A sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

An investigation on the existence of projective objects and projective resolutions has been carried out in some categories more general than the category of compact spaces and continuous maps.

A churning transformation can be defined on probability measures by an infinite sequence of finite permutations of mass. Continuity and absolute continuity of measures are invariants for such transformations but it is shown that certain probability measures whose Fourier-Stieltjes transforms fail to vanish at infinity may be churned into measures whose transforms do vanish in this sense.

One of Ramanujan's unpublished, unproven identities has excited considerable interest over the years. Indeed, no fewer than four proofs have appeared in the literature. The object of this note is to present yet another proof, simpler than the others, relying only on Jacobi's triple product identity.

We examine positive semigroups acting on Banach lattices and operator algebras. In the lattice framework we characterize strict positivity and strict ordering of holomorphic semigroups by irreducibility criteria. In the algebraic setting we derive ergodic criteria for irreducibility and discuss various aspects of strict positivity. Finally we examine invariant states of a C*-dynamical system in which the automorphism group is replaced by a strongly positive semigroup. We demonstrate that ergodic states are characterized by a cluster property despite the absence of a covariant implementation law for the semigroup.

A recursive construction for orthogonal diagonal latin squares, using group divisible designs, is presented. In consequence the numbers of orders for which the existence of such squares is in question is reduced to 72.

Let T: [0,1] → [0,1] be a map which is given piecewise as a linear fractional map such that T0 = T] = 0 and T'0 < 1. Then T is ergodic and admits an invariant measure which can be calculated explicitly.

The class ϕ+(X, Y) of semi-Fredholm transformations consists of those transformations T: X → Y for which α(T) = dim ker T < ∞ and for which T(X) is closed. It forms an open subset of B (X, Y) closed under perturbation by compact transformations and is a particularly important class of transformation since T is Fredholm if and only if T ∈ ϕ+ (X, Y) and T′ ∈ ϕ+ (Y′. X′). The realization that elements of ϕ+ (X, Y) have very simple nonstandard characterizations lead the author to consider the possibility of finding an analogous open class of transformation which is closed under perturbation by weakly compact transformations. Consequently this paper investigates two related classes which contain ϕ+ (X, Y). The first such class coincides with the class of Tauberian transformations whilst the second consists of those transformations which have Tauberian extensions on the nonstandard hulls. The Tauberian transformations are closed under perturbation by weakly compact transformations but in general are not open. The “super” Tauberian transformations are closed under perturbation by super weakly compact transformations and in fact form an open subset of B(X, Y).

Suppose the elementary abelian group A acts on the group G where A and G have relatively prime orders. If CG(a) belongs to some formation F for all non-identity elements a in A, does it follow that G belongs to F? For many formations, the answer is shown to be yes provided that the rank of A is sufficiently large.

This paper establishes an important link between the class of abelian l-groups and the class of distributive lattices with a distinguished element. This is accomplished by describing the distributive lattice free product of a family of abelian l-groups as a naturally generated sublattice of their abelian l-group free product.

Let the word “graph” be used in the sense of a countable, connected, simple graph with at least one vertex. We write Qn and Ocn for the graphs associated with the n-cube Qn and the n-octahedron Ocn respectively. In a previous paper (Dekker, 1981) we generalized Qn and Qn to a graph QN and a cube QN, for any nonzero recursive equivalence type N. In the present paper we do the same for Ocn and Ocn. We also examine the nature of the duality between QN and OcN, in case N is an infinite isol. There are c RETs, c denoting the cardinality of the continuum.

We define the notion of a best Diophantine approximation vector to a set of linear forms. This generalizes definitions of a best approximation vector to a single linear form and of a best simultaneous Diophantine approximation vector. We derive necessary and sufficient conditions for the existence of an infinite set of best Diophantine approximation vectors. Finally, we prove that such approximation vectors are spaced far apart in an appropriate sense.

Let n be a natural with largest component sd. We prove that if xσ(n) = yn + z (x, y, z given positive integers), n is not primitive (y/x)-abundant and n/s is not (y/x)-perfect, then n < 4(z + ½)3/27y (if z ≥ 175). All solutions are tabled for the equation xσ(n) - yn when x = 1, y ≥ 2, 1 ≤ z ≤ 210, and n is not primitive y-abundant. We also prove that if n is primitive (y/x)-abundant, then s3d < (yn/2)2. A number of results are proved concerning the range of σ(n)/n when n is primitive αabundant, for any real number α > 1. For example, then α(n/n < α + min {½ 3 α e-5a/9 /2) and σ(n)/n < α /1.6α/log n. All primitive abundant numbers n with α(n)/ ≥ 2.05 are listed.

A recent article of G. Chang shows that an n × n partial latin square with prescribed diagonal can always be embedded in an n × n latin square except in one obvious case where it cannot be done. Chang's proof is to show that the symbols of the partial latin square can be assigned the elements of the additive abelian group Zn so that the diagonal elements of the square sum to zero. A theorem of M. Halls then shows this to be embeddable in the operation table of the group. In this paper, we show that when n is a prime one can determine exactly the number of distinct ways in which this assignment can be made. The proof uses some graph theoretic techniques.