On distributions of functionals of anomalous diffusion paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.Comment: 25 pages, 4 figure

Evolutionarily Stable Association of Intronic snoRNAs and microRNAs with Their Host Genes

Small nucleolar RNAs (snoRNAs) and microRNAs (miRNAs) are integral to a range of processes, including ribosome biogenesis and gene regulation. Some are intron encoded, and this organization may facilitate coordinated coexpression of host gene and RNA. However, snoRNAs and miRNAs are known to be mobile, so intron-RNA associations may not be evolutionarily stable. We have used genome alignments across 11 mammals plus chicken to examine positional orthology of snoRNAs and miRNAs and report that 21% of annotated snoRNAs and 11% of miRNAs are positionally conserved across mammals. Among RNAs traceable to the bird–mammal common ancestor, 98% of snoRNAs and 76% of miRNAs are intronic. Comparison of the most evolutionarily stable mammalian intronic snoRNAs with those positionally conserved among primates reveals that the former are more overrepresented among host genes involved in translation or ribosome biogenesis and are more broadly and highly expressed. This stability is likely attributable to a requirement for overlap between host gene and intronic snoRNA expression profiles, consistent with an ancestral role in ribosome biogenesis. In contrast, whereas miRNA positional conservation is comparable to that observed for snoRNAs, intronic miRNAs show no obvious association with host genes of a particular functional category, and no statistically significant differences in host gene expression are found between those traceable to mammalian or primate ancestors. Our results indicate evolutionarily stable associations of numerous intronic snoRNAs and miRNAs and their host genes, with probable continued diversification of snoRNA function from an ancestral role in ribosome biogenesis

“The Eastern Enlargement of the European Union and Environmental Policy: Challenges, Expectations, Speeds and Flexibility”

Having sketched the general problems and opportunities raised by Eastern enlargement, the remainder of this paper will concentrate on the implications of enlargement for European environmental policy. In this context, the conclusion reached by the European Commission that, as far as Community environmental law is concerned, no accession country will, in the near future, be in a position to meet in total the obligations laid down in the Acquis might give cause for scepticism, particularly as this level of pessimism is unparalleled for any other area of the Acquis. This begs the question of which problems specific to the environmental sector lead the Commission to such a prognosis and further: What are the possibilities of ensuring-in spite of these problems-the greatest possible level of compliance with Community environmental legislation in an expanded European Union