Itô diffusions, modified capacity and harmonic measure. Applications to Schrödinger operators.

We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R^d, d>1. We introduce Feynman-Kac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better understand the analytic properties of the processes, we construct and study a special version of the potential theory. The modified capacity and harmonic measure play an important role in these considerations. Various applications to Schrödinger operators are also given. For example, we relate the presence of the absolutely continuous spectrum to the geometric properties of the support of the potential

On the growth of the polynomial entropy integrals for measures in the Szegö class

Let be a probability Borel measure on the unit circle T and f ng be the orthonormal polynomials with respect to . We say tRhat is a Szego measure, if it has an arbitrary singular part s, and T log 0dm > -∞, 1, where 0 is the density of the absolutely continuous part of , m being the normalized Lebesgue measure on T. The entropy integrals for n are de ned as n = Z T j nj2 log j njd It is not di cult to show that n = o( p n). In this paper, we construct a measure from the Szego class for which this estimate is sharp (over a subsequence of n's)

The Transcriptional and Epigenomic Foundations of Ground State Pluripotency

SummaryMouse embryonic stem (ES) cells grown in serum exhibit greater heterogeneity in morphology and expression of pluripotency factors than ES cells cultured in defined medium with inhibitors of two kinases (Mek and GSK3), a condition known as “2i” postulated to establish a naive ground state. We show that the transcriptome and epigenome profiles of serum- and 2i-grown ES cells are distinct. 2i-treated cells exhibit lower expression of lineage-affiliated genes, reduced prevalence at promoters of the repressive histone modification H3K27me3, and fewer bivalent domains, which are thought to mark genes poised for either up- or downregulation. Nonetheless, serum- and 2i-grown ES cells have similar differentiation potential. Precocious transcription of developmental genes in 2i is restrained by RNA polymerase II promoter-proximal pausing. These findings suggest that transcriptional potentiation and a permissive chromatin context characterize the ground state and that exit from it may not require a metastable intermediate or multilineage priming

Identification of novel functional TBP-binding sites and general factor repertoires

Our current knowledge of the general factor requirement in transcription by the three mammalian RNA polymerases is based on a small number of model promoters. Here, we present a comprehensive chromatin immunoprecipitation (ChIP)-on-chip analysis for 28 transcription factors on a large set of known and novel TATA-binding protein (TBP)-binding sites experimentally identified via ChIP cloning. A large fraction of identified TBP-binding sites is located in introns or lacks a gene/mRNA annotation and is found to direct transcription. Integrated analysis of the ChIP-on-chip data and functional studies revealed that TAF12 hitherto regarded as RNA polymerase II (RNAP II)-specific was found to be also involved in RNAP I transcription. Distinct profiles for general transcription factors and TAF-containing complexes were uncovered for RNAP II promoters located in CpG and non-CpG islands suggesting distinct transcription initiation pathways. Our study broadens the spectrum of general transcription factor function and uncovers a plethora of novel, functional TBP-binding sites in the human genome