Numerical integration of ordinary differential equations on manifolds

Summary This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.     

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.     

Otpically illuminated dielectric interfaces as high-speed far-infrared modulators

A new device for the gigahertz modulation of far-infrared radiation is analytically and numerically analyzed. It consists of a thin layer of a high-mobility, direct-bandgap semiconductor, such as GaAs, in which a high-density electron-hole plasma is rapidly created and destroyed, thereby rapidly changing the free-carrier reflectivity of the active layer. Illumination by a high-power, near-infrared laser diode array generates the plasma through intrinsic photoconduction. It is shown that this device acis primarily as an amplitude modulator, and that its efficiency increases sharply with increasing far-IR frequency, in contrast to a Schottky diode, which acts primarily as a phase modulator, and whose efficiency falls off sharply with far-IR frequency. The breakeven frequency lies at about 1.5 THz, depending slightly on the assumed device parameters. The relative advantage of the new device increases rapidly with increasing far-infrared frequency. At an operating frequency of 2.5 THz (119 m), for example, a 1 GHz modulation bandwidth may be achieved with a single-sideband conversion loss of only-21 db, versus a Schottky's loss of-39 db, assuming a laser diode power of 1 W, which is readily available from recently developed laser diode arrays.     

FLUORESCENCE SPECTRAL CHANGES OF HEMATOPORPHYRIN DERIVATIVE UPON BINDING TO LIPID VESICLES, Staphylococcus aureus AND Escherichia coli CELLS

Abstract— The binding of hematoporphyrin derivated (Hpd) to lipid vesicles and bacterial membranes was determined by fluorescence spectroscopy. The fluorescence measurements of Hpd in aqueous solutions showed two bands at 613 and 677 nm. In lipid environments of lecithin vesicles the fluorescence spectrum was shifted to 631 and 692 nm, respectively. Hpd was rapidly bound to the cell membrane of Staphylococcus aureus while much less binding occurred in the presence of Escherichia coli. At the same time, spheroplasts of both bacteria were shown to bind Hpd to a similar extent. These results are well correlated with the photoinactivation of the gram positive bacteria with Hpd while the gram negative cells were shown to be resistant. The pH dependence of both Hpd binding to S. aureus as well as the photodynamic inhibitory effect of the same bacteria are similar. It is concluded that the segregation of Hpd to the cell membrane is a prerequisite for its photodynamic effect.     

Hypoxanthine,xanthine and theobromine complexes with palladium(II) and platinum(IV) chlorides

Summary Complexes of the general types Pd(L)(LH)Cl (LH=hxH, xnH, or tbH) and Pt(L)(LH)Cl₃ (LH=hxH, or xnH) are formed by boiling under reflux 21 molar mixtures of hypoxanthine (hxH), xanthine (xnH) or theobromine (tbH) and PdCl₂ or PtCl₄ in ethanol-triethyl orthoformate. These complexes appear to be linear chain polymeric species, characterized by single monoanionic L^– ligands bridging between adjacent Pd²⁺ or Pt⁴⁺ ions. Inclusion of one terminal neutral LH and one terminal chloro-ligand completes the coordination sphere in the square-planar Pd²⁺ complexes, while the Pt⁴⁺ complexes aretrans-octahedral, involving three terminal chloro and one terminal LH ligand per platinum. The possible binding sites of the bidentate bridging L^– and the unidentate terminal LH are discussed.     

INAA determination of holmium in submilligram samples of cosmochemical and geochemical interest and the second-order activation interference

Second-order neutron activation becomes a serious interference in NAA determinations of REE in submilligram samples of meteoritic and terrestrial materials when neutron fluences reach levels of 10²⁰ n·cm^–2 or more. The Dy-interference to Ho determination and the Eu-interference to Gd determination are quite, large. We discuss the importance of second-order activation interferences in REE determinations, and present an experimental method to correct the Dy-interference to Ho determination by INAA. Ho concentrations in splits of less than 0.2 mg of BCR-1 and SP can be accurately determined by INAA with a neutron fluence of 2·10²⁰ n·cm^–2 in spite of large Dy interferences of up to 200%. It is necessary for NAA determinations of REE with high neutron fluences that each REE standard be separated from its neighboring REE, in order to correct the second-order activation interferences experimentally.     

Gas-phase acidities of disubstituted methanes and of enols of carboxamides substituted by electron-withdrawing groups

The gas-phase acidities DeltaG degrees (acid) of some 20 amides/enols of amides RNHCOCHYY'/RNHC(OH)=CYY' [R = Ph, i-Pr; Y, Y' = CO(2)R', CO(2)R' ', or CN, CO(2)R', R', R' ' = Me, CH(2)CF(3), CH(CF(3))(2)], the N-Ph and N-Pr-i amides of Meldrum's acid, 1,3-cyclopentanedione, dimedone, and 1,3-indanedione, and some N-p-BrC(6)H(4) derivatives and of nine CH(2)YY' (Y, Y' = CN, CO(2)R', CO(2)R' '), including the cyclic carbon acids listed above, were determined by ICR. The acidities were calculated at the B3LYP/6-31+G//B3LYP/6-31+G level for both the enol and the amide species or for the carbon acid and the enol on the CO in the CH(2)YY' series. For 12 of the compounds, calculations were also conducted with the larger base sets 6-311+G and G-311+G. The DeltaG degrees (acid) values changed from 341.3 kcal/mol for CH(2)(CO(2)Me)(2) to 301.0 kcal/mol for PhNHC(OH)=C(CN)CH(CF(3))(2). The acidities increased for combinations of Y and Y' based on the order CO(2)Me < CO(2)CH(2)CF(3) < CN, CO(2)CH(CF(3))(2) for a single group and reflect the increased electron-withdrawal ability of Y,Y' coupled with the ability to achieve planarity of the crowded anion. The acidities of corresponding YY'-substituted systems follow the order N-Ph enols > N-Pr-i enols > CH(2)YY'. Better linear relationships between DeltaG degrees (acid) values calculated for the enols and the observed values than those for the values calculated for the amides suggest that the ionization site is the enolic O-H of most of the noncyclic trisubstituted methanes. The experimental DeltaG degrees (acid) value for Meldrum's acid matches the recently reported calculated value. The calculated structures and natural charges of all species are given, and the changes occurring in them on ionization are discussed. Correlations between the DeltaG degrees (acid) values and the pK(enol) values, which are linear for the trisubstituted methanes, excluding YY' = (CN)(2) and nonlinear for the CH(2)YY' systems, are discussed.