New results on a classical operator

It is remarkable how the classical Volterra integral operator, which was one of the first operators which attracted mathematicians' attention, is still worth of being studied. In this essentially survey work, by collecting some of the very recent results related to the Volterra operator, we show that there are new (and not so new) concepts that are becoming known only at the present days. Discovering whether the Volterra operator satisfies or not a given operator property leads to new methods and ideas that are useful in the setting of Concrete Operator Theory as well as the one of General Operator Theory. In particular, a wide variety of techniques like summability kernels, theory of entire functions, Gaussian cylindrical measures, approximation theory, Laguerre and Legendre polynomials are needed to analyze different properties of the Volterra operator. We also include a characterization of the commutator of the Volterra operator acting on L-P[0, 1], 1

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.

Dendrochronological evidence for a lower water-table on peatland around 3200-3000 BC from subfossil pine in northern Scotland

Tree-ring analysis of subfossil Pinus sylvestris L., from nine new peatland sites located beyond the species’ current northern limit in Scotland, established a regional chronology called WRATH-9. The chronology has been provisionally dated against Irish pine chronologies and provides the first annual resolution picture of Scots pine expansion from c. 3200 bc and subsequent demise from c. 3000 bc. Pine germination and growth is suggested to be associated with a widespread fall in bog water-tables that indicates a regional climatic control. Bog pines progressively declined in number, rather than died out in a single event, reflecting their growth in a marginal habitat, close to a critical ecological threshold. The use of tree-ring sequences from in situ bog pine macrofossils provides a higher resolution insight into past conditions than possible with existing radiocarbon and pollen-based chronologies.

Development of a genotyping microarray for Usher syndrome

Usher syndrome, a combination of retinitis pigmentosa (RP) and sensorineural hearing loss with or without vestibular dysfunction, displays a high degree of clinical and genetic heterogeneity. Three clinical subtypes can be distinguished, based on the age of onset and severity of the hearing impairment, and the presence or absence of vestibular abnormalities. Thus far, eight genes have been implicated in the syndrome, together comprising 347 protein-coding exons.