This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Let B, S, and T be subsets of a (left) near-ring R with B and T nonempty. We say B is (S, T)-distributive if s(b1+b2)t = sb1t + sb2t, for each s ∈ S, b1, 2 ∈ B, t ∈ T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals. Theorem. If I is a minimal ideal of R and Ik is (Im, In)-distributive for some k, n ≧ 1, m ≧ 0, then either I2 = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let Rk be (Rm, Rn)-distributive, for some k, n ≧ 1, m ≧ 0; if R is semiprime or is a subdirect product of simple near-rings, then R is a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. If R → A → 0 is an exact sequence of near-rings, then conditions on A are given which will impose conditions on the minimal ideals of R.

We prove that every Banach space X with characteristic of uniform convexity less than 2 has the fixed point property whenever X satisfies a certain orthogonality condition.

A class of functional differential equations in some Hilbert space are studied. The results are applicable to many quasi-linear parabolic paratial differential equations with (possibly) countably many discrete delays and finitely many distributed delays in the highest order spatial derivatives. For the linear case, an evolution operator on the underline space H is introduced, via which a variation of constant formula for the solution of the equation in the underline space H is derived. Some spectral properties of the generator of the solution semigroup defined on some appropriate space are discussed as well.

Crossed products of C*-algebras by *-endomorphisms are defined in terms of a universal property for covariant representations implemented by families of isometries and some elementary properties of covariant representations and crossed products are obtained.

Let X be a complex Banach space, G a compact abelian metrizable group and Λ a subset of Ĝ, the dual group of G. If X has the Radon-Nikodym property and is separable then has the Radon-Nikodym property. One consequence of this is that CΛ(G, X) has the Radon-Nikodym property whenever X has the Radon-Nikodym property and the Schur property and Λ is a Rosenthal set. A partial stability property for products of Rosenthal sets is also obtained.

A method for computing the number of contours for a twistor diagram, using Grothendieck's algebraic de Rham theorem, is described and some examples are given.

We establish a one-to-one “group-like” correspondence between congruences on a free monoid X* and so-called positively self-conjugate inverse submonoids of the polycyclic monoid P(X). This enables us to translate many concepts in semigroup theory into the language of inverse semigroups.

The existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.

For unbounded operators A1, …, Ad, Gevrey spaces Sλ1, …, λd (A1, …, Ad) of order (λ1, …, λd) are introduced, where the orders λ1, …, λd need not be equal. These extend the notion of Gevrey space defined by Goodman and Wallach where λ1 = … = λd. Several mild conditions on the operators A1, … Ad and the orders λ1, …, λd are presented such that the equality is valid. Examples are included.