Subextension of plurisubharmonic functions with weak singularities

We prove several results showing that plurisubharmonic functions with various bounds on their Monge-Ampere masses on a bounded hyperconvex domain always admit global plurisubharmonic subextension with logarithmic growth at infinity

Birth Order and BMI in Teenage Girls

The goal of this study was to investigate the relation of birth order to relative weight and prevalence of obesity in a group of 13–15 years old girls. In 1997, 1458 girls were examined. The height and weight measured by trained staff were recorded. Family size and birth order were obtained by a questionnaire. For the purpose of the present study, 776 and 250 girls coming from two- and three-child full families, respectively, were selected from the total sample on the basis of complete information. The Body Mass Index (kg/m2) was adjusted to reference US population (NCHS) by means of the LMS parameters. Prevalence of overweight and obesity was defined according to recommendation of the International Obesity Task Force. The effect of birth order on BMI was tested by one-way analysis of variance. Prevalence of obesity was tested by the means of Pearson chi-square. First and second born girls from two-sibling families did not show significant differences in average standardized BMI. Relative weight significantly differs among girls coming from three sibling families, decreasing along with the birth order. The first-born girls were 1.5 times at higher risk of obesity in comparison to later- born girls. Differences in the proportion of overweight girls among birth order groups showed a high significance within three sibling families

Weighted pluricomplex energy

We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\mathcal{E}_\chi (\Omega)$ in the general case ($\chi(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(dd^c \cdot)^n$ on the classes $\mathcal{E}\chi(\Omega).$Comment: Contrary to what we claimed in the previous version, in Theorem 5.1 we generalize some Theorem of Urban Cegrell but we do not give a new proof. To appear in Potenial Analysi

The frequency map for billiards inside ellipsoids

The billiard motion inside an ellipsoid Q \subset \Rset^{n+1} is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Besides, any billiard trajectory inside $Q$ is tangent to $n$ caustics $Q_{\lambda_1},...,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,...,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda \mapsto \omega$ is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes "exponentially sharp" at some of them. Finally, we study triaxial ellipsoids of \Rset^3. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of \Rset^2.Comment: 50 pages, 13 figure

Hyperbolic outer billiards : a first example

We present the first example of a hyperbolic outer billiard. More precisely we construct a one parameter family of examples which in some sense correspond to the Bunimovich billiards.Comment: 11 pages, 8 figures, to appear in Nonlinearit

The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized K\"ahler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for K\"aher surfaces with a numerical effective canonical line bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page

Aerothermal Performance Constraints for Small Radius Leading Edges Operating at Hypervelocity

Small radius leading edges and nosetips were used to minimize wave drag in early hypervelocity vehicle concepts until further analysis demonstrated that extreme aerothermodynamic heating blunted the available thermal protection system materials. Recent studies indicate that ultra-high temperature composite (UHTC) materials are shape stable at temperatures approaching 3033 K and will be available for use as sharp leading edge components in the near future. Steady-state aerothermal performance constraints for UHTC components are presented in this paper to identify their non-ablating operational capability at altitudes from sea level to 90 km. An integrated design tool was developed to estimate these constraints. The tool couples aerothermodynamic heating with material response using commercial finite element analysis software and is capable of both steady-state and transient analysis. Performance during entry is analyzed by transient thermal analysis along the trajectory. The thermal load condition from the transient thermal analysis is used to estimate thermal stress. Applying the tool to UHTC materials shows that steady-state, non-ablating operation of a HfB2/SiC(A-7) (A-7) component is possible at velocities approaching Earth's circular orbital velocity of 7.9 km/s at altitudes approaching 70 km