The shadow of a collapsing dark star

The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it has existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static observer who is watching the collapse from a certain distance and then for an observer who is falling towards the centre following the collapsing star.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Evaluation of Morphology and Deformation Micromechanisms in Multilayered PP/PS Films: An Electron Microscopy Study

Nanostructured polymers and ultra-thin polymer layers are used more and more in technical applications like nanotechnology and microelectronics. Therefore, it is really important to understand the size-scale dependent properties as bulk polymers become thinner and more two-dimensional. Here the morphology as well as the macroscopic and the microscopic deformation behaviour of multilayered films of polypropylene (PP) and polystyrene (PS) have been investigated. For investigation different microscopic techniques and tensile testing were used. The films were prepared by multilayer coextrusion, whereas the composition of PP and PS and the film thickness – and therefore the thickness of each layer – varied. The thinnest calculated thickness of a single layer was about 5 nm. It is shown that the PP/PS films consist usually of homogeneous layers with only few defects. As the composition of PP/PS deviates strongly from 50/50 or the films get thinner the number of defects increases and the layered system turns to irregular lamellar system. In macroscopic tensile tests the small PS layers affect the elongation at break enormously: Most of the samples are brittle. For the films with a composition of PP/PS 90/10 and the film PP/PS 70/30 with a film thickness of 25 µm an elongation at break of 66% and higher could be reached. Transmission electron microscopy on these samples shows that the layers are characterized by plastic yielding in local deformation zones.     

K4[PbSe4]⋅en⋅NH3: A Non‐Oxide,Non‐Halide Inorganic Lead(IV) Compound

The first inorganic lead(IV) compound without oxygen, nitrogen or halogen ligands attached to the lead atom was obtained as the potassium salt of the tetraselenidoplumbate(IV) anion [Pb^IVSe₄]^4?. It is stable under inert conditions which may enable the transfer of the chemistry of chalcogenidogermanate(IV) or chalcogenidostannate(IV) materials, to the lead homologues.     

Synthesis, characterization, and substitution chemistry of [Bu4N]2[W6Cl8(OSO2CF3)6]. A versatile precursor for axially substituted clusters containing the (W6Cl8)4+ core

The new cluster [Bu4N]2[W6Cl8(OSO2CF3)6] (1) has been prepared and structurally characterized. This material is an effective precursor for the generation of cluster ions with the general formula [W6C18L6]n (L = Cl-, Br-, I-, NCS-, NCO-, NCSe-, and O=PPh3; n = 2- or 4+). The last three clusters are new. The products have been characterized by IR spectroscopy, NMR spectroscopy, and FAB mass spectrometry. In addition to 1, the products [Bu4N]2[W6C18(NCS)6] (5) and [Bu4N]2[W6C18(NCO)6] (7) were structurally characterized. Crystal data for 1: space group, P2(1/c) (No. 14); a = 11.116(5) A; b = 27.952(1) A; c = 24.516(1) A; beta = 95.182(9) degrees; V = 7586.3(5) A3; Z = 4. Crystal data for 5: space group, P2(1/n) (No. 14); a = 11.3323(9) A; b = 12.3404(9) A; c = 44.583(3) A; beta = 97.089(1) degrees ; V = 6187.1(7) A3; Z = 4. Crystal data for 7: space group, P1 (No. 2); a = 11.8009(8) A; b = 11.9332(8) A; c = 11.9522(8) A; alpha = 77.904(1) degrees; beta = 95.182(9) degrees; gamma = 62.574(1) degrees V = 1450.5(2) A3; Z = 1.     

A heterometallic, heterovalent Cu(I)/Sn(II/IV)/S cluster with an unprecedented Cu4Sn core and stannacyclopentane units

[(Ph3PCu(I))6{(CH2)4Sn(IV)S2}6Cu(I)4Sn(II)], a walnut-type, heterometallic, heterovalent cluster, as confirmed by density functional theory calculations, was obtained under reductive solvothermal conditions and features an unprecedented metalloid core within a metal sulfide shell and an organic surface.     

The iromycins, a new family of pyridone metabolites from Streptomyces sp. I. Structure, NOS inhibitory activity, and biosynthesis

The iromycins A-D are members of a new family of rare alpha-pyridone metabolites. The isolation and structure elucidation of these microbial secondary metabolites from Streptomyces sp. Dra 17 revealed a N-heterocyclic core structure with two unusual side chains. Iromycins act as inhibitors of nitric oxide synthases (NOS), a protein family, which produces the crucial second messenger nitric oxide (NO). Importantly, these compounds inhibit selectively endothelial NOS rather than neuronal NOS and thus set prospects for both medical therapy and basic research. Feeding experiments with 13C- and 15N -labeled precursors indicated an uncommon type of polyketide biosynthesis and clearly ruled out an isoprenoid origin. A detoxification pathway of a particular secondary metabolite in the host strain is a rare observation and here we demonstrate it with the iromycin family.     

Very Facile Polarity Umpolung and Noncovalent Functionalization of Inorganic Nanoparticles: A Tool Kit for Supramolecular Materials Chemistry

The facile assembly of shell‐by‐shell (SbS)‐coated nanoparticles [TiO₂?PAC₁₆]@shell 1 – 7 (PAC₁₆=hexadecylphosphonic acid), which are soluble in water and can be isolated as stable solids, is reported. In these functional architectures, an umpolung of dispersibility (organic apolar versus water) was accomplished by the noncovalent binding of ligands 1 – 7 to titania nanoparticles [TiO₂?PAC₁₆] containing a first covalent coating with PAC₁₆. Ligands 1 – 7 are amphiphilic and form the outer second shell of [TiO₂?PAC₁₆]@shell 1 – 7 . The tailor‐designed dendritic building blocks 3 – 5 contain negative and positive charges in the same molecule, and ligands 6 and 7 contain a perylenetetracarboxylic acid dimide (PDI) core ( 6 / 7 ) as a photoactive reporter component. In the redox and photoactive system [TiO₂?PAC₁₆]@shell 7 , electronic communication between the inorganic core to the PDI ligands was observed.