On the deprojection of triaxial galaxies with St\"ackel potentials

A family of triaxial St\"ackel potential-density pairs is introduced. With the help of a Quadratic Programming method, a linear combination of potential-density pairs of this family which fits a given projected density distribution can be built. This deprojection strategy can be used to model the potentials of triaxial elliptical galaxies with or without dark halos. Besides, we show that the expressions for the St\"ackel triaxial density and potential are considerably simplified when expressed in terms of divided differences, which are convenient numerically. We present an example of triaxial deprojection for the galaxy NGC~5128 whose photometry follows the de Vaucouleurs law.Comment: 8 pages, to appear in A&A, postscript file with figures available at ftp://naos.rug.ac.be/pub/deprojection.ps.

Probing the halo of Centaurus A: a merger dynamical model for the PN population

Photometry and kinematics of the giant elliptical galaxy NGC~5128 (Centaurus~A) based on planetary nebulae observations (Hui~\etal 1995) are used to build dynamical models which allow us to infer the presence of a dark matter halo. To this end, we apply a Quadratic Programming method. Constant mass-to-light ratio models fail to reproduce the major axis velocity dispersion measurements at large radii: the profile of this kind of models falls off too steeply when compared to the observations, clearly suggesting the necessity of including a dark component in the halo. By assuming a mass-to-light ratio which is increasing with radius, the model satisfactorily matches the observations. The total mass for the best fit model is $\sim4\times10^{11}M_\odot$ of which about 50\% is dark matter. However, models with different total masses and dark halos are also consistent with the data; we estimate that the total mass of Cen~A within 50~kpc may vary between $3\times10^{11}M_\odot$ and $5\times10^{11}M_\odot$. The best fit model consists of 75\% of stars rotating around the short axis $z$ and 25\% of stars rotating around the long axis $x$. Finally, the morphology of the projected velocity field is analyzed using Statler's classification criteria (Statler 1991). We find that the appearance of our velocity field is compatible with a type 'Nn' or 'Nd'.Comment: 13 pages, uuencoded compressed postscript, without figures. The full postscript version, including all 14 figures, is available via anonymous ftp at ftp://naos.rug.ac.be/pub/cena.ps.

Statistical Mechanics of the Uniform Electron Gas

In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.Comment: Final version to appear in J. Ec. polytech. Mat

Structure and enumeration of (3+1)-free posets

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to clarify the construction of skeleta and the enumeration. An extended abstract of this paper appears as arXiv:1212.535

The Boltzmann Equation in Classical Yang-Mills Theory

We give a detailed derivation of the Boltzmann equation, and in particular its collision integral, in classical field theory. We first carry this out in a scalar theory with both cubic and quartic interactions and subsequently in a Yang-Mills theory. Our method is not relied on a doubling of the fields, rather it is based on a diagrammatic approach representing the classical solution to the problem.Comment: 24 pages, 7 figures; v2: typos corrected, reference added, published in Eur. Phys. J.

Biological networks and epistasis in genome-wide association studies

Over the last few years, technological improvements have made possible the genotyping of hundreds of thousands of SNPs, enabling whole-genome association studies. The first genome-wide association studies have recently been completed to detect causal variant for complex traits. Although increasing evidence suggests that interaction between loci, such as epistasis between two loci, should be considered, most of these studies proceed by considering each SNP independently. One reason for this choice is that looking at all pairs of SNPs increases dramatically the number of tests (approximatively 50 billions of tests for a 300,000 SNPs data set) that faces with computational limitation and strong multiple testing correction.
We proposed to reduce the number of tests by focusing on pairs of SNPs that belong to genes known to interact in some metabolic network. Although some interactions might be missed, these pairs of genes are good candidates for epistasis. Furthermore the use of protein interaction databases (such as the STRING database) may reduce the number of tests by a factor of 5,000.
Results using this approach will be presented on simulated data sets and on public data sets.
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DNA methylation and DNA methyltransferases

The prevailing views as to the form, function, and regulation of genomic methylation patterns have their origin many years in the past, at a time when the structure of the mammalian genome was only dimly perceived, when the number of protein-encoding mammalian genes was believed to be at least five times greater than the actual number, and when it was not understood that only ~10% of the genome is under selective pressure and likely to have biological function. We use more recent findings from genome biology and whole-genome methylation profiling to provide a reappraisal of the shape of genomic methylation patterns and the nature of the changes that they undergo during gametogenesis and early development. We observe that the sequences that undergo deep changes in methylation status during early development are largely sequences without regulatory function. We also discuss recent findings that begin to explain the remarkable fidelity of maintenance methylation. Rather than a general overview of DNA methylation in mammals (which has been the subject of many reviews), we present a new analysis of the distribution of methylated CpG dinucleotides across the multiple sequence compartments that make up the mammalian genome, and we offer an updated interpretation of the nature of the changes in methylation patterns that occur in germ cells and early embryos. We discuss the cues that might designate specific sequences for demethylation or de novo methylation during development, and we summarize recent findings on mechanisms that maintain methylation patterns in mammalian genomes. We also describe the several human disorders, each very different from the other, that are caused by mutations in DNA methyltransferase genes

Strichartz inequality for orthonormal functions

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schr\"odinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.Comment: Final version to appear in the Journal of the European Mathematical Societ

A Collection of Problems on Spectrally Bounded Operators

We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights.Comment: 15 pages,; submitte