This paper considers the question of when a quasi-uniform space has a compatible uniform structure. Typical of the sufficient conditions provided is the result that a quasi-uniform space whose conjugate topology is compact and R0 is uniformizable.

It is shown that in the universal classes of

(i) all commutative algebras,

(ii) all anti-commutative algebras and

(iii) all algebras satisfying x2 = 0 (over any commutative, associative, unital ring)

the only radical classes with hereditary semi-simple classes are those for which membership is determined by additive structure. Some examples of non-hereditary semi-simple classes in the class of all power-associative algebras are also presented.

Finite groups G possessing a proper subgroup U such that for each element g of G there exists an automorphism of G mapping g into U are considered. The question of how the structure of U determines the structure of G is examined. For example, if G is soluble and U is nilpotent then G is nilpotent.

A group is potent if for any element of the group and any prescribed positive integer (dividing its order if this order is finite) there corresponds a finite homomorphic image of the group in which the element has the prescribed integer as its order. The finite potent groups form a finite variety that contains all finite nilpotent groups, all finite metabelian groups, and precisely one simple group, A5.

Of concern is the best constant K in the inequality ‖Ax‖2 ≤ K‖A2x‖‖x‖ where A generates a strongly continuous contraction semigroup in a Hilbert space. Criteria are obtained for approximate extremal vectors x when K = 2 (K ≤ 2 always holds). By specializing A + I to be a shift operator on a sequence space, very simple proofs of Copson's recent results on series inequalities follow. Inequalities of the above type are also studied on LP spaces, and earlier results of the authors and of Holbrook are improved.

Let Un be the open unit polydisc, and let T be a mapping from Un into itself. Then the composition transformation CT is a mapping on the Hardy space H2(Un) into the space of complex functions on Un defined as CTf = f ∘ T for every f ∈ H2(Un). An attempt is made to study some properties of CT in this note. A partial generalization of a result of Schwartz, and a relation between intertwining analytic Toeplitz operators and composition operators are reported.

In 1958, Gagliardo showed that if u is a locally integrable function on a domain Ω satisfying the cone condition, with all weak derivatives belonging to the Lebesgue space Lp (Ω) (1 ≤ p < ∞), then u belongs to Lp (Ω) also. We extend this result to Orlicz spaces, and use it to extend a result of Marcus and Mizel on Nemitsky operators between Sobolev spaces to Orlicz-Sobolev spaces.

In this note we characterize strongly regular graphs which are stable.

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.

In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

The barotropic motion of a viscous fluid in a laboratory simulation of ocean circulation may be modelled by Beards ley's vorticity equations. It is established here that these equations have unique smooth solutions which depend continuously on initial conditions. To avoid a boundary condition which involves an integral operator, the vorticity equations are replaced by an equivalent system of momentum equations. The system resembles the two-dimensional incompressible Navier-Stokes equations in a rotating reference frame. The existence of unique generalized solutions of the system in a square domain is established by modifying arguments used by Ladyzhenskaya for the Navier-Stokes equations. Smoothness of the solutions is then established by modifying Golovkin's arguments, again originally for the Navier- Stokes equations. A numerical procedure for solving the vorticity equations is discussed, as are the effects of reentrant corners in the domain modelling islands and peninsulae.

It Is known that a ring R with Krull dimension is an order in an Artinian ring if R is K-homogeneous and the prime radical N of R is weakly ideal invariant. The notion of weak ideal invariance can be interpreted in torsion theoretic terms, yielding a shorter and more conceptual proof of this result. In addition, it is shown that the orders in Artinian rings which arise in this fashion are precisely those for which R/N is K-homogeneous.

A class of graphs called randomly k-axial graphs is introduced, which generalizes randomly traceable graphs. The problems of determining which bipartite graphs and which complete n-partite graphs are randomly k-axial are studied.

A question asked by G. Kowol is answered by the construction, to an arbitrarily given natural number n, of groups G with automorphisms that agree with inner automorphisms on each set of fewer than n elements of G, but fail to agree with any inner automorphism on at least one set of n elements.

An operator T on a complex Hilbert space is d-symmetric if , where is the uniform closure of the range of the derivation operator δT(X)=TX−XT. It is shown that if the commutator ideal of the inclusion algebra for a d-symmetric operator is the ideal of all compact operators then T has countable spectrum and T is a quasidiagonal operator. It is also shown that if for a d-symmetric operator I(T) is the double commutant of T then T is diagonal.

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

Given a ŕeal-valued function f on [−1, 1], n ∈ N, and the following partition of [−1, 1[:

there exists a unique polynomial R4n−1(f; x) of degree not exceeding 4n − 1 such that

and, for j = 1, 2 and 3,

As they stand, the two papers [1], [2] are vacuous by the virtue of the fact that it is not possible to choose any functions a, k and constant β satisfying all the necessary hypotheses.

In this paper we study the exponential dichotomy property for linear systems, the evolution of which can be described by a semigroup of class C0 on a Banach space. We define the class of (p, q) dichotomic semigroups and establish the connections between the dichotomy concepts and admissibility property of the pair (Lp, Lq) for linear control systems. The obtained results are generalizations of well-known results of W.A. Coppel, J.L. Massera and J.J. Schäffer, K.J. Palmer.