An operational category is a category of models for an equational theory where the interpretation of some operations is predetermined. Examples include the equational and co-equational categories of Linton, categories of functors preserving some class of limits, and algebras for a prop as defined by MacLane. The chief result is a characterisation of the operational categories and functors in terms of their internal structure.

A symmetry of an ordered Banach space is an order and norm isomorphism which commutes with its ideal centre. A class of ordered Banach spaces is introduced to show that, for a space in this class, the group of symmetries is trivial if and only if the space is lattice-ordered. When this group becomes larger, the space approaches an antilattice. This phenomenon is also investigated.

The purpose of this note is to give extensions of an inequality of Jensen for analytic functions in an unit disc. We investigate functions which satisfy equalities in the inequalities.

We give a complete description of holomorphic curves in the complex two quadric via the method of moving frames. For compact curves a Morse theory type integral formula is derived.

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

For an arbitrary field K there are two related questions that can be asked:

(1) Is there a proper subfield, L, of K such that K is countably generated over L?

(2) Given a proper subfield M of K is there a proper subfield, L, of K containing M such that K is countably generated over L?

We give an affirmative answer to (1) in characteristic p ≠ 0 and provide counterexamples to (2) for arbitrary characteristic ≠ 2.

Amply boundedness of collections of analytic mappings is proved to be equivalent to equicontinuity of the corresponding collections of adjoints, for certain classes of locally convex spaces which have good analytic properties.

In this paper a single server preemptive priority queueing system, consisting of two types of units, with unlimited Poisson inputs and exponential service time distributions, is studied. The higher priority units are served in batches according to a general bulk service rule and they have preemptive priority over lower priority units. Steady state queue length distributions, stability condition and the mean queue lengths are obtained.

It is proved that the simple group PSL (2,q) satisfies a law [x,3y] = [x, sy], s > 3, if and only if q = 4, 5, 8.

For 0 < P,q < ∞, α> −1, Ap,q,α denotes the space of all holomorphic functions in the unit disc satisfying

where

In this paper, we find a sufficient condition for the multipliers from AP,q,α into ls, 1 ≤ s ≤ ∞, 1 ≤ q ≤ 2, which interpolates the results of Patrick Ahern and Miroljub Jevtic. As a corollary, we can calculate

for q′ ≤ s ≤ ∞, 1/q + 1/q′ = 1. Also, we can find a sharp coefficient condition for HP functions.

It is shown that a semidirect product of an inverse semigroup and a group, in that order, contains an inverse subsemigroup that is a retract and that, together with the retraction mapping, forms free inverse morphic image of the semidirect product. The congruence determined by the retraction mapping is shown to be determined by the semigroup of idempotents of the semidirect product.

Our main concern is the existence of maximal compact subgroups in a locally compact topological group. If G is a locally compact group such that P(G/G o), the set of periodic points of G/Go, is a compact subgroup of G/Go, than G has maximal compact subgroups K such that G/N is a Lie group where N = ∩K, the intersection of the collection K of all maximal compact subgroups of G. Also every compact subgroup of G is contained in a maximal compact subgroup. We given an example of a discrete group which has maximal finite subgroup and has finite subgroups not contained in maximal finite subgroups. We note that the above result on P(G/Go) is an extension of the well-known corresponding result for almost connected groups.

Two new types of containment of operators on Hilbert space, namely partial containment and semi-containment are introduced. We show in Proposition 11 that A is semi-contained in B if and only if the map P(B) → P(A) for polynomials P extends to be an ultra-weakly continuous completely positive map from all bounded operators on the underlying Hilbert space of B. We show in Theorem 15 that if an isometry is semi-contained in a contraction T, then T has a non-zero invariant subspace on which T is isometric. The semi-equivalence class of the simple unilateral shift is characterized in Theorem 18, and we show that a unilateral positive-weighted shift semi-contains the unilateral shift if and only the weights are eventually 1.

Let J ⊆ C∞ (Rn) be any ideal. Since a function of the variables = (t1,…,tn) is a function of the variables which does not depend on , we have J ⊆ C∞ (IRn+P). Of course, J is not an ideal of C∞ (IRn+P), but it generates an ideal that we call . Consider the following statement (1) on J: “Given any if and only if for every fixed .

In this paper we show that statement (1) holds for a large class of finitely generated ideals although not for all of them. We say that ideals satisfying statement (1) have line determined extensions. We characterize these ideals to be closed ideals J() (in the sense of Whitney) such that for all p ∈ ℕ, the ideal is also closed. Finally, some non-trivial examples are developed.

For a locally compact topological group admitting a weight function, we establish necessary and sufficient criteria for all the weighted continuous functions to be weakly almost periodic. Among other results, we show that weak almost periodicity of all ω-weighed continuous functions on a discrete semigroup S, can be very different drom the phenomenon of regularity of multiplication in the weighted algebra ℓ1 (S, w).