Interpolation estimate for a finite-element space with embedded discontinuities

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming Formula space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in Formula, the interpolant Formula defined by this new space satisfies Graphic where h is the mesh size, Formula is the domain, Formula, Formula, Formula and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across Γ, which would yield an error in the Formula-norm of order Graphic. These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.

Transiting exoplanets from the CoRoT space mission XX. CoRoT-20b: A very high density, high eccentricity transiting giant planet

We report the discovery by the CoRoT space mission of a new giant planet, CoRoT-20b. The planet has a mass of 4.24 +/- 0.23 M-Jup and a radius of 0.84 +/- 0.04 R-Jup. With a mean density of 8.87 +/- 1.10 g cm(-3), it is among the most compact planets known so far. Evolutionary models for the planet suggest a mass of heavy elements of the order of 800 M-circle plus if embedded in a central core, requiring a revision either of the planet formation models or both planet evolution and structure models. We note however that smaller amounts of heavy elements are expected by more realistic models in which they are mixed throughout the envelope. The planet orbits a G-type star with an orbital period of 9.24 days and an eccentricity of 0.56. The star's projected rotational velocity is v sin i = 4.5 +/- 1.0 km s(-1), corresponding to a spin period of 11.5 +/- 3.1 days if its axis of rotation is perpendicular to the orbital plane. In the framework of Darwinian theories and neglecting stellar magnetic breaking, we calculate the tidal evolution of the system and show that CoRoT-20b is presently one of the very few Darwin-stable planets that is evolving toward a triple synchronous state with equality of the orbital, planetary and stellar spin periods.

Structure and luminescent investigation of the Ln(III)-beta-diketonate complexes containing tertiary amides

The synthesis and photoluminescent properties of Ln(III)-thenoyltrifluoroacetonate and dibenzoylmethanate complexes (Ln = Eu(III) and Gd(III) ions) containing tertiary amides such as dimethylacetamide (DMA), dimethylformamide (DMF), and dimethylbenzamide (DMB) as neutral ligands are reported. The Ln complexes were characterized by elemental analysis, complexometric titration with EDTA, and infrared spectroscopy. Single-crystal X-ray structure data of the [Eu(DBM)(3).(DMA)] compound indicates that this complex crystallizes in the triclinic system, space group PT with the following cell parameters: a = 10.2580(3) angstrom, b = 10.3843(2) angstrom, c= 22.3517(5) angstrom, alpha = 78.906(2)degrees, beta = 78.049(2)degrees, lambda= 63.239(2)degrees, V= 2066.41(9) angstrom(3), and Z = 2. The coordination polyhedron for the Eu(III) complex may be described as an approximate C-2v distorted monocapped trigonal prism. The optical properties of the Eu(III) complexes were studied based on the intensity parameters and luminescence quantum yield (q). The values of the ohm(2) parameter of the Eu-DBM complexes are larger than those for the Eu-TTA complexes, indicating that the Eu(III) ion is in a more polarizable chemical environment in the former case. The geometries of the complexes have been optimized by using the Sparkle Model, and the results have been used to perform theoretical predictions of the ligand-to-metal energy transfer via direct and exchange Coulomb mechanisms. (C) 2012 Elsevier Ltd. All rights reserved.