Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati

Prevalence and risk of Down syndrome in monozygotic and dizygotic multiple pregnancies in Europe: implications for prenatal screening.

OBJECTIVE: To determine risk of Down syndrome (DS) in multiple relative to singleton pregnancies, and compare prenatal diagnosis rates and pregnancy outcome. DESIGN: Population-based prevalence study based on EUROCAT congenital anomaly registries. SETTING: Eight European countries. POPULATION: 14.8 million births 1990-2009; 2.89% multiple births. METHODS: DS cases included livebirths, fetal deaths from 20 weeks, and terminations of pregnancy for fetal anomaly (TOPFA). Zygosity is inferred from like/unlike sex for birth denominators, and from concordance for DS cases. MAIN OUTCOME MEASURES: Relative risk (RR) of DS per fetus/baby from multiple versus singleton pregnancies and per pregnancy in monozygotic/dizygotic versus singleton pregnancies. Proportion of prenatally diagnosed and pregnancy outcome. STATISTICAL ANALYSIS: Poisson and logistic regression stratified for maternal age, country and time. RESULTS: Overall, the adjusted (adj) RR of DS for fetus/babies from multiple versus singleton pregnancies was 0.58 (95% CI 0.53-0.62), similar for all maternal ages except for mothers over 44, for whom it was considerably lower. In 8.7% of twin pairs affected by DS, both co-twins were diagnosed with the condition. The adjRR of DS for monozygotic versus singleton pregnancies was 0.34 (95% CI 0.25-0.44) and for dizygotic versus singleton pregnancies 1.34 (95% CI 1.23-1.46). DS fetuses from multiple births were less likely to be prenatally diagnosed than singletons (adjOR 0.62 [95% CI 0.50-0.78]) and following diagnosis less likely to be TOPFA (adjOR 0.40 [95% CI 0.27-0.59]). CONCLUSIONS: The risk of DS per fetus/baby is lower in multiple than singleton pregnancies. These estimates can be used for genetic counselling and prenatal screening

Oka's inequality for currents and applications

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46241/1/208_2005_Article_BF01446636.pd

A Renormalization Group Approach to Relativistic Cosmology

We discuss the averaging hypothesis tacitly assumed in standard cosmology. Our approach is implemented in a "3+1" formalism and invokes the coarse graining arguments, provided and supported by the real-space Renormalization Group (RG) methods. Block variables are introduced and the recursion relations written down explicitly enabling us to characterize the corresponding RG flow. To leading order, the RG flow is provided by the Ricci-Hamilton equations studied in connection with the geometry of three-manifolds. The properties of the Ricci-Hamilton flow make it possible to study a critical behaviour of cosmological models. This criticality is discussed and it is argued that it may be related to the formation of sheet-like structures in the universe. We provide an explicit expression for the renormalized Hubble constant and for the scale dependence of the matter distribution. It is shown that the Hubble constant is affected by non-trivial scale dependent shear terms, while the spatial anisotropy of the metric influences significantly the scale-dependence of the matter distribution.Comment: 57 pages, LaTeX, 15 pictures available on request from the Author

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2 \mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $\mu$ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If $(B,g_B)$ is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.Comment: 32 pages, 3 figure

Beta-Blocker Use in Pregnancy and Risk of Specific Congenital Anomalies: A European Case-Malformed Control Study.

The prevalence of chronic hypertension is increasing in pregnant women. Beta-blockers are among the most prevalent anti-hypertensive agents used in early pregnancy. The objective of this study was to investigate whether first-trimester use of beta-blockers increases the risk of specific congenital anomalies in offspring. A population-based case-malformed control study was conducted in 117,122 registrations of congenital anomalies from 17 European Concerted Action on Congenital Anomalies and Twins (EUROCAT) registries participating in EUROmediCAT with data for all or part of the period between 1995 and 2013. Associations previously reported in the literature (signals) were tested and an exploratory analysis was performed to identify new signals. Odds ratios of exposure to any beta-blocker or to a beta-blocker subgroup were calculated for each signal anomaly compared with two control groups (non-chromosomal, non-signal anomalies and chromosomal anomalies). The exploratory analyses were performed for each non-signal anomaly compared with all the other non-signal anomalies. The signals from the literature (congenital heart defects, oral clefts, neural tube defects and hypospadias) were not confirmed. Our exploratory analysis revealed that multi-cystic renal dysplasia had significantly increased odds of occurring after maternal exposure to combined alpha- and beta-blockers (adjusted odds ratio 3.8; 95% confidence interval 1.3-11.0). Beta-blocker use in the first trimester of pregnancy was not found to be associated with a higher risk of specific congenital anomalies in the offspring, but a new signal between alpha- and beta-blockers and multi-cystic renal dysplasia was found. Future large epidemiological studies are needed to confirm or refute our findings

Comparative genomics of isolates of a pseudomonas aeruginosa epidemic strain associated with chronic lung infections of cystic fibrosis patients

Pseudomonas aeruginosa is the main cause of fatal chronic lung infections among individuals suffering from cystic fibrosis (CF). During the past 15 years, particularly aggressive strains transmitted among CF patients have been identified, initially in Europe and more recently in Canada. The aim of this study was to generate high-quality genome sequences for 7 isolates of the Liverpool epidemic strain (LES) from the United Kingdom and Canada representing different virulence characteristics in order to: (1) associate comparative genomics results with virulence factor variability and (2) identify genomic and/or phenotypic divergence between the two geographical locations. We performed phenotypic characterization of pyoverdine, pyocyanin, motility, biofilm formation, and proteolytic activity. We also assessed the degree of virulence using the Dictyostelium discoideum amoeba model. Comparative genomics analysis revealed at least one large deletion (40-50 kb) in 6 out of the 7 isolates compared to the reference genome of LESB58. These deletions correspond to prophages, which are known to increase the competitiveness of LESB58 in chronic lung infection. We also identified 308 non-synonymous polymorphisms, of which 28 were associated with virulence determinants and 52 with regulatory proteins. At the phenotypic level, isolates showed extensive variability in production of pyocyanin, pyoverdine, proteases and biofilm as well as in swimming motility, while being predominantly avirulent in the amoeba model. Isolates from the two continents were phylogenetically and phenotypically undistinguishable. Most regulatory mutations were isolate-specific and 29% of them were predicted to have high functional impact. Therefore, polymorphism in regulatory genes is likely to be an important basis for phenotypic diversity among LES isolates, which in turn might contribute to this strain's adaptability to varying conditions in the CF lung