Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Atomic-absorption determination of lead in geological materials

A method is described for the determination by atomic-absorption spectrophotometry of lead, tip to the milligram level, in samples of geological materials. After attack with perchloric-hydrofluoric acid mixture and the removal of perchlorate ion by precipitation as potassium perchlorate, lead is separated from matrix elements by means of anion-exchange in 2M hydrobromic acid on the strongly basic anion-exchange resin Dowex 1 x 8. Lead is adsorbed on the resin column while practically all other accompanying elements pass into the effluent. For the elution of lead 6(M) hydrochloric acid is used and after evaporation of the eluate lead is determined by atomic-absorption spectrophotometry. The method was tested by analysing numerous samples with contents ranging from a few ppm to milligram amounts of lead. In most cases very good agreement of results was obtained.     

Syntheses of Heterocycles by Intramolecular Acylation of Nitrile-Hydrogen Halide Adducts

Five-membered, six-membered, and seven-membered aza-, diaza-, and thiazaheterocycles can be prepared by cyclization of. ω-cyano carboxylic acid halides in the presence of hydrogen halides in aprotic solvents. Nitrile-hydrogen halide adducts occur as intermediates in this novel heterocycle synthesis of wide application. The acylating cyclizations of nitriles in protonic media, which proceed via imidic esters or amides, are not discussed.     

Struktur des [C3H5]+-Ions im Cl(CH4)-Plasma

Analysis of collisional activation spectra proves that [C₃H₅]⁺ ions present in the chemical ionization plasma of methane, possess the allyl cation structure.     

The impact parameter dependence of the Pb + Pb K − MO spectrum

Molecular orbital X-rays are emitted during ion-atom collisions. The theoretical treatment, numerical results for the Pb + Pb system and a comparison with background radiation like NNB, SEB and γ-decay of Coulomb excited nuclei are presented.     

194Pt(12C, 12C′) reaction and the triaxial-rotor model

Elastic and inelastic scattering differential cross sections for the ¹⁹⁴Pt(¹²C, ¹²C′) reaction at a bombarding energy of 78 MeV have been measured. Data have been obtained for 0⁺, 2⁺, 4⁺, and 2^+′, states in ¹⁹⁴Pt. The data have been analyzed using a rigid asymmetric-rotor model in coupled-channels reaction calculations. It is found that satisfactory fits to all data can be obtained but only if the hexadecapole deformation is included in the asymmetric-rotor model.     

Irreduzible und treue Darstellungen unendlichdimensionaler Cliffordalgebren

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