TIC: A Stokes inversion code for scattering polarization with partial frequency redistribution and arbitrary magnetic fields

We present the Tenerife Inversion Code (TIC), which has been developed to infer the magnetic and plasma properties of the solar chromosphere and transition region via full-Stokes inversion of polarized spectral lines. The code is based on the HanleRT forward engine, which takes into account many of the physical mechanisms that are critical for a proper modeling of the Stokes profiles of spectral lines originating in the tenuous and highly dynamic plasmas of the chromosphere and transition region: quantum level population imbalance and interference (atomic polarization), frequency coherence effects in polarized resonance scattering (partial frequency redistribution), and the impact of arbitrary magnetic fields on the atomic polarization and the radiation field. We present first results of atmospheric and magnetic inversions, and discuss future developments for the project.Comment: 17pages, 7 figures. Accepted for publication in The Astrophysical Journa

Stationary solutions of the nonlinear Schr\"odinger equation with fast-decay potentials concentrating around local maxima

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$},$$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential $V$ without any restriction on the potential.Comment: 25 pages, reformatted the abstract for MathJa

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena

The patterns of population differentiation in a Brassica rapa core collection

With the recent advances in high throughput profiling techniques the amount of genetic and phenotypic data available has increased dramatically. Although many genetic diversity studies combine morphological and genetic data, metabolite profiling has yet to be integrated into these studies. For our study we selected 168 accessions representing the different morphotypes and geographic origins of Brassica rapa. Metabolite profiling was performed on all plants of this collection in the youngest expanded leaves, 5 weeks after transplanting and the same material was used for molecular marker profiling. During the same season a year later, 26 morphological characteristics were measured on plants that had been vernalized in the seedling stage. The number of groups and composition following a hierarchical clustering with molecular markers was highly correlated to the groups based on morphological traits (r = 0.420) and metabolic profiles (r = 0.476). To reveal the admixture levels in B. rapa, comparison with the results of the programme STRUCTURE was needed to obtain information on population substructure. To analyze 5546 metabolite (LC–MS) signals the groups identified with STRUCTURE were used for random forests classification. When comparing the random forests and STRUCTURE membership probabilities 86% of the accessions were allocated into the same subgroup. Our findings indicate that if extensive phenotypic data (metabolites) are available, classification based on this type of data is very comparable to genetic classification. These multivariate types of data and methodological approaches are valuable for the selection of accessions to study the genetics of selected traits and for genetic improvement programs, and additionally provide information on the evolution of the different morphotypes in B. rapa. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s00122-010-1516-1) contains supplementary material, which is available to authorized users

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

Comparative Methods for Association Studies: A Case Study on Metabolite Variation in a Brassica rapa Core Collection

Background Association mapping is a statistical approach combining phenotypic traits and genetic diversity in natural populations with the goal of correlating the variation present at phenotypic and allelic levels. It is essential to separate the true effect of genetic variation from other confounding factors, such as adaptation to different uses and geographical locations. The rapid availability of large datasets makes it necessary to explore statistical methods that can be computationally less intensive and more flexible for data exploration. Methodology/Principal Findings A core collection of 168 Brassica rapa accessions of different morphotypes and origins was explored to find genetic association between markers and metabolites: tocopherols, carotenoids, chlorophylls and folate. A widely used linear model with modifications to account for population structure and kinship was followed for association mapping. In addition, a machine learning algorithm called Random Forest (RF) was used as a comparison. Comparison of results across methods resulted in the selection of a set of significant markers as promising candidates for further work. This set of markers associated to the metabolites can potentially be applied for the selection of genotypes with elevated levels of these metabolites. Conclusions/Significance The incorporation of the kinship correction into the association model did not reduce the number of significantly associated markers. However incorporation of the STRUCTURE correction (Q matrix) in the linear regression model greatly reduced the number of significantly associated markers. Additionally, our results demonstrate that RF is an interesting complementary method with added value in association studies in plants, which is illustrated by the overlap in markers identified using RF and a linear mixed model with correction for kinship and population structure. Several markers that were selected in RF and in the models with correction for kinship, but not for population structure, were also identified as QTLs in two bi-parental DH populations

Gradient flows and instantons at a Lifshitz point

I provide a broad framework to embed gradient flow equations in non-relativistic field theory models that exhibit anisotropic scaling. The prime example is the heat equation arising from a Lifshitz scalar field theory; other examples include the Allen-Cahn equation that models the evolution of phase boundaries. Then, I review recent results reported in arXiv:1002.0062 describing instantons of Horava-Lifshitz gravity as eternal solutions of certain geometric flow equations on 3-manifolds. These instanton solutions are in general chiral when the anisotropic scaling exponent is z=3. Some general connections with the Onsager-Machlup theory of non-equilibrium processes are also briefly discussed in this context. Thus, theories of Lifshitz type in d+1 dimensions can be used as off-shell toy models for dynamical vacuum selection of relativistic field theories in d dimensions.Comment: 19 pages, 1 figure, contribution to conference proceedings (NEB14); minor typos corrected in v

On the structure of phase transition maps for three or more coexisting phases

This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.Comment: 23 pages, 6 figure