Time operators for one parameter semigroups

We develop two simple approaches to the construction of time operators for semigroups of continuous linear operators in Hilbert spaces provided that the generators of these semigroups are normal operators. The first approach enables us to give explicit formulas (in the spectral representations) both for the time operators and for their eigenfunctions. The other approach provides no explicit formula. However, it enables us to find necessary and sufficient conditions for the existence of time operators for semigroups of continuous linear operators in separable Hilbert spaces with normal generators. Time superoperators corresponding to unitary groups are also discussed.

On linear extension operators

We construct a countable-dimensional Hausdorff locally convex topological vector space $E$ and a stratifiable closed linear subspace $F$ subset of $E$ such that any linear extension operator from $C_b(F)$ to $C_b(E)$ is unbounded (here $C_b(X)$ stands for the Banach space of continuous bounded real-valued functions on $X$).

Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.

Time operators and shift representation of dynamical systems

We introduce and characterise time operators for unilateral shifts and exact endomorphisms. The associated shift representation of evolution is related to the spectral representation by a generalized Fourier transform. We illustrate the results for a simple exact system, namely the Renyi map.

Invasion by the amphipod Gammarus pulex alters community composition of native freshwater macroinvertebrates

Assessing the effects of invading species on native community structure is often confounded by environmental factors and weakened by lack of replicated, long-term pre- and post-invasion monitoring. Here, we uncouple the community effects of a freshwater amphipod invader from environmental differences. In Irish rivers, the introduced Gammarus pulex replaces the native Gammarus duebeni celticus. However, the River Lissan in Northern Ireland is dissected by a weir that has slowed the upstream invasion by G. pulex. This allowed us in 2000 to sample three contiguous 150-m reaches that were (1) G. pulex dominated; (2) mixed Gammarus spp.; and (3) G. duebeni celticus only. In 2003, we resampled these reaches and one additional of mixed Gammarus species and one with only G. duebeni celticus further upstream. In temperature, conductivity, and pH, there were statistically significant but no biologically relevant differences among the five reaches of 2003, and between the three reaches surveyed in both years. Although there was evidence of recovery in macroinvertebrate diversity and richness in invaded reaches between years, continued upstream invasion was associated with sustained reductions in these community metrics as compared to un-invaded sites. Community ordination indicated (1) different associations of community composition attributed to the distribution, abundance, and biomass of the invader; and (2) increasing similarity of invaded communities over time. The impact mechanisms of G. pulex on macroinvertebrate community composition may include predation and competition. The consequences of the observed community changes for ecosystem functioning require further investigation.

Differences in composition of macroinvertebrate communities with invasive and native Gammarus spp. (Crustacea : Amphipoda)

1. Assessing the effects on communities of invasive species is often confounded by environmental factors. In Irish rivers, the introduced amphipod Gammarus pulex replaces the native G. duebeni celticus in lowland stretches. The two amphipods are associated with different macroinvertebrate communities, which may in part be the result of natural longitudinal physicochemical change. However, this hinders assessment of any direct community impacts of the invasive as compared with the native species. Here, we report on a fortuitous circumstance that allowed us to uncouple the community effects of Gammarus species from environmental differences.

Strasheela: Design and usage of a music composition environment based on the Oz programming model

Strasheela provides a means for the composer to create a symbolic score by formally describing it in a rule-based way. The environment defines a rich music representation for complex polyphonic scores. Strasheela enables the user to define expressive compositional rules and then to apply them to the score. Compositional rules can restrict many aspects of the music - including the rhythmic structure, the melodic structure and the harmonic structure - by constraining the parameters (e.g. duration or pitch) of musical events according to some numerical or logical relation. Strasheela combines this expressivity with efficient search strategies.