Precise tail asymptotics of fixed points of the smoothing transform with general weights

We consider solutions of the stochastic equation $R=_d\sum_{i=1}^NA_iR_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent both from $A_i$'s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}|A_1|^{\alpha}=1/N$ and $\mathbb{E}|A_1|^{\alpha}\log|A_1|>0$, the limit $\lim_{t\to\infty}t^{\alpha}\mathbb{P}[|R|>t]=K$ exists. In the present paper, we prove positivity of $K$.Comment: Published at http://dx.doi.org/10.3150/13-BEJ576 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices

Tissue Doppler imaging following paediatric cardiac surgery : early patterns of change and relationship to outcome

In this study, tissue Doppler imaging (TDI) was used to assess changes in ventricular function following repair of congenital heart defects. The relationship between TDI indices, myocardial injury and clinical outcome was explored. Forty-five children were studied; 35 withcardiac lesions and 10 controls. TDI was performed preoperatively, on admission to paediatric intensive care unit (PICU) and day 1. Regional myocardial Doppler signals were acquired from the right ventricle (RV), left ventricle (LV) and septum. TDI indices included: peak systolicvelocities, isovolumetric velocities (IVV) and isovolumetric acceleration (IVA). Preoperatively, bi-ventricular TDI velocities in the study groupwere reduced compared with normal controls. Postoperatively, RV velocities were significantly reduced and this persisted to day-1 (PreOp vs. PICU and day-1: 7.7+2.2 vs. 3.4+1.0, P < 0.0001 and 3.55+1.29, P < 0.0001). LV velocities initially declined but recovered towards baseline by day-1 (PreOp vs. PICU: 5.31+1.50 vs. 3.51+1.23, P < 0.0001). Isovolumetric parameters in all regions were reduced throughout the postoperative period. Troponin-I release correlated with longer X-clamp times (r=0.82, P < 0.0001) and reduced RV velocities (r=0.42, P=0.028). Reduced pre- and postoperative LV velocities correlated with longer ventilation (PreOp: r=0.54, P=0.002; PostOp: r=0.42, P=0.026). This study identified reduced postoperative RV velocities correlated with myocardial injury while reduced LV TDI correlated with longer postoperative ventilation