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

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times

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

A new strategy for isolating genes controlling dosage compensation in Drosophila using a simple epigenetic mosaic eye phenotype

<p>Abstract</p> <p>Background</p> <p>The <it>Drosophila </it>Male Specific Lethal (MSL) complex contains chromatin modifying enzymes and non-coding <it>roX </it>RNA. It paints the male X at hundreds of bands where it acetylates histone H4 at lysine 16. This epigenetic mark increases expression from the single male X chromosome approximately twofold above what gene-specific factors produce from each female X chromosome. This equalises X-linked gene expression between the sexes. Previous screens for components of dosage compensation relied on a distinctive male-specific lethal phenotype.</p> <p>Results</p> <p>Here, we report a new strategy relying upon an unusual male-specific mosaic eye pigmentation phenotype produced when the MSL complex acts upon autosomal <it>roX1 </it>transgenes. Screening the second chromosome identified at least five loci, two of which are previously described components of the MSL complex. We focused our analysis on the modifier alleles of MSL1 and MLE (for 'maleless'). The MSL1 lesions are not simple nulls, but rather alter the PEHE domain that recruits the MSL3 chromodomain and MOF ('males absent on first') histone acetyltransferase subunits to the complex. These mutants are compromised in their ability to recruit MSL3 and MOF, dosage compensate the X, and support long distance spreading from <it>roX1 </it>transgenes. Yet, paradoxically, they were isolated because they somehow increase MSL complex activity immediately around <it>roX1 </it>transgenes in combination with wild-type MSL1 subunits.</p> <p>Conclusions</p> <p>We propose that these diverse phenotypes arise from perturbations in assembly of MSL subunits onto nascent <it>roX </it>transcripts. This strategy is a promising alternative route for identifying previously unknown components of the dosage compensation pathway and novel alleles of known MSL proteins.</p

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