The paper is concerned with the determination of the degree of convergence of a sequence of linear operators connected with the Fourier series of a function of class Lp (p > 1) to that function and some inverse results in relating the convergence to the classes of functions. In certain cases one can obtain the saturation results too. In all cases Lp norm is used.

Let G be a primitive permutation group of finite degree n containing a subgroup H which fixes k points and has r orbits on Δ, the set of points it moves. An old and important theorem of Jordan says that if r = 1 and k ≥ 1 then G is 2-transitive; moreover if H acts primitively on Δ then G is (k + 1)-transitive. Three extensions of this result are proved here: (i) if r = 2 and k ≥ 2 then G is 2-transitive, (ii) if r = 2, n > 9 and H acts primitively on both of its two nontrivial orbits then G is k-primitive, (iii) if r = 3, n > 13 and H acts primitively on each of its three nontrivial orbits, all of which have size at least 3, then G is (k − 1)-primitive.

The results here concern bijective continous functions from one connected separable n-manifold M to another N. If M has the property that every such function is necessarily a homeomorphism, then M is said to be strongly reversible. Strongly reversible manifolds having only compact boundary components are completely charaterized.

If Ω is a ring region with starlike boundary components α and β, then we show for each λ > 0 there exists a ring region ω ⊂ Ω with ∂ ω = α ∪ ϒ, α ∩ ϒ = φ such that there is a harmonic function V in ω satisfying (a) V(z) = 0 for z ∈ α, (b) V(z) = 1 for z ∈ ϒ, (c) | grad V(z)| = λ for z ∈ ϒ ∩ Ω. Furthermore, we show when ω is not equal to Ω; that is, that is, there is a non-trival solution.

It is shown that a semigroup is right self-injective and a band of groups if and only if it is isomorphic to the spined product of a self-injective semilattice of groups and a right self-injective band. A necessary and sufficient condition for a band to be right self-injective is given. It is shown that a left [right] self-injective semigroup has the [anti-] representation extension property and the right [left] congruence extension property.

We derive some specific inequalities involving absolutely continuous functions and relate them to a norm inequality arising from Banach algebras of functions having bounded k th variation.

A joint spectral theorem for an n-tuple of doubly commuting unbounded normal operators in a Hilbert space is proved by using the techniques of GB*-algebras.

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.

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

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 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.

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.

For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).

For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.

1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.

2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.

3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.