Equiconvergence of spectral decompositions of Hill operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = -d (2)/dx (2) + v(x), x a L (1)([0, pi], with H (per) (-1) -potential and the free operator L (0) = -d (2)/dx (2), subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that parallel to S-N - S-N(0) : L-a -> L-b parallel to -> 0 if 1 < a <= b < infinity, 1/a - 1/b < 1/2, where S (N) and S (N) (0) are the N-th partial sums of the spectral decompositions of L and L (0). Moreover, if v a H (-alpha) with 1/2 < alpha < 1 and , then we obtain the uniform equiconvergence aEuro-S (N) -S (N) (0) : L (a) -> L (a)aEuro- -> 0 as N -> a

Networks of mRNA Processing and Alternative Splicing Regulation in Health and Disease

Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 1157)mRNA processing events introduce an intricate layer of complexity into gene expression processes, supporting a tremendous level of diversification of the genome's coding and regulatory potential, particularly in vertebrate species. The recent development of massive parallel sequencing methods and their adaptation to the identification and quantification of different RNA species and the dynamics of mRNA metabolism and processing has generated an unprecedented view over the regulatory networks that are established at this level, which contribute to sustain developmental, tissue specific or disease specific gene expression programs. In this chapter, we provide an overview of the recent evolution of transcriptome profiling methods and the surprising insights that have emerged in recent years regarding distinct mRNA processing events - from the 5' end to the 3' end of the molecule.info:eu-repo/semantics/publishedVersio