We consider the Abelian functional equation

where φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfy

We show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

Let λ1, λ2, λ3 be non-zero reals, not all of the same sign and such that at least one ratio λi/λj is irrational. Then it is proved that for any given integer k ≥ 1 and real η, the inequaltiy

is solvable for every ε > 0. More general and sharper results are also proved.

We improve S. Yamashita's hyperbolic version of the well-known Hardy-Littlewood theorem. Let f be holomorphic and bounded by one in the unit disc D. If (f#)p has a harmonic mojorant in D for some p, p > 0, then so does σ(f)q for all q, 0 < q < ∞. Here

A theorem is given on the interchange of integrals for the product of a Radon polymeasure and a measure. Examples show that if the conditions of the theorem are not satisfied, then the conclusions may not hold.

An algorithm which has been developed to solve the problem of determining an optimal path of the hand of a robot is applied to various classical problems in the calculus of variations.

We investigate the set of limit points of the continued fractions

where x1, x2, … is a given sequence of positive integers. We show that this set is closed, and that it may include any given countable subset of [0, 1] if the integers xk are chosen appropriately. Our main result, which has applications in transcendence theory, is that the sequence of continued fractions has no rational limit point when the sequence {xk} of partial quotients is bounded.

In this note the recent result of Corduneanu on almost periodicity of L2(G)-bounded solutions of nonlinear parabolic equations is extended to the case when the nonlinear growth rate is beyond the first eigenvalue of the associated elliptic boundary value problem.

The generic closure of the set of primes contracted from the complete ring of quotients of a reduced commutative ring is shown to be just the set of those primes not containing a finitely generated dense ideal. It is also the smallest generically closed, quasi-compact set containing the minimal primes.

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

Let A be a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebra B with | · | such that ‖ · ‖ majorises | · | on A. The purpose of this paper is to investigate the annihilator property between the algebras A and B.

Let (Xn)n≥1 be a sequence of random variables with zero means and uniformly bounded variences. Let τn be the stopping time defined on a given Brownian motion (Bt)t≥0, B0 = 0, by Dubins' method such that B(τn) has the same distribution as Xn. We prove that Xn converging to 0 in distribution implies that τn converges to 0 in probability. Examples are presented to illustrate the result is the best possible.

Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

In this paper the oscillatory and asymptotic properties of solutions of the equation

are investigated where δ = ±1, τ > 0, σ > 0, the functions r(s) and are non- decreasing and .

We obtain a topological zero-one law for sets with the Baire property which are invariant under a semigroup of open continuous maps acting on a topological space.

In this paper certain estimates of the rate of convergence of triangular matrix means of the Fourier Stieltjes series and its conjugate series are obtained.

The multivariate stochastic ordering induced by the convex nondecreasing functions compares a combination of size and variability of random vectors. Closely following methods developed by Strassen, we show that two probability measures are ordered in this way if and only if they are the marginals of some submartingale. The implications of this in majorisation theory are discussed.

A complete description is given of rings isomorphic to their unbounded left (right) ideals. The same problem for 2-sided ideals remains open.

In this paper we prove that if R is a ring with unity satisfying [xy − xn, ym = 0, for all x, y ∈ R and fixed integers m ≥ 1, n ≥ 1, then R is commutative.

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

In this note two theorems of Ma and Fan concerning sets with convex sections are generalised.