Price's Law on Nonstationary Spacetimes

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.Comment: 32 pages, no figures, typos correcte

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as $t^{-2\ell-3}$ for $t \to \infty$ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be $t^{-2\ell-4}$. We give a proof of $t^{-2\ell-2}$ decay for general data in the form of weighted $L^1$ to $L^\infty$ bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain $t^{-2\ell-3}$. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.Comment: 43 pages, statement of Price's Law clarified and some minor corrections, to appear in Adv. Mat

Strichartz estimates on Schwarzschild black hole backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place

Genetic contributions to visuospatial cognition in Williams syndrome: insights from two contrasting partial deletion patients

Background Williams syndrome (WS) is a rare neurodevelopmental disorder arising from a hemizygotic deletion of approximately 27 genes on chromosome 7, at locus 7q11.23. WS is characterised by an uneven cognitive profile, with serious deficits in visuospatial tasks in comparison to relatively proficient performance in some other cognitive domains such as language and face processing. Individuals with partial genetic deletions within the WS critical region (WSCR) have provided insights into the contribution of specific genes to this complex phenotype. However, the combinatorial effects of different genes remain elusive. Methods We report on visuospatial cognition in two individuals with contrasting partial deletions in the WSCR: one female (HR), aged 11 years 9 months, with haploinsufficiency for 24 of the WS genes (up to GTF2IRD1), and one male (JB), aged 14 years 2 months, with the three most telomeric genes within the WSCR deleted, or partially deleted. Results Our in-depth phenotyping of the visuospatial domain from table-top psychometric, and small- and large-scale experimental tasks reveal a profile in HR in line with typically developing controls, albeit with some atypical features. These data are contrasted with patient JB’s atypical profile of strengths and weaknesses across the visuospatial domain, as well as with more substantial visuospatial deficits in individuals with the full WS deletion. Conclusions Our findings point to the contribution of specific genes to spatial processing difficulties associated with WS, highlighting the multifaceted nature of spatial cognition and the divergent effects of genetic deletions within the WSCR on different components of visuospatial ability. The importance of general transcription factors at the telomeric end of the WSCR, and their combinatorial effects on the WS visuospatial phenotype are also discussed

Missense variants in the N-terminal domain of the A isoform of FHF2/FGF13 cause an X-linked developmental and epileptic encephalopathy

Fibroblast growth factor homologous factors (FHFs) are intracellular proteins which regulate voltage-gated sodium (Na v) channels in the brain and other tissues. FHF dysfunction has been linked to neurological disorders including epilepsy. Here, we describe two sibling pairs and three unrelated males who presented in infancy with intractable focal seizures and severe developmental delay. Whole-exome sequencing identified hemi- and heterozygous variants in the N-terminal domain of the A isoform of FHF2 (FHF2A). The X-linked FHF2 gene (also known as FGF13) has alternative first exons which produce multiple protein isoforms that differ in their N-terminal sequence. The variants were located at highly conserved residues in the FHF2A inactivation particle that competes with the intrinsic fast inactivation mechanism of Na v channels. Functional characterization of mutant FHF2A co-expressed with wild-type Na v1.6 (SCN8A) revealed that mutant FHF2A proteins lost the ability to induce rapid-onset, long-term blockade of the channel while retaining pro-excitatory properties. These gain-of-function effects are likely to increase neuronal excitability consistent with the epileptic potential of FHF2 variants. Our findings demonstrate that FHF2 variants are a cause of infantile-onset developmental and epileptic encephalopathy and underline the critical role of the FHF2A isoform in regulating Na v channel function

Further delineation of the clinical spectrum of White-Sutton syndrome: 12 new individuals and a review of the literature

White–Sutton syndrome (WHSUS) is a neurodevelopmental disorder caused by heterozygous loss-of-function variants in POGZ. Through the Deciphering Developmental Disorders study and clinical testing, we identified 12 individuals from 10 families with pathogenic or likely pathogenic variants in POGZ (eight de novo and two inherited). Most individuals had delayed development and/or intellectual disability. We analyzed the clinical findings in our series and combined it with data from 89 previously reported individuals. The results demonstrate WHSUS is associated with variable developmental delay or intellectual disability, increased risk of obesity, visual defects, craniofacial dysmorphism, sensorineural hearing loss, feeding problems, seizures, and structural brain malformations. Our series includes further individuals with rod-cone dystrophy, cleft lip and palate, congenital diaphragmatic hernia, and duplicated renal drainage system, suggesting these are rare complications of WHSUS. In addition, we describe an individual with a novel, de novo missense variant in POGZ and features of WHSUS. Our work further delineates the phenotypic spectrum of WHSUS highlighting the variable severity of this disorder and the observation of familial pathogenic POGZ variants

Measurement of the cross-section and charge asymmetry of WW bosons produced in proton-proton collisions at s=8\sqrt{s}=8 TeV with the ATLAS detector

This paper presents measurements of the $W^+ \rightarrow \mu^+\nu$ and $W^- \rightarrow \mu^-\nu$ cross-sections and the associated charge asymmetry as a function of the absolute pseudorapidity of the decay muon. The data were collected in proton--proton collisions at a centre-of-mass energy of 8 TeV with the ATLAS experiment at the LHC and correspond to a total integrated luminosity of 20.2~\mbox{fb^{-1}}. The precision of the cross-section measurements varies between 0.8% to 1.5% as a function of the pseudorapidity, excluding the 1.9% uncertainty on the integrated luminosity. The charge asymmetry is measured with an uncertainty between 0.002 and 0.003. The results are compared with predictions based on next-to-next-to-leading-order calculations with various parton distribution functions and have the sensitivity to discriminate between them.Comment: 38 pages in total, author list starting page 22, 5 figures, 4 tables, submitted to EPJC. All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/STDM-2017-13