Let r be a given prime. Then a monoid M is the endomorphism monoid of a field of characteristic r if and only if either M is a finite cyclic group or M is a right cancellative monoid and M has an element of infinite order in its centre. The main lemma is the technical base of the present and other papers.

Let SΓ be a vector space graph. A graphic subspace of SΓ need not be a direct summand with a graphic complement. A necessary and sufficient condition for the existence of a graphic complement is given. Also, it is shown that every graphic subspace possesses an o-special basis which extends to an o-special basis of SΓ.

The topology of the Čech fundamental group of the one-point compactification of an appropriate space Y induces a topology on the fundamental group of Y. We describe this topology in terms of a topological group introduced by Higman.

A result by the author on the elimination of randomization (or relaxation) for variational problems is partially extended and then used to obtain a very general result on the denseness of the set of original control functions in the set of relaxed control functions. Also, a slight extension of Aumann's theorem on the integrals of multifunctions is shown to follow directly from the elimination result.

An elementary proof is given of a theorem of Castro and Lazer that the degree of the gradient of a coercive functional on a large ball of Rn is one.

In this paper we give necessary and sufficient conditions for exponential dichotomy of a general class of strongly continuous semigroups of operators defined on a Banach space. As a particular case we obtain a Datko theorem for exponential stability of a strongly continuous semigroup of class C0 defined on a Banach space.

Let Γ be a graph with isomorphic subgraphs G and H, and let θ: G → H be an isomorphism. If θ can be extended to an automorphism of Γ, we call θ a partial automorphism of Γ.

We consider the application of partial automorphisms to the graph reconstruction conjecture, in particular, to the problem of reconstructing graphs with two vertices of degree k – 1 and the remaining vertices of degree k.

We show that a certain local pinching condition on a compact minimal submanifold in a sphere leads to a condition on the codimension of the immersion.

Let B be the class of functions ω(z) regular in |z| < 1 and satisfying ω(0) = 0, |ω(z)|<1 in |z|<1. We denote by P(A, B), −1 ≤ B < A ≤1, the class of functions p(z) = l+p1z+… regular in |z| < 1 and such that p(z) = [1+Aω(z)]/[1+Bω(z)] for some ω(z) ∈ Β. This paper establishes sharp lower and upper bounds on |z| = r<1 for the functional

where p(z) varies in P(A, B). The results are then used to study certain geometric properties of the corresponding class of meromorphic starlike univalent functions

Sublinear elliptic differential inequalities with variable coefficients are studied. Sufficient conditions are given that all solutions are oscillatory in exterior domains. Riccati inequalities are used to establish sublinear oscillation criteria.

Connexions are sought between the subvarieties of a variety U of groups and the subvarieties of the variety of all groups which are central extensions by groups in U, in the case when U has the form . Here , is the variety of abelian groups of exponent dividing r and Bis a variety of soluble groups of finite exponent.

Several procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

V. Krishnamurthy has shown that on a finite set X all topologies can be mapped into a certain set of matrices of zeros and ones. In this paper it is shown that all lattices, algebras and rings on a finite set X can be mapped onto particular sets of matrices of zeros and ones.

We prove that for a non-discrete space X, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i) X is paracompact and each point is Gδ, or (ii) X is a completely paracompact Morita k-space. Consequently, if X is a non-discrete completely paracompact space in which each point is a Gδ-set or it is also a Morita k-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a space X there exists a point x ∈ X and a family ϕ of supports on X such that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result for Rn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.

A verbal product is introduced for a particular class of varieties of inverse semigroups and this product is shown to be associative. As well, the structure of this class is examined.

It is well-known that every sufficiently large positive integer is the sum of seven cubes. Both proofs of this result, due to Linnik and Watson, are ineffective. Here we show that Watson's proof may be made effective.