Level sets of multiple ergodic averages

We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, Hausdorff dimensions of the level sets of multiple ergodic average limit are determined by using Riesz products.Comment: This note was refused by Proceedings of AMS although the referee said "In my opinion this is a nice application of the Riesz product technique to solve, in principle, a hard problem when considered in its full generality. Nevertheless, I think it needs some extra work to see how this example seats in a more general context and explore how far this technique can go." We should say that Riesz product works perfectly in the situation described in this note, but Riesz product has its limit--we don't think that Riesz product technique can solve the problem in its generalit

Penalized variable selection procedure for Cox models with semiparametric relative risk

We study the Cox models with semiparametric relative risk, which can be partially linear with one nonparametric component, or multiple additive or nonadditive nonparametric components. A penalized partial likelihood procedure is proposed to simultaneously estimate the parameters and select variables for both the parametric and the nonparametric parts. Two penalties are applied sequentially. The first penalty, governing the smoothness of the multivariate nonlinear covariate effect function, provides a smoothing spline ANOVA framework that is exploited to derive an empirical model selection tool for the nonparametric part. The second penalty, either the smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO penalty, achieves variable selection in the parametric part. We show that the resulting estimator of the parametric part possesses the oracle property, and that the estimator of the nonparametric part achieves the optimal rate of convergence. The proposed procedures are shown to work well in simulation experiments, and then applied to a real data example on sexually transmitted diseases.Comment: Published in at http://dx.doi.org/10.1214/09-AOS780 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The roles of endoglin gene in cerebrovascular diseases.

Endoglin (ENG, also known as CD105) is a transforming growth factor β (TGFβ) associated receptor and is required for both vasculogenesis and angiogenesis. Angiogenesis is important in the development of cerebral vasculature and in the pathogenesis of cerebral vascular diseases. ENG is an essential component of the endothelial nitric oxide synthase activation complex. Animal studies showed that ENG deficiency impairs stroke recovery. ENG deficiency also impairs the regulation of vascular tone, which contributes to the pathogenesis of brain arteriovenous malformation (bAVM) and vasospasm. In human, functional haploinsufficiency of ENG gene causes type I hereditary hemorrhagic telangiectasia (HHT1), an autosomal dominant disorder. Compared to normal population, HHT1 patients have a higher prevalence of AVM in multiple organs including the brain. Vessels in bAVM are fragile and tend to rupture, causing hemorrhagic stroke. High prevalence of pulmonary AVM in HHT1 patients are associated with a higher incidence of paradoxical embolism in the cerebral circulation causing ischemic brain injury. Therefore, HHT1 patients are at risk for both hemorrhagic and ischemic stroke. This review summarizes the possible mechanism of ENG in the pathogenesis of cerebrovascular diseases in experimental animal models and in patients