An extension (Theorem 1) of Schwarz and Pick's lemma motivates us to study the analogues for functions which are bounded in the sense of Bloch, normal, or yoshida. A typical result is that, for a function f holomorphic in D = {|z| < 1} and Bloch, that is, , with the expansion f(w) = c0 + cn (w − z)n + … (n ≧ 1) about 2 ε D, we have (1 − |z|2)n|f(n) (z)|/n! ≦ Anα, where An is an absclute constant; the estimate is sharp.

A set of conditions for the almost sure convergence of a stochastic iterative procedure is given. The conditions are framed in terms of the behaviour of the random adjustment made the n-th step rather than in terms of some underlying regression model.

This paper studies endomorphism monoids of Kleene algebras. The main result is that these algebras form an almost universal variety k, from which it follows that for a given monoid M there is a proper class of non-isomorphic Kleene algebras with endomorphism monoid M+ (where M+ denotes the extension of M by a single element that is a right zero in M+). Kleene algebras form a subvariety of de Morgan algebras containing Boolean algebras. Previously it has been shown the latter are uniquely determined by their endomorphisms, while the former constitute a universal variety, containing, in particular, arbitrarily large finite rigid algebras. Non-trivial algebras in K always have non-trivial endomorphisms (so that universality of K is ruled out) and unlike the situation for de Morgan algebras the size of End(L) for a finite Kleene algebra L necessarily increases as |L| does. The paper concludes with results on endomorphism monoids of algebras in subvarieties of the variety of MS-algebras.

Let {Qn} denote the orthogonal polynomials associated with the weight function p on [−1, 1] and let denote the zeros of (1−x2) Qn (x). Consider the Lagrange polynomials which interpolate a given continuous function at these points. It is shown that, as n → ∞, the Lagrange polynomial converges to the function in the W weighted mean square sense, where w (x) = ρ(x)\(1−x2), provided that W is integrable. An application to numerical product integration is noted.

Let S be an inverse semigroup. We prove that there is a ring with a proper involution * in which S is *-embeddable. The ring will be a natural one, R[S], the semigroup ring of S over any formally complex ring R; for example ℝ, Ȼ.

A ring R with centre Z (R) is called compressible if Z (eRe) = eZ (R) e for any idempotent e of R. In this paper we shall give some examples of compressible group algebras and of noncompressible group algebras. These examples show that it is very difficult to judge the compressibility of a group algebra.

In the context of synthetic differential geometry, we prove that group valued 1-forms on the unit interval are exact, provided the group in question is a Lie group. This exactness is the basic assumption in a previous paper by the author on differential forms with values in groups.

The following Theorem is proved: Let R be a semi-prime ring in which either (xy)n − xnyn or (xy)n − ynxn is central, for all x,y in R where n > 1 is a fixed integer. Then R is commutative.

Characterizations of efficiency and proper efficiency are given for classes of multiple objective fractional optimization problems. These results are then applied to the case of multiple objective fractional linear problems. A dual problem is given for the multiple objective fractional problem and it is shown that for a properly efficient primal solution the dual solution is also properly efficient.

A closed subset c of [0,∞]×[−∞,∞] is called a barrier if

(i) (∞,x) ∈ C, x,

(ii) (t, ±∞) ∈ C, t,

(iii) (t, x) ∈ C implies (s, x) ∈ C, S ≥ t.

Given a Brownian motion (B (t)) Starting at the origin and a barrier C, let τ(C) be inf{t: (t, B (t)) ∈ C}. A random variable x (or a distribution F) is called achievable if there exists a barrier C so that B (τ(C)) is distributed as x (F). In this paper we shall show that if x is bounded above or below with finite mean or if x has zero mean and E (|x| log+ |x|) < ∞ then x is achievable. This result gives a partial answer to a problem raised by Loynes [7].

For 0 < p < 1, let Sp denote the class of functions f (z) which are meromorphic and univalent in the unit disk U, with the normalisations f (0) = 0, f′(0) = 1 and f (p) = ∞, and let Sp (a) denote subclass of Sp consisting of those functions in Sp whose residue at the pole in equal to a. In this paper, we determine, for values of the residue a in a certain disk Δp, the greatest possible outer area over all functions in the class Sp (a). We also determine additional information concerning extremal function if the reside a dose not lie in Δp.

If S is a (multiplicative) Semigroup and a ∈ S, the binary operation ∘ defined on the set S by x ∘ y = x a y is associative and the resulting semigroup is called a variant of S. We study the congruence a defined on S by saying that two elements are α-related if and only if they determine the same variant of S. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involving D-equivalence).

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.

We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:

If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

It is shown that a non-discrete locally compact Hausdorff group has each of its proper closed subgroups finite (respectively, discrete) if and only if it is topologically isomorphic to the circle group (respectively, the circle group or the group of real numbers).