Self-dual weak Hopf algebras

In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.     

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

We consider a singular perturbation of the generalized viscous Cahn–Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter ρ, which represents the density of atoms, and it is given on a n‐rectangle (n?3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter ε>0, as ε goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n=1, 2, then we also construct a family of inertial manifolds that is continuous with respect to ε. These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the ?ojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd.     

Division et extension dans l'algèbreA ∞ (Ω) d'un ouvert pseudo-convexe à bord lisse de ℂ n

Sans résumé     

A gravitational field in the Plebanski spacetime

The gravitational wave equation in the Plebanski spacetime has been reduced to a one-dimensional wave equation with a real short-range potential. We then find that the superradiance phenomenon occurs.     

Thermal transformations in soils studied by Mössbauer spectroscopy and x-ray diffraction

Mossbauer spectroscopy and X-ray diffraction techniques were used to investigate the iron oxides and other mineral phase developments in calcareous and noncalcareous soil samples after firing at ≈ 1000°C. Well crystalline haematite with larger particle size was produced in the fired non-calcareous soil. On the other hand, while goethite in the strongly calcareous natural material was transformed into haematite at the oven drying temperature, non of the latter was detected after firing at 1050°C. This could be related to the reaction between CaO, quartz and aluminosilicates at 800°C to produce calcium aluminosilicates which may have hindered the growth of haematite above this temperature.     

Two-dimensional percolation and classical string theory

Both two-dimensional percolation and classical string theory are shown to correspond toc=0 in the 2d conformal algebra. Therefore, despite being different theories, they have the same underlying conformal structure.     

Energy levels for the Coulomb potential with cut-off

Zusammenfassung Die Absorptionsbanden der Alkalihalogenidkristalle werden auf Elektronen zurückgeführt, die an leeren Gitterstellen eingefangen sind. Als Feld wird ein bei kleinen Distanzen horizontal abgeschnittenes Coulomb-Feld angesetzt. Die Eigenwerte werden im ganzen Energiebereich aus einer genauen Reihenentwicklung der Lösungen der Schrödinger-Gleichung bestimmt.