Knowledge about factors that are important in coral reef growth help us to understand how reef ecosystems react following major anthropogenic and environmental disturbances. In addition, they may help the industry understand how aquarists can improve the health of their corals. I have studied environmental and climate effects on corals on fringing reefs in Jamaica. Radial growth rates (mm/yr) of non-branching corals calculated on an annual basis from 2000–2008 showed few significant differences either spatially or temporally along the north coast, although growth rates tended to be higher on reefs of higher rugosity and lower macroalgal cover. I have also reconstructed recruitment patterns, using growth modelling, for non-branching corals at sites on the north coast of Jamaica near Discovery Bay, and near Kingston Harbour, on the south coast. For all the sites, recruitment of non-branching corals was lowered due to hurricanes or severe storms. For 1560 non-branching corals at sites along the north coast of Jamaica, from Rio Bueno to Pear Tree, there was a significant difference in estimated coral recruitment in years when there were no storms or hurricanes by comparison to years when storms and hurricanes impacted the area. For 347 non-branching corals at sites in the Port Royal Cays on the south coast, there was a significant difference in estimated coral recruitment in years when there were no storms or hurricanes by comparison to years when storms and hurricanes impacted the area. Interestingly, recruitment of Siderastrea siderea on to the side of the ship channel at Rackham's Cay (~100 m from the path taken by large ships) outside Kingston Harbour had been consistent since its construction. These findings have important implications for better understanding the impacts of tropical storms on coral reefs and for aquarists to better maintain coral reef species in artificial environments.

Based on the work of Adem and Cohen, this note describes an explicit stable decomposition of the space of commuting n-tuples in SU(2) as a wedge of indecomposable summands.

Let S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

A class of extremal Banach algebras has Arens irregular multiplication.

Throughout this paper X will be a finite connected CW-complex of dimension m, and ξ will be a real (n + l)-plane bundle over X(n >0) equipped with a Riemannian metric. We aim to give a systematic account of the space ГSξ of sections of the sphere-bundle Sξ.

Let V and W be finite dimensional real vector spaces, k≧0 an integer. We write L(V, W) for the space of all linear maps V→W and Lk(V, W) for the subspace of maps with kernel of dimension k; in particular, L0(V, W) is the open subspace of injective linear maps. Thus Lk(ℝn, ℝn) is the space of n × n-matrices of rank n – k in the title. We also need the notation Gk(V) for the Grassmann manifold of K-dimensional subspaces of V.

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

The loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds.

It is shown that the complex semigroup algebra of a free monoid of rank at least two is *-primitive, where * denotes the involution on the algebra induced by word-reversal on the monoid.

Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).

A generalization due to Gessel [3] of Miki's identity between Bernoulli numbers is shown to be a direct consequence of a functional equation for the generating function.

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1. We show that: Ea(u,v)={f+gu:f,g\in C(𝕋)} , where [] is the unit circle; Ea(u,v) is C^*-equivalent to C^*({\cal G}), where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v).

Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension. The stable decomposition is related to a description of the integral cohomology ring.

Methods from fibrewise homology theory are illustrated by computations of cohomology rings of certain mapping spaces arising in the geometry of loop groups, specifically the spaces of maps from S^1 to the classifying space BSO(n) of SO(n) and maps from S{\hskip1}^2 to BSU(n).

We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions.

For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ is finite; indeed this remains true when these gauge groups are classified by $H$-equivalence, that is, homotopy equivalences which respect multiplication up to homotopy.A case study is given for $K = S^4$, $G = \text{SU}(2)$:there are eighteen$H$-equivalence classes of gauge group in this case.These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve $K$-theories and $e$-invariants. 1991 Mathematics Subject Classification:54C35, 55P15, 55R10.

This paper asks: given a vector bundle ξ and a line bundle λ over the same base space, are λ[otimes]ξ and ξ equivalent? We concentrate on real bundles ξ. Although the question is sensible in its own right, we explain in Section 2 our immediate motivation for studying it. In Section 3 we make some general comments about the question, the most significant being that under certain restrictions the answer depends on the stable class of ξ rather than on ξ itself (Proposition 3·4).

The rest of the paper tackles an interesting special case. To state the main result, let P(ℝn+1) denote n-dimensional real projective space, H the Hopf line bundle over it, and an+1 the order of the reduced Grothendieck group [wavy overbar]KO0(P(ℝn+1)).