The conformal ‘bratwurst’ mapsand associated Faber polynomials

Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of ‘bratwurst’ shape sets in the complex plane are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A ‘bratwurst’ shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis onto the unit circle. Such sets are well suited to serve as inclusion sets for sets associated with a matrix, for example the spectrum, field of values or a pseudospectrum. Hence, the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems. The main advantage of the approach is that the conformal maps are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic convergence factor and Faber polynomials. Numerical examples are given. Received October 7, 1998 / Revised version received March 15, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for d‐variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d‐dimensional torus or the d‐dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci‐lattice for bivariate periodic functions from in the case , and . A non‐periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst‐case error.     

A portable laser system for high-precision atom interferometry experiments

We present a modular rack-mounted laser system for the cooling and manipulation of neutral rubidium atoms which has been developed for a portable gravimeter based on atom interferometry that will be capable of performing high-precision gravity measurements directly at sites of geophysical interest. This laser system is constructed in a compact and mobile design so that it can be transported to different locations, yet it still offers improvements over many conventional laboratory-based laser systems. Our system is contained in a standard 19″ rack and emits light at five different frequencies simultaneously on up to 12 fibre ports at a total output power of 800 mW. These frequencies can be changed and switched between ports in less than a microsecond. The setup includes two phase-locked diode lasers with a phase noise spectral density of less than 1 μrad/Hz^1/2 in the frequency range in which our gravimeter is most sensitive to noise. We characterise this laser system and evaluate the performance limits it imposes on an interferometer.     

Stereochemistry of the Reaction Intermediates of Prolinol Ether Catalyzed Reactions Characterized by Vibrational Circular Dichroism Spectroscopy

Spectroscopic characterizations of key reaction intermediates are often considered the final confirmation of a reaction mechanism. This proof-of-principle study showcases the application of vibrational circular dichroism (VCD) spectroscopy for the characterization of in situ generated reaction intermediates using the key intermediates of enamine catalysis of Jørgensen–Hayashi-type prolinol ether catalysts as model system. By comparison of experimental and computed spectra, the enamines are shown to preferentially adopt an anti-conformation with E-configured C=C bond. For the parent prolinol catalyst, the structure and stereochemistry of the oxazolidine side product is determined as well. This study thus demonstrates that VCD spectra can provide insights into structural preferences of organocatalysts that utilize a covalent activation mechanism. Thereby it outlines new fields of applications for VCD spectroscopy and finally adds the technique to the toolbox of physical organic chemistry for in-depth mechanistic studies.     

Ultrasonic attenuation spectroscopy of emulsions with droplet sizes greater than 10 microm

Ultrasonic attenuation measurement is a frequently used tool for non-destructive determination of dispersion characteristics. Useful information like particle or droplet size and their concentration can be obtained, if the relation between size and attenuation of the dispersion is known. In this work, the theoretical model by Faran for the intermediate sound wave regime (IWR) is presented in combination with experimental data. In the IWR, the acoustic behavior is governed by elastic scattering rather than by dissipative effects. Experiments with emulsion of droplet sizes greater than 10 mum were carried out. Silicone oil, sunflower oil and olive oil were selected for the disperse phase of the oil-in-water emulsions. First, emulsions having droplets in the micrometer range were created. Afterwords, attenuation measurements of different concentrated emulsion were carried out. Some adjustments reflecting concentration influence were performed to outline the agreement between calculations and measurements. The validity of the model can be confirmed, if the volume fraction of the disperse phase is considered as a variable. Finally, droplet size distributions from theoretical attenuation spectra could be calculated based on a log-normal distribution.     

The Infrared Multiphoton-Dissociation Spectra of Bromopropene Isomeric Cations

The infrared quasi-resonance stepwise-multiphoton-dissociation spectra of the molecular ions of 1-bromo-propene, 2-bromopropene, 3-bromopropene, and bromocyclopropane have been obtained at selected wavelengths between 9. 2 and 10. 7 μm in a Fourier -transform mass spectrometer. The resulting spectra and the photo fragmentation kinetics suggest that these molecular ions retain their identity upon ionization, and can, thus, provide fingerprint spectral identifications for the ions.     

Spectres de masse des composés organiques. 5e communication. Perte de méthane et d'éthylène à partir des ions butyle

The loss of methane and ethylene in the mass-spectrometric fragmentation of different isomeric butyl ions which originate from butyl halides has been studied. The different carbon atoms in n-butyl are already equivalent after 10^?7 s, whereas the statistical distribution of the hydrogen atoms within the molecule can only be observed for the metastable peaks. A protonated cyclobutane structure is proposed as an intermediate product in the fragmentation of the n-butyl ion. The fragmentation of this model has been simulated by a computer. This allows prediction as to the time-scale of fragmentation. The comparison of this model fragmentation with that of isomeric butyl ions shows that, even in the decay of the tertiary butyl ion, the formation of the proposed rearranged cyclic structure competes favorably with the direct fragmentation.     

Optical Absorption Spectra for the Solvated Electron Observed in the Pulse Radiolysis of cis- and trans- Isomers of 3- and 4-Methylcyclohexanols

The absorption band maximum of solvated electrons, λ_max(e^?_s), in 3- or 4-methylcyclohexanols is observed at longer wavelengths (818–837 nm), if the OH group is axial, and at shorter wavelengths (721–723 nm), if it is equatorial. It is surmised that the size of cavity for the solvated electron is larger in the former case and smaller in the latter.     

The Gas-Phase Ion/Molecule Reactions of C3H5X (X = C1, Br,I)

The ion/molecule reactions of the molecular ion, the C₃H ion, and the C₃H ion obtained from 3-chloropropene. 1-bromopropene, 2-bromopropene, 3-bromopropene, bromocyclopropane. and 3-iodopropene have been studied with their neutral precursor in a Fourier-transform mass spectrometer (FT/ICR). The molecular ions react to yield primarily C₆H except for the ion derived from 1-bromopropene that is unreactive. The kinetics of the 3-bromopropene molecular ion reveals that 18% of these ions must possess a different structure which is unreactive. The fact that C₃H ions obtained from 2-bromopropene are the only ones to undergo proton transfer is taken as evidence that only this parent compound gives rise to 2-propenyl cations by low-energy electron impact. The C₃H ions generated in these systems are shown to be roughly an equal mixture of propargylium ions that react to yield C₆H and unreactive cyclopropenium ions.