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.

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.

In an earlier paper we showed that the set ψ+ (X, Y) of super Tauberian transformations between two Banach spaces X and Y forms an open subset of B(X, Y) which is closed under perturbation by super weakly compact transformations. In this note we characterize a class dual to ψ+ (X, Y) which we denote by ψ-(X, Y). We show that T∈ψ+(X, Y) if and only if T′ ∈ ψ-(Y′, X′) and that T′∈ψ+(Y′, X′) if and only if T ∈ ψ-(X, Y) and provide standard and nonstandard characterizations of elements of ψ-(X, Y). These two classes thus play in some ways analogous roles to the sets of semi-Fredholm transforms ϕ+ (X, Y) and ϕ-(X, Y).

Moreover en forms an open subset of B(X, Y) closed under the taking of adjoints, under the taking of nonstandard hull extensions, and under perturbation by super weakly compact transformations.

The problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0. Conditions for the existence and uniqueness of the solution are given. The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition.

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.

A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.

Different methods are used to show that a finite or countable product of Lindelöf scattered spaces is Lindelöf. Also, a technique of Kunen is modified to yield results concerning the Lindelöf degree of the Gδ and Gα-topologies on the countable product of compact scattered spaces.

We find the atoms of certain subclasses of varieties of finite semigroups and the corresponding varieties of languages. For example we give a new description of languages whose syntactic monoids are R-trivial and idempotent. We also describe the least variety containing all commutative semigroups and at least one non-commutative semigroup. Finally we extend to varieties of finite semigroups some classical results about semilattice decomposition of semigroups.

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.

A space X is said to be D1 provided each closed set has a countable basis for the open sets containing it. It is said to be D2 provided there is a countable base {Un} such that each closed set has a countable base for the open sets containing it, which is a subfamily of {Un}. In this paper, we give a separation theorem for D1 spaces, and provide a characterization of D1 and D2 spaces in terms of maps.

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.

Let f and g denote immersions of the n-manifolds M and N, respectively, in Rn+1. We say that f is athwart to g if f(M) and g(N)m have no tangent hyperplane in common. In this paper necessary conditions for athwartness are obtained.

Let A be an Archimedean, uniformly complete, semiprime f-algebra and F(X1,…Xn) ∈ R+ [X1,…Xn] a homogeneous polynomial of degree p (p∈ N). It is shown that (F(u 1…unn))1/p exists in A+ for all u1…un ∈ A+.

Let S(m, M) be the set of functions regular and satisfying │zf′(z)/f(z) – m│< M in │z│ <1, where│m –│ <M;≦ m; and let S*(p) be the set of starlike functions of order p, 0≦ p <1. In this paper we obtain integral operators which map S(m, M) into S(mM) and S* (p) × S(mM) into S*(p). Our results improve and generalize many recent results.

An operator form of the Euler-Maclaurin sum formula is obtained, expressing the sum of the Euler-Maclaurin infinite series in a closed derivation, whose spectrum is compact, not equal to {0}, and does not have 0 as a clusterpoint, as the difference between a summation operator and an antiderivation which is the local inverse of the derivation.