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.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Free and projective Banach lattices

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T: P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such thatT = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

GPCR structure, function, drug discovery and crystallography: report from Academia-Industry International Conference (UK Royal Society) Chicheley Hall, 1-2 September 2014

G-protein coupled receptors (GPCRs) are the targets of over half of all prescribed drugs today. The UniProt database has records for about 800 proteins classified as GPCRs, but drugs have only been developed against 50 of these. Thus, there is huge potential in terms of the number of targets for new therapies to be designed. Several breakthroughs in GPCRs biased pharmacology, structural biology, modelling and scoring have resulted in a resurgence of interest in GPCRs as drug targets. Therefore, an international conference, sponsored by the Royal Society, with world-renowned researchers from industry and academia was recently held to discuss recent progress and highlight key areas of future research needed to accelerate GPCR drug discovery. Several key points emerged. Firstly, structures for all three major classes of GPCRs have now been solved and there is increasing coverage across the GPCR phylogenetic tree. This is likely to be substantially enhanced with data from x-ray free electron sources as they move beyond proof of concept. Secondly, the concept of biased signalling or functional selectivity is likely to be prevalent in many GPCRs, and this presents exciting new opportunities for selectivity and the control of side effects, especially when combined with increasing data regarding allosteric modulation. Thirdly, there will almost certainly be some GPCRs that will remain difficult targets because they exhibit complex ligand dependencies and have many metastable states rendering them difficult to resolve by crystallographic methods. Subtle effects within the packing of the transmembrane helices are likely to mask and contribute to this aspect, which may play a role in species dependent behaviour. This is particularly important because it has ramifications for how we interpret pre-clinical data. In summary, collaborative efforts between industry and academia have delivered significant progress in terms of structure and understanding of GPCRs and will be essential for resolving problems associated with the more difficult targets in the future.

Effect of habitat fragmentation on levels and patterns of genetic diversity in natural populations of the peat moss Polytrichum commune

Peat bogs represent unique ecosystems that are under particular threat from fragmentation due to peat harvesting, with only 38% of the original peatland in Europe remaining intact and unaffected by peat cutting, drainage and silviculture. In this study, we have used microsatellite markers to determine levels and patterns of genetic diversity in both cut and uncut natural populations of the peat moss Polytrichum commune. Overall diversity levels suggest that there is more genetic variation present than had previously been assumed for bryophytes. Despite this, diversity values from completely cut bogs were found to be lower than those from uncut peatlands (average 0.729 versus 0.880). In addition, the genetic diversity was more highly structured in the cut populations, further suggesting that genetic drift is already affecting genetic diversity in peat bogs subjected to fragmentation.

Fitness reduction and potential extinction of wild populations of Atlantic salmon, Salmo salar, as a result of interactions with escaped farm salmon.

The high level of escapes from Atlantic salmon farms, up to two million fishes per year in the North Atlantic, has raised concern about the potential impact on wild populations. We report on a twogeneration experiment examining the estimated lifetime successes, relative to wild natives, of farm, F1 and F2 hybrids and BC1 backcrosses to wild and farm salmon. Offspring of farm and hybrids (i.e. all F1 , F2 and BC1 groups) showed reduced survival compared with wild salmon but grew faster as juveniles and displaced wild parr, which as a group were significantly smaller. Where suitable habitat for these emigrant parr is absent, this competition would result in reduced wild smolt production. In the experimental conditions, where emigrants survived downstream, the relative estimated lifetime success ranged from 2% (farm) to 89% (BC1 wild) of that of wild salmon, indicating additive genetic variation for survival . Wild salmon primarily returned to fresh water after one sea winter (1SW) but farm and hybrids produced proportionately more 2SW salmon. However, lower overall survival means that this would result in reduced recruitment despite increased 2SW fecundity. We thus demonstrate that interaction of farm with wild salmon results in lowered fitness, with repeated escapes causing cumulative fitness depression and potentially an extinction vortex in vulnerable populations.

Molecular documentation of polyembryony and the micro-spatial dispersion of clonal sibships in the nine-banded armadillo, Dasypus novemcinctus.

A battery of allelic markers at highly polymorphic microsatellite loci was developed and employed to confirm genetically the clonal nature of sibships in nine-banded armadillos. This phenomenon of consistent polyembryony, otherwise nearly unknown among the vertebrates, then was capitalized upon to describe the micro-spatial distributions of numerous clonal sibships in a natural population of armadillos. Adult clonemates were significantly more dispersed than were juvenile sibs, suggesting limited opportunities for altruistic behavioral interactions among mature individuals. These results, and considerations of armadillo natural history, suggest that evolutionary explanations for polyembryony in this species may not reside in the kinds of ecological and kin selection theories relevant to some of the polyembryonic invertebrates. Rather, polyembryony in armadillos may be associated evolutionarily with other reproductive peculiarities of the species, including delayed uterine implantation of a single egg.