A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page

On completeness of orbits of Killing vector fields

A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space--times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold. This result gives a significant strengthening of the uniqueness theorems for black holes.Comment: 16 pages, Latex, preprint NSF-ITP-93-4

Singular kernels, multiscale decomposition of microstructure, and dislocation models

We consider a model for dislocations in crystals introduced by Koslowski, Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $\Gamma$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound

Analysis of Nematic Liquid Crystals with Disclination Lines

We investigate the structure of nematic liquid crystal thin films described by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the elasticity constant goes to zero a limiting uniaxial texture forms with disclination lines corresponding to a finite number of defects, all of degree 1/2 or all of degree -1/2. We also state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs model without magnetic field that follows from a similar proof.Comment: 40 pages, 1 figur

Concentration analysis and cocompactness

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions

Earthworm activity and availability for meadow birds is restricted in intensively managed grasslands

Earthworms are an important prey for the endangered meadow birds of northwest Europe. Although intensive grassland management with high manure inputs generally promotes earthworm abundance, it may reduce the effective food availability for meadow birds through desiccation of the topsoil, which causes earthworms to remain deeper in the soil. We studied the response of Red Worm Lumbricus rubellus, a detritivore, and Grey Worm Aporrectodea caliginosa, a geophage, to soil moisture profiles in the field and under experimental conditions. Surfacing earthworms were counted weekly in eight intensively managed grasslands (treated with high inputs of slurry by slit injection) with variable groundwater tables in the Netherlands. At each count, soil penetration resistance, soil moisture tension and groundwater level were measured, while air temperature and humidity were obtained from a nearby weather station. The response to variation in the vertical distribution of soil moisture was also experimentally studied in the two earthworm species. In the field, earthworms’ surfacing activity at night was negatively associated with soil moisture tension and positively by relative air humidity. Surprisingly, there was no effect of groundwater level; an important management variable in meadow bird conservation. Under experimental conditions, both L. rubellus and A. caliginosa moved to deeper soil layers (&gt;20 cm) in drier soil moisture treatments, avoiding the upper layer when moisture levels dropped below 30%. Synthesis and applications. We propose that in intensively managed grasslands with slurry application, topsoil desiccation reduces earthworm availability for meadow birds. This can be counteracted by keeping soil moisture tensions of the top soil above −15 kPa. We suggest that the late raising of groundwater tables in spring and the disturbance of the soil by slit injection of slurry increase topsoil desiccation. This decreases earthworm availability when it matters most for breeding meadow birds. Meadow bird conservation will benefit from revised manure application strategies that promote earthworm activity near or at the soil surface.</p

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

Codon Size Reduction as the Origin of the Triplet Genetic Code

The genetic code appears to be optimized in its robustness to missense errors and frameshift errors. In addition, the genetic code is near-optimal in terms of its ability to carry information in addition to the sequences of encoded proteins. As evolution has no foresight, optimality of the modern genetic code suggests that it evolved from less optimal code variants. The length of codons in the genetic code is also optimal, as three is the minimal nucleotide combination that can encode the twenty standard amino acids. The apparent impossibility of transitions between codon sizes in a discontinuous manner during evolution has resulted in an unbending view that the genetic code was always triplet. Yet, recent experimental evidence on quadruplet decoding, as well as the discovery of organisms with ambiguous and dual decoding, suggest that the possibility of the evolution of triplet decoding from living systems with non-triplet decoding merits reconsideration and further exploration. To explore this possibility we designed a mathematical model of the evolution of primitive digital coding systems which can decode nucleotide sequences into protein sequences. These coding systems can evolve their nucleotide sequences via genetic events of Darwinian evolution, such as point-mutations. The replication rates of such coding systems depend on the accuracy of the generated protein sequences. Computer simulations based on our model show that decoding systems with codons of length greater than three spontaneously evolve into predominantly triplet decoding systems. Our findings suggest a plausible scenario for the evolution of the triplet genetic code in a continuous manner. This scenario suggests an explanation of how protein synthesis could be accomplished by means of long RNA-RNA interactions prior to the emergence of the complex decoding machinery, such as the ribosome, that is required for stabilization and discrimination of otherwise weak triplet codon-anticodon interactions

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

The Jumonji-C oxygenase JMJD7 catalyzes (3S)-lysyl hydroxylation of TRAFAC GTPases

Biochemical, structural and cellular studies reveal Jumonji-C (JmjC) domain-containing 7 (JMJD7) to be a 2-oxoglutarate (2OG)-dependent oxygenase that catalyzes (3S)-lysyl hydroxylation. Crystallographic analyses reveal JMJD7 to be more closely related to the JmjC hydroxylases than to the JmjC demethylases. Biophysical and mutation studies show that JMJD7 has a unique dimerization mode, with interactions between monomers involving both N- and C-terminal regions and disulfide bond formation. A proteomic approach identifies two related members of the translation factor (TRAFAC) family of GTPases, developmentally regulated GTP-binding proteins 1 and 2 (DRG1/2), as activity-dependent JMJD7 interactors. Mass spectrometric analyses demonstrate that JMJD7 catalyzes Fe(ii)- and 2OG-dependent hydroxylation of a highly conserved lysine residue in DRG1/2; amino-acid analyses reveal that JMJD7 catalyzes (3S)-lysyl hydroxylation. The functional assignment of JMJD7 will enable future studies to define the role of DRG hydroxylation in cell growth and disease.Fil: Markolovic, Suzana. University of Oxford; Reino UnidoFil: Zhuang, Qinqin. University Of Birmingham; Reino UnidoFil: Wilkins, Sarah E.. University of Oxford; Reino UnidoFil: Eaton, Charlotte D.. University Of Birmingham; Reino UnidoFil: Abboud, Martine I.. University of Oxford; Reino UnidoFil: Katz, Maximiliano Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigaciones Bioquímicas de Buenos Aires. Fundación Instituto Leloir. Instituto de Investigaciones Bioquímicas de Buenos Aires; ArgentinaFil: McNeil, Helen E.. University Of Birmingham; Reino UnidoFil: Leśniak, Robert K.. University of Oxford; Reino UnidoFil: Hall, Charlotte. University Of Birmingham; Reino UnidoFil: Struwe, Weston B.. University of Oxford; Reino UnidoFil: Konietzny, Rebecca. University of Oxford; Reino UnidoFil: Davis, Simon. University of Oxford; Reino UnidoFil: Yang, Ming. The Francis Crick Institute; Reino Unido. University of Oxford; Reino UnidoFil: Ge, Wei. University of Oxford; Reino UnidoFil: Benesch, Justin L. P.. University of Oxford; Reino UnidoFil: Kessler, Benedikt M.. University of Oxford; Reino UnidoFil: Ratcliffe, Peter J.. University of Oxford; Reino Unido. The Francis Crick Institute; Reino UnidoFil: Cockman, Matthew E.. The Francis Crick Institute; Reino Unido. University of Oxford; Reino UnidoFil: Fischer, Roman. University of Oxford; Reino UnidoFil: Wappner, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigaciones Bioquímicas de Buenos Aires. Fundación Instituto Leloir. Instituto de Investigaciones Bioquímicas de Buenos Aires; ArgentinaFil: Chowdhury, Rasheduzzaman. University of Stanford; Estados Unidos. University of Oxford; Reino UnidoFil: Coleman, Mathew L.. University Of Birmingham; Reino UnidoFil: Schofield, Christopher J.. University of Oxford; Reino Unid