Sharp norm estimates of layer potentials and operators at high frequency

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency λ2→∞. For single layer potentials, we find that the L2(∂Ω)→L2(Ω) norms decay in λ. The rate of decay depends on the curvature of ∂Ω: The norm is λ−3/4 in general domains and λ−5/6 if the boundary ∂Ω is curved. The double layer potential, however, displays uniform L2(∂Ω)→L2(Ω) bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. The appendix by Galkowski gives bounds L2(∂Ω)→L2(∂Ω) for the single and double layer operators at high frequency that are sharp modulo logλ. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary

Recurrent Modification of a Conserved Cis-Regulatory Element Underlies Fruit Fly Pigmentation Diversity

The development of morphological traits occurs through the collective action of networks of genes connected at the level of gene expression. As any node in a network may be a target of evolutionary change, the recurrent targeting of the same node would indicate that the path of evolution is biased for the relevant trait and network. Although examples of parallel evolution have implicated recurrent modification of the same gene and cis-regulatory element (CRE), little is known about the mutational and molecular paths of parallel CRE evolution. In Drosophila melanogaster fruit flies, the Bric-à-brac (Bab) transcription factors control the development of a suite of sexually dimorphic traits on the posterior abdomen. Female-specific Bab expression is regulated by the dimorphic element, a CRE that possesses direct inputs from body plan (ABD-B) and sex-determination (DSX) transcription factors. Here, we find that the recurrent evolutionary modification of this CRE underlies both intraspecific and interspecific variation in female pigmentation in the melanogaster species group. By reconstructing the sequence and regulatory activity of the ancestral Drosophila melanogaster dimorphic element, we demonstrate that a handful of mutations were sufficient to create independent CRE alleles with differing activities. Moreover, intraspecific and interspecific dimorphic element evolution proceeded with little to no alterations to the known body plan and sex-determination regulatory linkages. Collectively, our findings represent an example where the paths of evolution appear biased to a specific CRE, and drastic changes in function were accompanied by deep conservation of key regulatory linkages. © 2013 Rogers et al

Strichartz estimates for non-unitary energy bounds and eigenfunction estimates

M. Tacyhttp://waves-2009.bordeaux.inria.fr/index.php?lang=e

Semiclassical L p estimates of quasimodes on curved hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L 2 normalized family of functions such that P(h)u(h) is O(h) in L 2(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch −δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator. In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved (in the sense of Definition 2.6 below) with respect to the bicharacteristic flow of P. Under this assumption we improve the estimate from δ=1/4 to 1/6, generalizing work of Burq–Gérard–Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the Melrose–Taylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.Andrew Hassell, Melissa Tac

Chronic antepartum maternal hyperoxygenation in a case of severe fetal Ebstein's anomaly with circular shunt physiology

Perinatal mortality remains high among fetuses diagnosed with Ebstein's anomaly of the tricuspid valve. The subgroup of patients with pulmonary valve regurgitation is at particularly high risk. In the setting of pulmonary valve regurgitation, early constriction of the ductus arteriosus may be a novel perinatal management strategy to reduce systemic steal resulting from circular shunt physiology. We report the use of chronic antepartum maternal oxygen therapy for constriction of the fetal ductus arteriosus and modulation of fetal pulmonary vascular resistance in a late presentation of Ebstein's anomaly with severe tricuspid valve regurgitation, reversal of flow in the ductus arteriosus, and continuous pulmonary valve regurgitation