Justification of the nonlinear Schr?dinger equation in spatially periodic media

The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schr?dinger equation. Here we prove estimates for the error made by this formal approximation using Bloch wave analysis, normal form transformations, and Gronwall’s inequality.     

Zum Hydratisierungs- und Dehydratisierungsverhalten von Magnesiumhydrogensulfat

On the Hydration and Dehydration Behaviour of Magnesium Hydrogensulfate Mg(HSO₄)₂ · H₂O has been prepared by crystallization from diluted sulfuric acid. The thermal destruction leads to the formation of siccative Mg(HSO₄)₂. This process is reversible, i. e., Mg(HSO₄)₂ rehydrates during cooling under formation of Mg(HSO₄)₂. H₂O. Due to further water reception Mg(HSO₄)₂ · H₂O decomposes at a temperature of 50°C forming Kieserite (MgSO₄ · H₂), MgSO₄ · 1,25H₂O, sulfuric acid, and water. During gentle warming of this heterogenous phase mixture it reacts under reconstruction of Mg(HSO₄)₂ · H₂O. The thermal measurements have been carried out in dynamic as well as in static gas atmosphere. The strong dependence of the decomposition temperatures from the partial pressure of the gaseous products could be demonstrated by in situ X-ray measurements.     

Compton scattering by the proton through θcms = 75° and 90° in the Δ-resonance region

Differential cross sections for Compton scattering by the proton have been measured in the energy interval between 200 and 500 MeV at scattering angles of θcms = 75° and θcms = 90° using the CATS, the CATS/TRAJAN, and the COPP setups with the Glasgow Tagger at MAMI (Mainz). The data are compared with predictions from dispersion theory using photo-meson amplitudes from the recent VPI solution SM95. The experiment and the theoretical procedure are described in detail. It is found that the experiment and predictions are in agreement as far as the energy dependence of the differential cross sections in the Δ-range is concerned. However, there is evidence that a scaling down of the resonance part of the M1+3/2 photo-meson amplitude by (2.8 ± 0.9)% is required in comparison with the VPI analysis. The deduced value of the M1+3/2-photoproduction amplitude at the resonance energy of 320 MeV is: |M1+3/2| = (39.6 ± 0.4) × 10−3 mπ+−1.     

Time-resolved emission spectra and anisotropy profiles for symmetric diacyl- and dietherphosphatidylcholines

The solvent relaxation behavior of Patman (6-palmitoyl-2-[[2-(trimethylammonium) ethyl]methylamino]naphthalene chloride) was investigated in small unilamellar vesicles composed of symmetric diacyl( 1,2-dipalmitoylphosphatidylcholine; DPPC) and diether lipids (l,2-dihexadecylphosphatidylcholine; DHPC), calculating time-resolved emission spectra (TRES) and correlation functions. Both the steady-state spectra as a function of temperature and excitation wavelength and the TRES of Patman in DPPC are blue-shifted compared to those in DHPC. The solvent relaxation at three temperatures above and below the phase transition is considerably faster in DHPC than in DPPC. As the steady-state anisotropies of Patman and TMA-DPH [l-(4-trimethylammoniumphenyl)-6-phenyl-l,3,5-hexatriene] are similar in both lipids as a function of both temperature and emission wavelength, we conclude that the introduction of ether linkages allows more efficient water penetration in the glycerol region, leading to a more polar environment and therefore faster solvent relaxation of the incorporated dyes. Using a series ofn-(9-anthroyloxy) fatty acids (n = 2, 3, 6, 9, 12; 16-AP), we show that anisotropy profiles can be used to distinguish between noninterdigitated (DPPC) and fully interdigitated (DHPC) gel-phase structures. 16-(9Antroyloxy) palmitic acid (16-AP) is an especially useful probe exhibiting pronounced differences in the steady-state anisotropies in non- and fully interdigitated gel phases.     

Observation of sequential electron capture to individual highly-charged Th ions

Electron transfer from H₂ to Th⁷⁹⁺ was studied in a cryogenic Penning trap by non-destructively observing the sequential development with time of the charge state of one or a few stored particles. The ratio of the true double capture cross section to the total capture cross section was found to be 0.21 ± 0.11 at a mean center-of-mass energy near 6 eV. This revised version was published online in August 2006 with corrections to the Cover Date.     

Franz–Keldysh oscillations at the miniband edge in a GaAs/AlxGa1 −xAs superlattice

We have systematically measured the electroreflectance spectra of a GaAs (7.0 nm)/Al_0.1Ga_0.9As (3.5 nm) superlattice at various electric fields to investigate Franz–Keldysh (FK) oscillations. In the low-field regime, we clearly observe the FK oscillations toward the low-energy side of theM₁critical point (mini-Brillouin-zone edge). As the electric field is increased, the direction of the FK oscillations is reversed, then the oscillations disappear. The change of the oscillation direction correlates with the transformation of the electronic structures from the miniband to the Stark-ladder states in the Wannier-Stark localization. We discuss these experimental results on the basis of a theory of the FK oscillations and envelope-function forms calculated by a transfer matrix method with Airy functions.     

How fair is an equitable distribution?

Envy is a rather complex and irrational emotion. In general, it is very difficult to obtain a measure of this feeling, but in an economical context envy becomes an observable which can be measured. When various individuals compare their possessions, envy arises due to the inequality of their different allocations of commodities and different preferences. In this paper, we show that an equitable distribution of goods does not guarantee a state of fairness between agents and in general that envy cannot be controlled by tuning the distribution of goods.