Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

Synthesis of Tricyclic beta-Methylene Spiro Lactones Related to Bakkenolides by Successive Radical Cyclization-High Pressure Diels-Alder Reactions

The synthesis of some novel bakkenolides and their epi-spiro analogues was achieved by a new approach. Photolysis of allyl 1-(phenylseleno)-2-oxocyclopentanecarboxylates 7-9 afforded the corresponding spiro lactones 10-12 by radical cyclization via group transfer of the phenylseleno group. Selenoxide elimination of 11 and 12 produced the corresponding beta-methylene lactones 14 and 15. Diels-Alder cycloaddition of lactone11 with piperylene failed at ambient pressure, but proceeded in generally good yield in the presence of various Lewis acids at pressures of ca. 16 kbar, to give mixtures of beta-exo, alpha-endo, and beta-endo cycloadducts 19, 21, and 23, respectively. The preponderance of endo products 21 and 23, formed via highly hindered, but more compact, transition states was attributed to the high pressure and resulted in trans-dimethyl configurations of the products. The facial selectivity was dependent upon the Lewis acids, and the greatest alpha:beta ratio was observed with catalysts of the type TiCl(2)(OR)(2). Epimerization of the C-4 methyl group in 21 and 23 to furnish the corresponding cis-dimethyl analogues was achieved via exo-epoxidation, regioselective reduction, oxidation to the corresponding 3-keto derivatives, and base-catalyzed equilibration, thereby affording (+/-)-3,6-dioxobakkenolide-A (39) and its epi-spiro derivative 28, respectively. When the radical cyclization step was performed subsequent to the Diels-Alder cycloaddition by photolysis of perhydrindane 43, only the epi-spiro product 44 was obtained.     

Raman spectroscopic investigation on the microenvironment of the liver tissues of Zebrafish (Danio rerio) due to titanium dioxide exposure

Nano titanium dioxide (nTiO₂), generally considered to be toxicologically inert, is manufactured in large quantities and extensively applied in consumer products. The small size and large surface area endow them with an active group or intrinsic toxicity. Advances in instrumentation are making Raman spectroscopy the tool of choice for an increasing number of (bio) chemical applications. One of the great advantages of this technique is its ability to provide information on the concentration, structure and interaction of biochemical molecules in their microenvironments within intact cells and tissues, non-destructively. Zebrafish (Danio rerio), one of the most important vertebrate model organisms used in developmental biology, are increasingly used in biomedical research, particularly as a model of human disease. In the present work, an attempt is made to study the effect of titanium dioxide, both nano and bulk, on the microenvironment of the liver tissues of Zebrafish using FT-Raman spectroscopy. The results of the present study suggest that TiO₂ exposure demonstrate a marked influence on the microenvironments of the liver tissues of Zebrafish. A shift to a higher wavenumber and an increase in the intensity of the band at ∼1087 cm⁻¹ in the TiO₂ exposed tissues suggest that some of the conformational changes resulting from the alkali recovery process takes place due to TiO₂ exposure. The decreased intensity ratio (I₃₂₂₀/I₃₄₀₀) observed in the titanium-exposed tissues suggests a decreased water domain size, which could be interpreted in terms of weaker hydrogen-bonded molecular species of water in the TiO₂ exposed tissues. The observed shift of COO⁻ bands to higher frequencies shows the disruption of salt bridges as a result of a change in the oppositely charged partners and due to the enhanced random coil conformation. The variation in the intensity ratio of the tyrosyl doublet (I₈₅₈/I₈₂₅) indicates variation in the hydrogen bonding of the phenolic hydroxyl group due to TiO₂ exposure. The results further suggest that the microenvironments are greatly altered due to titanium nano exposure when compared to titanium bulk. In conclusion, the results indicate that FT-Raman spectroscopy might be a useful tool for rapid assessment of nano particle biological interactions.     

Rearrangement in the mass spectrometer via a 3,3,1-bicyclic reaction path

The fragmentation of some azoacylals showed a 3,3,1 bicyclic rearrangement which could be demonstrated by a deuterium labelling study.     

Some relations between additive functions,II

Aequationes mathematicae -     

On the system of functional equationsf(x + y) = f(x) + f(y) andf(xy) = p (x)f(y) + q(y)f(x)

Aequationes mathematicae -