Criteria for a holomorphic function f defined in |z| < 1 to be Bloch, namely, , are obtained in terms of the area and the length of the images of non-Euclidean disks and non-Euclidean circles, respectively.

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

It was shown by Bryce and Cossey that each Hall π-subgroup of a group in the smallest normal Fitting class S* necessarily lies in S*, for each set of primes π. We prove here that for each set of primes π such that |π| ≥ 2 and π′ is not empty, there exists a normal Fitting class without this closure property. A characterisation is obtained of all normal Fitting classes which do have this property.

Let F be a normal Fitting class closed under taking Hall π-subgroups, in the sense of the paragraph above, and let Sπ denote the Fitting class of all finite soluble π-groups, for some set of primes π. The second main theorem is a characterisation of the groups in the smallest Fitting class containing F and Sπ in terms of their Hall π-subgroups.

During the 1920's and 30's, two distinct theories of “completions” for topological spaces were being developed: the French school of mathematics was describing the familiar notion of “complete relative to a uniformity”, and the Russian school the less well-known idea of “absolutely closed”. The two agree precisely for compact spaces.

The first part of this article describes these two notions of completeness; the remainder is a presentation of the interesting, but apparently unrecorded, fact that the two ideas coincide when put in the context of topological groups.

In this paper we prove a G-invariant implicit function theorem and indicate how it can be used to improve an earlier result of the author on the bifurcation of solutions of nonlinear equations in the presence of continuous groups of symmetries. We also use our theorem to show that, under reasonable hypotheses, the method of looking for solutions in invariant subspaces yields all solutions. This can be used to answer a question raised by Sattinger [J. Math. Phys. 19 (1978), 1729]. The abstract result is also of interest because it provides a theorem which should be of use in other symmetric situations.

Singular inner functions are in many respects the most important and difficult type of functions in the Banach algebra H∞ of bounded analytic functions in the unit disc. This paper is concerned with ideals generated by singular inner functions. In particular, conditions on the associated measures are given so that the ideal spans the whole algebra H∞. To this end the local boundary behavior of a singular inner function is studied and the results obtained there may be of independent interest.

Several problems arising from a functional analytic study on Omori's inverse mapping theorem are considered arriving at an inverse mapping theorem in locally convex spaces.

We prove, in normed linear spaces, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm. Also, we prove that under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.

The aim of this paper is to study the relationship between the nature of an infinite series of real terms in which the general term tends to zero and the derived set (set of limit points) of the aggregate of the fractional parts of its partial sums. For all types of series (in which the nth term tends to zero) we determine the derived set.

We denote by F the set of the fractional parts of the partial sums of the series, and by F′ the derived set of F. The principal results of the paper can be stated as follows:

1. the series is convergent if and only if F has at most one limit point in [0, 1] or F′ ⊆ {0, 1};

2. the series, if non-oscillatory, is divergent if and only if F′ = [0, 1];

3. if the series is oscillatory, F′ is a closed sub-interval of [0, 1] or [0, 1] - (a, b), where (a, b) ⊊ (0, 1).

In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: let n ε N* and αl, …, αn be distinct non-zero rational numbers; there exists a constant c = c(n, αl, …, αn) ≥ 0 such that

for every non-zero integer point (qo, ql, …, qn)and q = max {|ql|, … |qn|, 3}.

Convergence notions are used extensively in the areas of probability and statistics. Many times proofs can be simplified by considering an appropriate convergence structure on the space and using well-known results from the theory of convergence spaces; for example, compactness arguments are sometimes simplified by using a generalized Ascoli theorem in the convergence space setting. The theory of convergence spaces is also used to generalize some results in probability and statistics.

We prove a Baire category theorem for continuous lattices and derive category theorems for non-Hausdorff spaces which imply a category theorem of Isbell's and have applications to the spectral theory of C*-algebras. The same lattice theoretical methods yield a proof of de Groot's category theorem for regular subcompact spaces.

We are interested in the strict topology τ on , the set L(E, F) of all continuous linear mappings from E into a Banach space F endowed with the topology of pointwise convergence. The T3-completion of the convergence tensor product E ⊗cLc F is the set of all τ-continuous linear functionals on L(E, F) and τ is the topology of uniform convergence on the compact subsets of . In the case that E is a nuclear Fréchet space, a nuclear (DF)-space or a Banach space with the bounded approximation property the topology τ agrees with the topology of Lco (E, F).

In this note sufficient conditions for non-compactness of composition operators on two different functional Hilbert spaces have been obtained.

It is shown that if I is a finitely generated ideal of a commutative ring R such that the multiplication map I ⊗RI → I is an injection, then I is locally principal. As a corollary, one obtains a new homological characterization of Prüfer domains.