Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for (3/2)-\varep < p< 3+\epsilon. This gives an affirmative answer to a conjecture of M. Taylor.Comment: 28 page. Minor revision was made regarding the definition of the Stokes operator in Lipschitz domain

Some genus 3 curves with many points

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves

More Discriminants with the Brezing-Weng Method

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number

Equidistribution of Heegner Points and Ternary Quadratic Forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system

On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds

Asymptotic laws for mean multiplicities of lengths of closed geodesics in arithmetic hyperbolic three-orbifolds are derived. The sharpest results are obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o) and some congruence subgroups. Similar results hold for cocompact arithmetic quaternion groups, if a conjecture on the number of gaps in their length spectra is true. The results related to the groups above give asymptotic lower bounds for the mean multiplicities in length spectra of arbitrary arithmetic hyperbolic three-orbifolds. The investigation of these multiplicities is motivated by their sensitive effect on the eigenvalue spectrum of the Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as the Hamiltonian of a three-dimensional quantum system being strongly chaotic in the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT

A Novel Pzg-NURF Complex Regulates Notch Target Gene Activity

The Putzig (Pzg) protein is associated with the NURF nucleosome remodeling complex, thereby promoting Notch target gene expression. Our findings suggest a novel Pzg-NURF complex that is responsible for the epigenetic regulation of Notch target genes

ISWI Regulates Higher-Order Chromatin Structure and Histone H1 Assembly In Vivo

Imitation SWI (ISWI) and other ATP-dependent chromatin-remodeling factors play key roles in transcription and other processes by altering the structure and positioning of nucleosomes. Recent studies have also implicated ISWI in the regulation of higher-order chromatin structure, but its role in this process remains poorly understood. To clarify the role of ISWI in vivo, we examined defects in chromosome structure and gene expression resulting from the loss of Iswi function in Drosophila. Consistent with a broad role in transcriptional regulation, the expression of a large number of genes is altered in Iswi mutant larvae. The expression of a dominant-negative form of ISWI leads to dramatic alterations in higher-order chromatin structure, including the apparent decondensation of both mitotic and polytene chromosomes. The loss of ISWI function does not cause obvious defects in nucleosome assembly, but results in a significant reduction in the level of histone H1 associated with chromatin in vivo. These findings suggest that ISWI plays a global role in chromatin compaction in vivo by promoting the association of the linker histone H1 with chromatin

RSF Governs Silent Chromatin Formation via Histone H2Av Replacement

Human remodeling and spacing factor (RSF) consists of a heterodimer of Rsf-1 and hSNF2H, a counterpart of Drosophila ISWI. RSF possesses not only chromatin remodeling activity but also chromatin assembly activity in vitro. While no other single factor can execute the same activities as RSF, the biological significance of RSF remained unknown. To investigate the in vivo function of RSF, we generated a mutant allele of Drosophila Rsf-1 (dRsf-1). The dRsf-1 mutant behaved as a dominant suppressor of position effect variegation. In dRsf-1 mutant, the levels of histone H3K9 dimethylation and histone H2A variant H2Av were significantly reduced in an euchromatic region juxtaposed with heterochromatin. Furthermore, using both genetic and biochemical approaches, we demonstrate that dRsf-1 interacts with H2Av and the H2Av-exchanging machinery Tip60 complex. These results suggest that RSF contributes to histone H2Av replacement in the pathway of silent chromatin formation

Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

Stress-Induced PARP Activation Mediates Recruitment of Drosophila Mi-2 to Promote Heat Shock Gene Expression

Eukaryotic cells respond to genomic and environmental stresses, such as DNA damage and heat shock (HS), with the synthesis of poly-[ADP-ribose] (PAR) at specific chromatin regions, such as DNA breaks or HS genes, by PAR polymerases (PARP). Little is known about the role of this modification during cellular stress responses. We show here that the nucleosome remodeler dMi-2 is recruited to active HS genes in a PARP–dependent manner. dMi-2 binds PAR suggesting that this physical interaction is important for recruitment. Indeed, a dMi-2 mutant unable to bind PAR does not localise to active HS loci in vivo. We have identified several dMi-2 regions which bind PAR independently in vitro, including the chromodomains and regions near the N-terminus containing motifs rich in K and R residues. Moreover, upon HS gene activation, dMi-2 associates with nascent HS gene transcripts, and its catalytic activity is required for efficient transcription and co-transcriptional RNA processing. RNA and PAR compete for dMi-2 binding in vitro, suggesting a two step process for dMi-2 association with active HS genes: initial recruitment to the locus via PAR interaction, followed by binding to nascent RNA transcripts. We suggest that stress-induced chromatin PARylation serves to rapidly attract factors that are required for an efficient and timely transcriptional response