that there is naturally associated to any compact G manifold M a compact G × (Z/2)p manifold on which G acts principally. Here Z/2 denotes the cyclic group of order two and p + 1 is the number of orbit types of the G action on M.

It is shown that the set of chaotic self-maps of the unit interval contains an open dense subset of the space of all continuous self-maps of the unit interval. Other aspects of chaotic behaviour are also considered together with some illustrative examples.

We prove in this paper several results on lattice related measures and images of such measures under mappings. If we apply these to various areas of point set topology we obtain as corollaries many known and new results on sequential compactness, repleteness, and measure repleteness – areas of recent considerable interest to mathematicians.

Suppose n competitors each compete in r races and a ranking function F assigns a score F(j) to the competitor finishing in the jth position in each race. The sum of the scores over r races gives each competitor a final ranking. If n is fixed, the ranking function F bifurcates as r increases. The complete bifurcation behaviour is determined for n = 3 and some information obtained for n > 3.

Michael A.B. Deakin (Bull. Austral. Math. Soc. 17 (1977), 161–175) states (on p. 173) a conjecture, which, if true, would lead to simpler proofs of a number of known theorems. This conjecture is, however, shown to be false by means of a counter-example.

The following is a correct proof of the main theorem of [1]. It should be substituted for the published Section 3, which, as pointed out by Professor B.L.S. Prakasa Rao, contains an error in the equations following (15) on page 49.