Anthony G. O'Farrell

Annihilators of rational modules

Abstract We apply the Cauchy transform to derive results which relate approximation problems in different Lipshitz norms, and in the uniform norm, to one another.

Approximation by a sum of two algebras. The lightning bolt principle

Abstract The measures on a compact Hausdorff space X orthogonal to the sum A 1 + A 2 of two subalgebras of C R ( X ), the real-valued continuous functions on X , are described. From this description, a geometric condition equivalent to the density of C ( x 1 ) + C ( x 2 ) in C R ( X ) is obtained, where X ⊂( x j ) and where C ( x j ) denotes the continuous functions depending only on the j th coordinate function.

Composition of Involutive Power Series, and Reversible Series

For each natural number n, we characterise the invertible series (under composition) that are the composition of n proper involutions. We work with formal power series in one variable over a field of characteristic zero. We also describe the reversible series (those conjugate to their own inverses), and the series that are the composition of n reversible series.

Reversibility in the group of homeomorphisms of the circle

The group of orientation-preserving homeomorphisms of the circle is simple, and, because there are non-trivial involutions in this group, it must be generated by its involutions. We show that, in this group of homeomorphisms, each element can be expressed as a product of three involutions. We also characterise those elements of the group that can be expressed as a composite of two involutions, and perform a similar characterisation in the full group of homeomorphisms of the circle.

Reversible maps in the group of quaternionic Möbius transformations

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Mobius transformations.

BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS

We introduce the concept of an E -valued function algebra, a type of Banach algebra that consists of continuous E -valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E -valued function algebras. We give some specific examples.

An Example on Sobolev Space Approximation

For each d ≥ 2 we construct a connected open set Ω ⊂ d such that Ω =int(clos(Ω)) and for each k ≥ 1 and each p ∈ [1,+ ∞), the subset Wk,∞(Ω) fails to be dense in the Sobolev space Wk,p(Ω), in the norm of Wk,p(Ω).

Formally-reversible maps of C^2

An element

DE PAEPE'S DISC HAS NONTRIVIAL POLYNOMIAL HULL

g

Bistability, Bifurcation and Chaos in a Laser System

of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group