Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

Determination of dichlobenil and its major metabolite (BAM) in onions by PTV–GC–MS using PARAFAC2 and experimental design methodology

The optimization of a GC–MS analytical procedure which includes derivatization, Quick Easy Cheap Effective Rugged and Safe (QuEChERS) and programmed temperature vaporization (PTV) using design of experiments is performed to determine 2,6-dichlorobenzonitrile (dichlobenil) and 2,6-dichlorobenzamide (BAM) in onions, using 3,5-dichlorobenzonitrile and 2,4-dichlorobenzamide as internal standards. The use of a central composite design and two D-optimal designs, together with the desirability function, makes it possible to significantly reduce the economic, time and environmental cost of the study. The usefulness of PARAFAC2 for solving problems as the interference of unexpected derivatization artifacts unavoidably linked to some derivatization agents, or the presence of coeluents from the complex matrix, which share m/z ratios with the target compounds, is shown. The limits of decision (CCα) of the optimized procedure, 5.00 μg kg− 1 for dichlobenil and 1.55 μg kg− 1 for BAM (α = 0.05), are below the maximum residue limit (MRL) established by the EU for dichlobenil (20 μg kg− 1) in this commodity.Ministerio de Economía y Competitividad (CTQ2011-26022) and Junta de Castilla y León (BU108A11-2

Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope

We analyze the timing of photons observed by the MAGIC telescope during a flare of the active galactic nucleus Mkn 501 for a possible correlation with energy, as suggested by some models of quantum gravity (QG), which predict a vacuum refractive index \simeq 1 + (E/M_{QGn})^n, n = 1,2. Parametrizing the delay between gamma-rays of different energies as \Delta t =\pm\tau_l E or \Delta t =\pm\tau_q E^2, we find \tau_l=(0.030\pm0.012) s/GeV at the 2.5-sigma level, and \tau_q=(3.71\pm2.57)x10^{-6} s/GeV^2, respectively. We use these results to establish lower limits M_{QG1} > 0.21x10^{18} GeV and M_{QG2} > 0.26x10^{11} GeV at the 95% C.L. Monte Carlo studies confirm the MAGIC sensitivity to propagation effects at these levels. Thermal plasma effects in the source are negligible, but we cannot exclude the importance of some other source effect.Comment: 12 pages, 3 figures, Phys. Lett. B, reflects published versio

Anisotropy studies around the galactic centre at EeV energies with the Auger Observatory

Data from the Pierre Auger Observatory are analyzed to search for anisotropies near the direction of the Galactic Centre at EeV energies. The exposure of the surface array in this part of the sky is already significantly larger than that of the fore-runner experiments. Our results do not support previous findings of localized excesses in the AGASA and SUGAR data. We set an upper bound on a point-like flux of cosmic rays arriving from the Galactic Centre which excludes several scenarios predicting sources of EeV neutrons from Sagittarius $A$. Also the events detected simultaneously by the surface and fluorescence detectors (the `hybrid' data set), which have better pointing accuracy but are less numerous than those of the surface array alone, do not show any significant localized excess from this direction.Comment: Matches published versio

The VLT-FLAMES Tarantula Survey

Context. The origin of massive runaway stars is an important unsolved problem in astrophysics. Two main scenarios have been proposed, namely: dynamical ejection or release from a binary at the first core collapse. However, their relative contribution remains heavily debated. Aims. Taking advantage of two large spectroscopic campaigns towards massive stars in 30 Doradus, we aim to provide observational constraints on the properties of the O-type runaway population in the most massive active star-forming region in the Local Group. Methods. We used radial velocity measurements of the O-type star populations in 30 Doradus obtained by the VLT-FLAMES Tarantula Survey and the Tarantula Massive Binary Monitoring to identify single and binary O-type runaways. Here, we discuss the rotational properties of the detected runaways and qualitatively compare the observations with expectations of ejection scenarios. Results. We identified 23 single and one binary O-type runaway objects, most of them located outside the main star-forming regions in 30 Doradus. We find an overabundance of rapid rotators (ve sin i > 200 km s−1) among the runaway population, thus providing an explanation for the observed overabundance of rapidly rotating stars in the 30 Doradus field. Considerations of the projected rotation rates and runaway line-of-sight velocities reveal a conspicuous absence of rapidly rotating (ve sin i > 210 km s−1), fast-moving (vlos > 60 km s−1) runaway stars in our sample, strongly suggesting the presence of two different populations of runaway stars: a population of rapidly spinning but slowly moving runaway stars and a population of fast-moving but slowly rotating ones. These are detected with a ratio close to 2:1 in our sample. Conclusions. We argue that slowly moving but rapidly spinning runaway stars result from binary ejections, while rapidly moving but slowly spinning runaways could result from dynamical ejections. Given that detection biases will more strongly impact the slow-moving runaway population, our results suggest that the binary evolution scenario dominates the current massive runaway star population in 30 Doradus

Cluster Density and the IMF

Observed variations in the IMF are reviewed with an emphasis on environmental density. The remote field IMF studied in the LMC by several authors is clearly steeper than most cluster IMFs, which have slopes close to the Salpeter value. Local field regions of star formation, like Taurus, may have relatively steep IMFs too. Very dense and massive clusters, like super star clusters, could have flatter IMFs, or inner-truncated IMFs. We propose that these variations are the result of three distinct processes during star formation that affect the mass function in different ways depending on mass range. At solar to intermediate stellar masses, gas processes involving thermal pressure and supersonic turbulence determine the basic scale for stellar mass, starting with the observed pre-stellar condensations, and they define the mass function from several tenths to several solar masses. Brown dwarfs require extraordinarily high pressures for fragmentation from the gas, and presumably form inside the pre-stellar condensations during mutual collisions, secondary fragmentations, or in disks. High mass stars form in excess of the numbers expected from pure turbulent fragmentation as pre-stellar condensations coalesce and accrete with an enhanced gravitational cross section. Variations in the interaction rate, interaction strength, and accretion rate among the primary fragments formed by turbulence lead to variations in the relative proportions of brown dwarfs, solar to intermediate mass stars, and high mass stars.Comment: 14 pages, 3 figures, to be published in ``IMF@50: A Fest-Colloquium in honor of Edwin E. Salpeter,'' held at Abbazia di Spineto, Siena, Italy, May 16-20, 2004. Kluwer Academic Publishers; edited by E. Corbelli, F. Palla, and H. Zinnecke

Early life risk factors and their cumulative effects as predictors of overweight in Spanish children

Objectives: To explore early life risk factors of overweight/obesity at age 6 years and their cumulative effects on overweight/obesity at ages 2, 4 and 6 years. Methods: Altogether 1031 Spanish children were evaluated at birth and during a 6-year follow-up. Early life risk factors included: parental overweight/obesity, parental origin/ethnicity, maternal smoking during pregnancy, gestational weight gain, gestational age, birth weight, caesarean section, breastfeeding practices and rapid infant weight gain collected via hospital records. Cumulative effects were assessed by adding up those early risk factors that significantly increased the risk of overweight/obesity. We conducted binary logistic regression models. Results: Rapid infant weight gain (OR 2.29, 99% CI 1.54–3.42), maternal overweight/obesity (OR 1.93, 99% CI 1.27–2.92), paternal overweight/obesity (OR 2.17, 99% CI 1.44–3.28), Latin American/Roma origin (OR 3.20, 99% CI 1.60–6.39) and smoking during pregnancy (OR 1.61, 99% CI 1.01–2.59) remained significant after adjusting for confounders. A higher number of early life risk factors accumulated was associated with overweight/obesity at age 6 years but not at age 2 and 4 years. Conclusions: Rapid infant weight gain, parental overweight/obesity, maternal smoking and origin/ethnicity predict childhood overweight/obesity and present cumulative effects. Monitoring children with rapid weight gain and supporting a healthy parental weight are important for childhood obesity prevention