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Abstract In this paper, we consider the fractional Hardy–Hénon equations with an isolated singularity. If the isolated singularity is located at the origin, we give a classification of solutions to this equation. If the isolated singularity is located at infinity, in the case of exterior domains, we provide decay estimates of solutions and their gradients at infinity. Our results are an extension of the classical work by Caffarelli, Gidas et al. [ABSTRACT FROM AUTHOR]

Humans are exposed to polybrominated diphenyl ethers (PBDEs) via ingestion of food, dust inhalation, and dermal absorption. Exposure to PBDEs via the placenta and breast milk is a special and important pathway in infants. This nested case-control study aimed to investigate the levels of PBDEs in maternal serum and colostrum, and to assess the association between the occurrence of fetal growth restriction (FGR) and prenatal exposure to PBDEs. We recruited 293 mother-newborn pairs, including 98 FGR cases and 195 healthy controls in Wenzhou, China. Maternal serum and colostrum samples were collected during pregnancy and after delivery, respectively, and the levels of PBDEs were measured by gas chromatography-tandem mass spectrometry. The total levels of PBDEs in maternal serum and colostrum were found to be in equilibrium, but congener profiles of PBDEs in these matrices were different. Increased BDE-207, BDE-209, ∑BDE196-209 and ∑PBDEs levels in maternal serum and BDE-99, ∑BDE17-154 and ∑PBDEs levels in colostrum were correlated with decreased birth weight Z score. Increased concentrations of higher brominated BDEs in maternal serum (odds ratio (OR) = 1.010, 95% CI = 1.003–1.018) and low-to moderately brominated BDEs in colostrum (OR = 1.004, 95% CI = 1.000–1.009) were associated with increased risk of FGR, which showed an exposure–response relationship. In addition, infants with FGR were more exposed to PBDEs in colostrum after birth than healthy infants. Longitudinal birth cohort studies are needed to determine the prolonged effect of PBDEs exposure on the growth of FGR infants in the future. • We measured polybrominated diphenyl ethers (PBDEs) concentrations of maternal serum and colostrum. • Higher concentrations of PBDEs in maternal serum and colostrum were associated with the risk of fetal growth restriction.. • Infants with fetal growth restriction are more exposed to PBDEs in colostrum after birth than who with normal birth.. [ABSTRACT FROM AUTHOR]

Abstract Heterogeneous longitudinal data have become prevalent in medical, biological, and social studies. This paper proposes a finite mixture of varying coefficient models for handling heterogeneous populations. Each component of the mixture is modeled by a varying coefficient mixed-effect model that characterizes the longitudinal relations among variables. The identifiability of the mixture model is studied. Regression splines with equally spaced knots are used to approximate the varying coefficient functions, and a nested expectation maximization algorithm is developed to obtain the maximum likelihood estimation. We propose a penalized likelihood method based on the smoothly clipped absolute deviation (SCAD) penalty for the component selection of finite mixture of varying coefficient model. A modified BIC-based criterion based on the SCAD penalty, the BIC SCAD , is proposed for selecting the penalty parameter and spline space simultaneously. The asymptotic properties of parameter estimation and component selection consistency are studied under mild conditions. Simulation studies are conducted to illustrate the component selection, parameter estimation, and inference of the proposed method. The model is then applied to a heterogeneous longitudinal data set from a study of the treatment effect on the use of heroin in the California Civil Addict Program. [ABSTRACT FROM AUTHOR]