A unified theory of continuous and certain non-continuous functions is proposed and developed. The proposed theory encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, almost continuous functions, c-continuous functions, c*-continuous functions, s-continuous functions, l-continuous functions, H-continuous functions, and the ε-continuous functions of Klee.

In this paper we consider classes of vector lattices over subfields of the real numbers. Among other properties we relate the archimedean condition of such a vector lattice to the uniqueness of scalar multiplication and the linearity of l-automorphisms. If a vector lattice in the classes considered admits an essential subgroup that is not a minimal prime, then it also admits a non-linear l-automorphism and more than one scalar multiplication. It is also shown that each l-group contains a largest archimedean convex l-subgroup which admits a unique scalar multiplication.

Let {λj}j≥0 be a sequence of positive integers such that λj+1/λj≥3 and {aj}j≥0 a sequence of complex numbers such that |aj|≤1. Let μ be the Riesz product πj≥0[1+ Re(ajeiλjx)], that is, the weak limit of measures on T the density of which are the partial products. Then if Σj≥0|aj|2≤∞ the series Σj≥0 aj(eiλjx - ½āj) converges for μ-almost every x. The μ-a.e. convergence of series Σ ajeinλjx is also investigated as well as the case of Riesz products on a compact commutative group.

Let G be a finite group of even order coprime to 3. If G admits a fixed-point-free automorphism group isomorphic to the symmetric group on three letters, then we prove that G is soluble.

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

An Eisenstein-like criterion is proved for power series with algebraic coefficients satisfying algebraic differential equations of a certain general kind. The proof is elementary and the result extends earlier results of Hurwitz, Pólya and Popken

Integrals related to Cauchy's integral formula and Huygens' principle are used to establish a link between domains of holomorphy in n complex variables and cells of harmonicity in one higher dimension. These integrals enable us to determine domains to which analytic functions on real analytic surface in Rn+1 may be extended to solutions to a Dirac equation.

We prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

Every invertible n-by-n matrix over a ring R satisfying the first Bass stable range condition is the product of n simple automorphisms, and there are invertible matrices which cannot be written as the products of a smaller number of simple automorphisms. This generalizes results of Ellers on division rings and local rings.

This paper deals with Hermite-Fejér interpolation of functions defined on a semi-infinite interval but the nodes are equally spaced. It is shown that, under certain conditions, the interpolation process has poor approximation properties.

Let G denote a locally compact Vilenkin group with dual group Γ. We give sufficient conditions for a function ϕ ∈ L∞ (Γ) to be a multiplier from the power-weighted Hardy space to itself or the corresponding power-weighted Lebesgue space

Two subgroups ME(G) and MI(G) of the Schur multiplier M(G) of a finite group G are introduced: ME(G) contains those cohomology classes [α] of M(G) for which every element of G is α-regular, and MI(G) consists of those cohomology classes of M(G) which contain a G-invariant cocycle. It is then shown that under suitable circumstances, such as when G has odd order, that each element of MI(G) can be expressed as the product of an element of ME(G) and an element of the image of the inflation homomorphism from M(G/G′) into M(G).

In this paper a generalization of the interpretation of Robinson of torsion products as homotopy groups is given. This generalization allows us to define right derived functors of a coproduct preserving functor defined on a small category with all finite colimits to the category of abelian groups. We show that when the category is additive the definition coincides with the definitions of the right derived functors of Cartan and Eilenberg.