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

Concerning the Wave equation on Asymptotically Euclidean Manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $x ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.Comment: Final version. To appear in Journal d'Analyse Mathematiqu

The Strauss conjecture on asymptotically flat space-times

By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.Comment: Final version, to appear in SIAM Journal on Mathematical Analysis. 17 page

Resolvent estimates for normally hyperbolic trapped sets

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schr\"odinger propagator as well as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5 and Lemma 4.1; this now also corrects hypotheses, explicitly requiring trapped set to be symplectic. Erratum follows references in this versio

Pharmacological Benefits of Herbal Formulations in the Management of <i>Psoriasis vulgaris<i>

Psoriasis is a chronic inflammatory skin disease, with an important impact on the patient's quality of life. Its incidence and prevalence are continuously increasing. The complex ethiopathology of this disorder is only partially known; there is a clear genetic predisposition, which associates a number of environmental triggering factors such as an unbalanced diet and lifestyle. The conventional therapeutic options are not always satisfactory in terms of efficiency and safety, therefore, complementary and alternative medicine approaches are frequently chosen by patients, mostly as self-medication. This review, based on recent literature flow data, outlines the pharmacological benefits of herbal formulations with antipsoriatic activity. It also reveals the molecules esponsible for their effects, as well as their interference with the metabolic and immunopathogenic mechanisms of this disease. An important number of plants have been proved to act as antipsoriatic agents, many botanical-based preparations containing key-phytochemical molecules (belonging mainly to phenolics, triterpenoids and phytosterols or unsaturated fatty acids, as mentioned in specific phyto-pharmaceutical databases). Specific mechanisms of action, which can explain their activity (such as lipoxygenase inhibition, antioxidant, anti-inflammatory, anti prostaglandin), were recently described. Only some of these formulations have been actively tested in vitro or in vivo . Most publications in the field agree on the need for more in vitro and in vivo studies, especially clinical assessment on patients with Psoriasis vulgaris . These would provide more accurate data on the efficacy and safety of such herbal formulations for this disease

L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $\int_{\gamma} |\phi_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.Comment: Correct some typos and added some more detail in section

World psychiatric association section of old age psychiatry consensus statement on ethics and capacity in older people with mental disorders

The World Psychiatric Association (WPA) Section of Old Age Psychiatry, since 1997, has developed Consensus Statements relevant to the practice of Old Age Psychiatry. Since 2006 the Section has worked to develop a Consensus Statement on Ethics and Capacity in older people with mental disorders, which was completed in Prague, September 2008, prior to the World Congress in Psychiatry. This Consensus meets one of the goals of the WPA Action Plan 2008–2011, ”to promote the highest ethical standards in psychiatric practice and advocate the rights of persons with mental disorders in all regions of the world“. This Consensus Statement offers to mental health clinicians caring for older people with mental disorders, caregivers, other health professionals and the general public the setting out of and discourse in ethical principles which can often be complex and challenging, supported by practical guidance in meeting such ethical needs and standards, and to encouraged good clinical practice. Copyright © 2009 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64443/1/2279_ftp.pd

Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schr\"odiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $\beta>0$ loss. This is in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension

Metabolomic Approaches for Detection and Identification of Biomarkers and Altered Pathways in Bladder Cancer

Metabolomic analysis has proven to be a useful tool in biomarker discovery and the molecular classification of cancers. In order to find new biomarkers, and to better understand its pathological behavior, bladder cancer also has been studied using a metabolomics approach. In this article, we review the literature on metabolomic studies of bladder cancer, focusing on the different available samples (urine, blood, tissue samples) used to perform the studies and their relative findings. Moreover, the multi-omic approach in bladder cancer research has found novel insights into its metabolic behavior, providing excellent start-points for new diagnostic and therapeutic strategies. Metabolomics data analysis can lead to the discovery of a “signature pathway” associated with the progression of bladder cancer; this aspect could be potentially valuable in predictions of clinical outcomes and the introduction of new treatments. However, further studies are needed to give stronger evidence and to make these tools feasible for use in clinical practice

Generalized and weighted Strichartz estimates

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied Analysis. 33 pages. 2 more references adde