Einstein equations under polarized U(1)\mathbb U(1) symmetry in an elliptic gauge

We prove local existence of solutions to the Einstein--null dust system under polarized $\mathbb U(1)$ symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations

High-frequency backreaction for the Einstein equations under polarized U(1)\mathbb U(1) symmetry

Known examples in plane symmetry or Gowdy symmetry show that given a $1$-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a non-trivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb U(1)$ symmetry - a much weaker symmetry than most of the known examples - such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized-$\mathbb U(1)$-symmetric solution to the Einstein-multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large, and appears to be the first construction of such examples with more than two families

Autour des équations d’Einstein dans le vide avec un champ de Killing spatial de translation.

This thesis aim sat studying vacuum Einstein equations with a space-like Killing vector field. With this symmetry, 3+1 vacuum Einstein equations reduce, in the polarized case, to Einstein equations coupled to a scalar field in 2+ 1 dimensions. In the first part of this thesis, we study the constraint equations in the asymptotically flat case. The constraint equations correspond to computability conditions that the initial data must satisfy. We show the existence of solutions for small data, and we introduce an asymptotic expansion involving quantities which are the 2 dimensional equivalents for the global charges. In the second part, we show the stability of Minkowski space-time with a translation space-like Killing vector field in exponential time with respect to the smallness of initial data. We introduce a family of Ricci flat metrics, and we impose the asymptotic behaviour of our solutions in the exterior of the light cone by picking the right element in the family. This choice allows for the convergence to Minkowski solution in the interior of the light cone. In the last part of this thesis, we study the constraint equations in the compact hyperbolic case. We show the existence of a limit equation associated to the constraint equations.Dans cette thèse, nous étudions les équations d’Einstein dans le vide avec un champ de Killing de translation. En présence de cette symétrie, les équations d’Einstein dans le vide en dimension 3+1 peuvent s’écrire, dans le cas polarisé, comme un système d’équations d’Einstein couplées à un champ scalaire en dimension 2+1. Dans la première partie de cette thèse, nous étudions les équations de contraintes dans le cas asymptotiquement plat. Les équations de contraintes sont des équations de compatibilité qui doivent être satisfaites par les données initiales. Nous montrons l’existence de solutions pour des données assez petites, et introduisons un développement asymptotique faisant intervenir des quantités correspondant aux charges globales. Dans une deuxième partie nous montrons la stabilité de l’espace-temps de Minkowski avec un champ de Killing de translation, en temps exponentiellement grand par rapport à la petitesse de la donnée initiale. Nous travaillons dans les coordonnées d’onde généralisées. Nous introduisons une famille de métriques Ricci plates, et imposons le comportement asymptotique de nos solutions à l’extérieur du cône de lumière en choisissant un élément de cette famille de manière adéquate. Ce choix permet la convergence de nos solutions à l’intérieur du cône de lumière vers la solution de Minkowski. Dans la dernière partie de cette thèse nous étudions les équations de contraintes dans le cas compact hyperbolique. Nous montrons l’existence d’une équation limite associée aux équations de contraintes

The global stability of the Kaluza-Klein spacetime

In this paper we show the classical global stability of the flat Kaluza-Klein spacetime, which corresponds to Minkowski spacetime in \m R^{1+4} with one direction compactified on a circle. We consider small perturbations which are allowed to vary in all directions including the compact direction. These perturbations lead to the creation of massless modes and Klein-Gordon modes. On the analytic side, this leads to a PDE system coupling wave equations to an infinite sequence of Klein-Gordon equations with different masses. The techniques we use are based purely in physical space using the vectorfield method.Comment: 80 page

New insights into the origin of the B genome of hexaploid wheat: Evolutionary relationships at the SPA genomic region with the S genome of the diploid relative Aegilops speltoides

<p>Abstract</p> <p>Background</p> <p>Several studies suggested that the diploid ancestor of the B genome of tetraploid and hexaploid wheat species belongs to the <it>Sitopsis </it>section, having <it>Aegilops speltoides </it>(SS, 2n = 14) as the closest identified relative. However molecular relationships based on genomic sequence comparison, including both coding and non-coding DNA, have never been investigated. In an attempt to clarify these relationships, we compared, in this study, sequences of the Storage Protein Activator (SPA) locus region of the S genome of <it>Ae. speltoides </it>(2n = 14) to that of the A, B and D genomes co-resident in the hexaploid wheat species (<it>Triticum aestivum, AABBDD</it>, 2n = 42).</p> <p>Results</p> <p>Four BAC clones, spanning the SPA locus of respectively the A, B, D and S genomes, were isolated and sequenced. Orthologous genomic regions were identified as delimited by shared non-transposable elements and non-coding sequences surrounding the SPA gene and correspond to 35 268, 22 739, 43 397 and 53 919 bp for the A, B, D and S genomes, respectively. Sequence length discrepancies within and outside the SPA orthologous regions are the result of non-shared transposable elements (TE) insertions, all of which inserted after the progenitors of the four genomes divergence.</p> <p>Conclusion</p> <p>On the basis of conserved sequence length as well as identity of the shared non-TE regions and the SPA coding sequence, <it>Ae speltoides </it>appears to be more evolutionary related to the B genome of <it>T. aestivum </it>than the A and D genomes. However, the differential insertions of TEs, none of which are conserved between the two genomes led to the conclusion that the S genome of <it>Ae. speltoides </it>has diverged very early from the progenitor of the B genome which remains to be identified.</p

The Nature of the Dietary Protein Impacts the Tissue-to-Diet 15N Discrimination Factors in Laboratory Rats

Due to the existence of isotope effects on some metabolic pathways of amino acid and protein metabolism, animal tissues are 15N-enriched relative to their dietary nitrogen sources and this 15N enrichment varies among different tissues and metabolic pools. The magnitude of the tissue-to-diet discrimination (Δ15N) has also been shown to depend on dietary factors. Since dietary protein sources affect amino acid and protein metabolism, we hypothesized that they would impact this discrimination factor, with selective effects at the tissue level. To test this hypothesis, we investigated in rats the influence of a milk or soy protein-based diet on Δ15N in various nitrogen fractions (urea, protein and non-protein fractions) of blood and tissues, focusing on visceral tissues. Regardless of the diet, the different protein fractions of blood and tissues were generally 15N-enriched relative to their non-protein fraction and to the diet (Δ15N>0), with large variations in the Δ15N between tissue proteins. Δ15N values were markedly lower in tissue proteins of rats fed milk proteins compared to those fed soy proteins, in all sampled tissues except in the intestine, and the amplitude of Δ15N differences between diets differed between tissues. Both between-tissue and between-diet Δ15N differences are probably related to modulations of the relative orientation of dietary and endogenous amino acids in the different metabolic pathways. More specifically, the smaller Δ15N values observed in tissue proteins with milk than soy dietary protein may be due to a slightly more direct channeling of dietary amino acids for tissue protein renewal and to a lower recycling of amino acids through fractionating pathways. In conclusion, the present data indicate that natural Δ15N of tissue are sensitive markers of the specific subtle regional modifications of the protein and amino acid metabolism induced by the protein dietary source

Stability in exponential time of Minkowski space-time with a space-like translation symmetry

International audienc