For a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.

Using the exponential map in multifunction context, the paper deduces a system of non-Hausdorff theorems which generalize all known Ascoli theorems for the space of continuous functions and the space of point-compact continuous multifunctions.

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).

The (2, 3, ν) bipacking number is determined for all integers ν, and the number of non-isomorphic bipackings is found for small values of ν. The general solution for lambada packings of pairs into triples is deduced from the results for λ = 1 and λ = 2.

Algebras (A, ∧, ∨, ~, γ, 0, 1) of type (2,2,1,1,0,0) such that (A, ∧, ∨, ~, γ 0, 1) is a De Morgan algebra and γ is a lattice homomorphism from A into its center that satisfies one of the conditions (i) a ≤ γa or (ii) a ≤ ~ a ∧ γa are considered. The dual categories and the lattice of their subvarieties are determined, and applications to Lukasiewicz algebras are given.

Under the standard assumptions, the formula for integration by parts is obtained by a straightforward calculation.

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.

We define and investigate H-prefrattini subgroups for Schunck classes H of finite soluble groups, and solve a problem of Gaschütz concerning the structure of H-prefrattini groups for H = {1}.

The separable cyclic extension of rings is generalized to a separable noncyclic extension of rings: a crossed product with a factor set over a ring (not necessarily commutative). A representation of separable idempotents for a separable crossed product is obtained, and simplifications for some special factor sets are also given.

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.

The elementary part of general topology is carried out in a system which is based on the arithmetically definable theory of the reals with definitions by definable induction (DDI), where a formal object is said to be definable if the quantifiers are restricted to the rationals, the names of the base members and the elements of the spaces.

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.

The order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

In this paper it is shown that aimost local connectedness is hereditary for the subspace that is the union of regular open sets and is preserved under almost-open (in the sense of Singal) θ-continuous surjections.

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.

In this note a formation U is considered which can be defined by a sequence of laws which ‘almost’ hold in every finite supersoluble group. The class U contains all finite supersoluble groups and each group in U has a Sylow tower.

It is shown that a finite group belongs to U if and only if all of its subgroups with nilpotent commutator subgroup are supersoluble. A more general result concerning classes of this type finally proves that U is a saturated formation.