Stochastic eigenvalue problems for differential equations

We consider an eigenvalue problem for stochastic ordinary differential operators with stochastic boundary conditions. By replacing the random variables and stochastic processes by their mean values one gets the so-called averaged problem corresponding to the given stochastic problem. Let λ_i(ω) denote the stochastic eigenvalues and μ_i the eigenvalues of the averaged problem. In this paper the differences 〈λ〉?μ_i are calculated. The motive of such investigations was the effect of “bowing” in mixed crystal theory.     

A VUV isochromat spectrometer for surface analysis

A vacuum ultraviolet isochromat spectrometer employing an energy selective Geiger-Müller photon counter is described. The mean energy of the spectrometer is 9.7 eV with a standard deviation of ±0.23 eV. Together with the thermal energy distribution of the electrons emitted from a directly heated tungsten cathode, an overall resolution of ±0.33 eV is obtained. Pulsed operation of the x-ray tube avoids additional broadening due to the voltage drop across the filament. Further, pulsed operation of the accelerating voltage is employed to avoid counting losses due to the relatively large dead time of the radiation detector. Operating frequencies of up to 2 kHz are possible. With sensitivities of typically 10⁷ pulses per Coulomb, the anode currents are in the range of 100–200 μA. These attractive properties suggest to employ the method to study surface electronic properties of solids.     

Structure and Dynamics of Molecular Rods in Membranes: Application of a Spin‐Labeled Rod

Molecular rods consisting of a hydrophobic backbone and terminally varying functional groups have been synthesized for applications for the functionalization of membranes. In the present study, we employ a spin‐labeled analogue of a recently described new class of molecular rods to characterize their dynamic interactions with membranes. By using the different approaches of ESR and NMR spectroscopy, we show that the spin moiety of the membrane‐embedded spin‐labeled rod is localized in the upper chain/glycerol region of membranes of different compositions. The rod is embedded within the membrane in a tilted orientation to adjust for the varying hydrophobic thicknesses of these bilayers. This orientation does not perturb the membrane structure. The water solubility of the rod is increased significantly in the presence of certain cyclodextrins. These cyclodextrins also allow the rods to be extracted from the membrane and incorporated into preformed membranes. The latter will improve the future applications of these rods in cellular systems as stable membrane‐associated anchors for the functionalization of membrane surfaces.     

3D Motions of Iron in Six‐Coordinate {FeNO}7 Hemes by Nuclear Resonance Vibration Spectroscopy

The vibrational spectrum of a six‐coordinate nitrosyl iron porphyrinate, monoclinic [Fe(TpFPP)(1‐MeIm)(NO)] (TpFPP=tetra‐para‐fluorophenylporphyrin; 1‐MeIm=1‐methylimidazole), has been studied by oriented single‐crystal nuclear resonance vibrational spectroscopy (NRVS). The crystal was oriented to give spectra perpendicular to the porphyrin plane and two in‐plane spectra perpendicular or parallel to the projection of the FeNO plane. These enable assignment of the FeNO bending and stretching modes. The measurements reveal that the two in‐plane spectra have substantial differences that result from the strongly bonded axial NO ligand. The direction of the in‐plane iron motion is found to be largely parallel and perpendicular to the projection of the bent FeNO on the porphyrin plane. The out‐of‐plane Fe‐N‐O stretching and bending modes are strongly mixed with each other, as well as with porphyrin ligand modes. The stretch is mixed with v₅₀ as was also observed for dioxygen complexes. The frequency of the assigned stretching mode of eight Fe‐X‐O (X=N, C, and O) complexes is correlated with the Fe?XO bond lengths. The nature of highest frequency band at ≈560 cm^?1 has also been examined in two additional new derivatives. Previously assigned as the Fe?NO stretch (by resonance Raman), it is better described as the bend, as the motion of the central nitrogen atom of the FeNO group is very large. There is significant mixing of this mode. The results emphasize the importance of mode mixing; the extent of mixing must be related to the peripheral phenyl substituents.     

Transient amplification of maximum vibration amplitudes

Many low damped structures as turbine blades or drill strings are exposed to high dynamical loads causing high vibration amplitudes. These applications comprise sub-critical eigenfrequencies. Hereby, the lower eigenfrequencies have to be passed before reaching the operating point. Most investigations of vibration amplitudes caused by a resonance passage deal with the computation of single degree of freedom systems. Thereby, it has been shown that the stationary vibration response provides the highest possible amplitude. Further it can be stated that the maximum vibration response of the resonance passage decreases with an increasing sweep velocity [3]. Isolated modes of linear systems can be represented by single degree of freedom systems. Subsequently a mode shape can be described by the multiplication of the amplification function of the mode and the belonging eigenvector. There are only some recent works that deal with resonance passages of vicinal modes, e. g. [1]. In this paper the resonance passage of a three dimensional system with nearby modes is studied. To calculate the transient vibration response an analytical approach is used. It is shown that the maximum amplitude of the stationary vibration response is not the upper limit for the maximum amplitude of the resonance passage. Thus, the maximum amplitude may rise while the sweep velocity increases. Hence, regarding a multi degree of freedom system the maximum amplitude of the resonance passage can exceed the maximum amplitude of the stationary vibration response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dye laser pumped, continuous-wave KTP optical parametric oscillators

4 (KTP) optical parametric oscillators (OPOs) with pump and idler resonant cavities. With a linear two-mirror cavity the pump power at threshold was 70 mW. The single-frequency signal and idler output wavelengths were tuned in the range of 1025 to 1040 nm and 1250 to 1380 nm by tuning the dye laser in the range of 565 to 588 nm. With a dual three-mirror cavity the threshold was 135 mW. Pumped by 500 mW of 578 nm radiation the 1040 nm single-frequency signal wave output power was 84 mW. Power and frequency stable operation with a spectral bandwidth of less than 9 MHz was obtained by piezo-electrically locking the length of the pump resonant cavity to the dye laser wavelength. Similar performance was achieved by placing the idler resonant OPO inside the resonator of the dye laser. With this system power stable and single-frequency operation was achieved with a spectral bandwidth of less than 11 MHz for the idler wave. Received: 3 February 1998/Revised version: 9 March 1998