Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

Feynman path integrals and the trace formula for the Schrödinger operators

We study Schrödinger operators of the form on ^d , whereA ² is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If _n are the eigenvalues ofH we show that the theta function is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of (t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt Unendlich dimensionale Analysis)     

Derivatives and reactions of glutacondialdehyde V : The crystal and molecular structure of 5-hydroxy-trans-2, trans-4-pentadienal acetate. The charge distribution in the glutacondialdehyde anion

The structure of 5-hydroxy-trans-2, trans-4-pentadienal acetate has been determined, using three-dimensional diffractometercollected X-ray data. The compound has the all-trans configuration with the atoms C-1 to C-5 in a plane. The charge distribution in the acetate and in the glutacondialdehyde anion have been calculated using the CNDO/2 approximation. In both compounds higher negative charges were found on carbon atoms C-2 and C-4 than on C-1, C-3 and C-5.     

Analysis of impact-excited HeI states by applying magnetic and electric depolarization techniques

The magnetic depolarization of the impact radiation at λ(4¹ D?2¹ P)=492 nm and λ(3¹ D?2¹ P) = 668 nm as well as the splitting of the magnetic depolarisation signals in electric fields were investigated for 10–25 keV H ₂ ⁺ (D ₂ ⁺ )-He collisions (only 492 nm line) and for 5-22.5 keV He-He collisions (both lines). The results are compared with analogous measurements for He⁺- He and Ne⁺ - He collisions and the dynamics of the excitation process are qualitatively discussed. In particular, we emphasize the importance of inertia for the evolution of the electron cloud during the final phase of the collision process.     

The real-pole sandwich for rational approximations and oscillation equations

Rational approximations to the exponential function with real poles only are studied with respect to stability at infinity and maximal order. Along each half line in the parameter space it is shown that these two properties occur in an alternating way (or do not occur at all). As an application of the general results the special approximations withq-fold poles at ± ^–1 only, real, are studied. A short proof of the superconvergence for collocation methods is also given.Dedicated to W. Gröbner on the occasion of his 80th birthday.     

Linear low density polyethylene: Microstructure and sealing properties correlation

The microstructures of four commercial linear low density polyethylenes (LLDPE) were evaluated and correlated with their sealing properties. Atomic Force Microscopy (AFM), Temperature Rising Elution Fractionation (TREF), Differential Scanning Calorimetry (DSC) and Crystallization Analysis Fractionation (CRYSTAF) experiments revealed that the comonomer distribution is one of the main factors that influence the sealing properties. The superior sealing performance showed by metallocene LLDPE samples in comparison to Ziegler-Natta LLDPE samples can be attributed to their well balanced chemical composition distribution.     

Stereocontrolled Organocatalytic Strategy for the Synthesis of Optically Active 2,3‐Disubstituted cis‐2,3‐Dihydrobenzofurans

An intramolecular, organocatalyzed Michael addition has been developed to obtain biologically important 2,3‐disubstituted cis‐2,3‐dihydrobenzofurans. By using mandelic acid salts of primary aminocatalysts, derived from cinchona alkaloids, the intramolecular cyclization reaction has been developed to proceed in high yield, with moderate to good diastereoselectivity, and up to 99 % ee. Based on the absolute configuration of the formed 2,3‐disubstituted‐cis‐2,3‐dihydrobenzofurans and by considering the observed substrate scope restrictions, a mechanistic rationalization has been presented.     

Simple,Safe, Large Scale Synthesis of 5-Arylmethyl-2,2-dimethyl-1,3-dioxane-4,6-diones and 3-Arylpropanoic Acids

Reaction of aryl aldehydes with Meldrum's acid 2 in the presence of formic acid and triethylamine gives 5-arylmethyl Meldrum's acid derivatives 4 at room temperature, whereas at 80–100°C 3-arylpropanoic acids 5 are formed.     

The Synthesis of 17-Ester of Corticosteroids Protection of 11β-Hydroxyl by the Trimethylsilyl Group

17–Esters of corticosteroids have received considerable attention due to their high topical anti-inflammatory activity. one of the best known is betamethasone 17-valerate which is widely used against a variety of skin diseases². The relatively simple compound, hydrocortisone 17–butyrate (3a)³ proved to be as active topically as the more sophisticated corticosteroids such as triamcinolone acetonide, while its systemic activity is of the same order as that of hydrocortisone⁴.