Slopes of vector bundles on projective curves and applications to tight closure problems

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous -primary ideal in a two-dimensional normal standard-graded algebra in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

     

Directional synthesis of a dye-linked conducting homopolymer

We report a new synthesis of a 10,20-diphenyl-5,15-bis(4-cyanophenyl)porphyrin (1) and its reduction to give 10,20-diphenyl-5,15-bis(4-formylphenyl)porphyrin (2). When the conducting polymer poly(2',5'-dioctyl-4,4' '-terphenylenecyanovinylene) was prepared in the presence of 2, we obtained a three-domain structure, compound 3, consisting of two homopolymer blocks (J domains) with the porphyrin dye molecule (P domain) in the middle. The JPJ assembly was subjected to photophysical studies where we showed that the J domain could serve as a light-harvesting antenna for the P domain.     

Synthetic studies and structural aspects of novel metallacyclic compoundsof titanium(IV) incorporating nitrogen,oxygen and sulphur: 1. Reactions of cis-dialkoxy-bis(acetylacetonato)titanium(IV) with alkoxyalkanols and the crystal structure of a new modification: di-μ-oxo-bis[diacetylacetonatotitanium(IV)]

Reactions of cis-dialkoxy-bis(acetylacetonato)titanium(IV), [(acac)₂Ti(OR)₂] (R = Et, Pr ⁱ ) with alkoxyalkanols (ROCH₂CH₂OH) (R = Me, Et, n-Bu) in 1:1 and 1:2 molar ratios in refluxing benzene under anhydrous conditions yield [(acac)₂Ti(OR)_2–n (OCH₂CH₂OR) _n ] (n = 1 or 2) complexes, which were purified by distillation under reduced pressure. On the basis of i.r. and n.m.r. (¹H- and ¹³C-) spectral studies, a cis-octahedral environment around Ti^IV is proposed. On keeping the distilled dark brown-red viscous liquid [(acac)₂Ti(OEt)(OCH₂CH₂OBu)] for 2 weeks, orange yellow crystals of [(acac)₂TiO]₂ were obtained. A single crystal X-ray diffraction study suggests the product is a new modification of [(acac)₂TiO]₂.     

The Os/Cu-Al-hydrotalcite catalysed hydroxylation of alkenes

A new Os/Cu-Al-hydrotalcite-like catalyst is described which, with N-methylmorpholine oxide as co-oxidant, heterogeneously catalyses the hydroxylation of olefins to give diols selectively and in high yield.     

? expansions in semiclassical theories for systems with smooth potentials and discrete symmetries

We extend a theory of first order ? corrections to Gutzwiller’s trace formula for systems with a smooth potential to systems with discrete symmetries and, as an example, apply the method to the two-dimensional hydrogen atom in a uniform magnetic field. We exploit the C_4v-symmetry of the system in the calculation of the correction terms. The numerical results for the semiclassical values will be compared with values extracted from exact quantum mechanical calculations. The comparison shows an excellent agreement and demonstrates the power of the ? expansion method.     

A statistical method for transforming temporal correlation functions from one-point measurements into longitudinal spatial and spatio-temporal correlation functions

The transformation of temporal, one-point correlation functions into longitudinal spatial and spatio-temporal correlation functions in turbulent flows using a simple statistical convection model is introduced. To illustrate and verify the procedure, experimental data (one-point and two-point) have been obtained with a laser Doppler system from a turbulent, round, free-air jet.     

Size and purity of gold nanoparticles changes with different types of thiolate ligands

Gold nanoparticles (Au-NPs) were prepared by a surfactant-free single-phase reduction of hydrogen tetrachloroaurate(III) hydrate in the presence of different organic thiol ligands. Sizes, size distributions, and crystallinity of the Au-NPs were determined by high-resolution transmission electron microscopy and powder X-ray diffraction, whereas thermogravimetric analysis provided information on the organic ligand-to-gold ratios as well as amounts of contaminants. A systematic decrease in size with increasing conical bulk of the thiolate ligand is observed but large size distributions and contamination of the generated Au-NPs prohibit detailed mechanistic studies. A first-generation Fréchet dendron thiol produced the smallest and cleanest Au-NPs of the narrowest size distribution.     

A vortex axis and vortex core border grid adaptation algorithm

In the design process of hydrodynamical and aerodynamical technical applications, the numerical simulation of massively separated vortical flow is crucial for predicting, for example, lift or drag. To obtain reliable numerical results, it is mandatory to accurately predict the physical behavior of vortices. Thus, the dominant vortical flow structures have to be resolved in detail, which requires a local grid refinement and certain adaptation techniques. In this paper, a vortex flow structure adaptation algorithm is presented, which is particularly designed for local grid refinement at vortex axes positions and associated vortex core border locations. To this end, a fast and efficient vortex axis detection scheme is introduced and the algorithm for the vortex core border determination is explained. As the interaction between vortices makes the assignment of grid points to a certain vortex axis difficult, a helicity‐based vortex distinction approach in combination with a geometrical rotational sensor is developed. After describing the combined different techniques in detail, the vortex feature adaptation algorithm is applied to analytical and more realistic examples, which show that the described grid adaptation algorithm is able to enhance the grid cell resolution locally such that all significant vortical flow phenomena are resolved. Copyright © 2008 John Wiley & Sons, Ltd.