The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.     

Pulse FT NMR of non-equilibrium states of half-integer spin quadrupolar nuclei in single crystals

For quadrupolar nuclei with spin quantum numbers equal to 3/2, 5/2 and 7/2, the intensities of the NMR transitions in a single crystal are examined as a function of the rf excitation flip angle. Single-quantum NMR intensities are calculated using density matrix theory beginning under various non-equilibrium conditions and are compared with those determined experimentally. As a representative spin-3/2 system, the flip-angle dependence of the ²³Na NMR intensities of a single crystal of NaNO₃ was investigated beginning with the inversion of the populations associated with one of the satellite transitions. Subsequently, the populations of both satellite transitions were inverted using highly frequency-selective hyperbolic secant pulses. Calculated and experimental intensities are in good agreement. As an example of a spin-5/2 system, the flip-angle dependence of the ²⁷Al NMR transition intensities was determined using a single crystal of sapphire, Al₂O₃, starting under different nuclear spin population conditions. The experimental trends mimicked those predicted by the density matrix calculations but the agreement was not as good as for the spin-3/2 case. Some SIMPSON simulations were also carried out to confirm the results generated by our density matrix calculations. The theoretical flip-angle behavior of the NMR transition intensities obtained from a spin-7/2 spin system is also discussed.     

A combined experimental and quantum chemistry study of selenium chemical shift tensors

A comprehensive investigation of selenium chemical shift tensors is presented. Experimentally determined chemical shift tensors were obtained from solid-state 77Se NMR spectra for several organic, organometallic, or inorganic selenium-containing compounds. The first reported indirect spin-spin coupling between selenium and chlorine is observed for Ph(2)SeCl(2) where 1J(77Se,35Cl)iso is 110 Hz. Selenium magnetic shielding tensors were calculated for all of the molecules investigated using zeroth-order regular approximation density functional theory, ZORA DFT. The computations provide the orientations of the chemical shift tensors, as well as a test of the theory for calculating the magnetic shielding interaction for heavier elements. The ZORA DFT calculations were performed with nonrelativistic, scalar relativistic, and scalar with spin-orbit relativistic levels of theory. Relativistic contributions to the magnetic shielding tensor were found to be significant for (NH4)2WSe4 and of less importance for organoselenium, organophosphine selenide, and inorganic selenium compounds containing lighter elements.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

The effectiveness of 1H decoupling in the 13C MAS NMR of paramagnetic solids: an experimental case study incorporating copper(II) amino acid complexes

The use of continuous-wave (CW) 1H decoupling has generally provided little improvement in the 13C MAS NMR spectroscopy of paramagnetic organic solids. Recent solid-state 13C NMR studies have demonstrated that at rapid magic-angle spinning rates CW decoupling can result in reductions in signal-to-noise and that 1H decoupling should be omitted when acquiring 13C MAS NMR spectra of paramagnetic solids. However, studies of the effectiveness of modern 1H decoupling sequences are lacking, and the performance of such sequences over a variety of experimental conditions must be investigated before 1H decoupling is discounted altogether. We have studied the performance of several commonly used advanced decoupling pulse sequences, namely the TPPM, SPINAL-64, XiX, and eDROOPY sequences, in 13C MAS NMR experiments performed under four combinations of the magnetic field strength (7.05 or 11.75T), rotor frequency (15 or 30kHz), and 1H rf-field strength (71, 100, or 140kHz). The effectiveness of these sequences has been evaluated by comparing the 13C signal intensity, linewidth at half-height, LWHH, and coherence lifetimes, T2('), of the methine carbon of copper(II) bis(dl-alanine) monohydrate, Cu(ala)(2).H2O, and methylene carbon of copper(II) bis(dl-2-aminobutyrate), Cu(ambut)(2), obtained with the advanced sequences to those obtained without 1H decoupling, with CW decoupling, and for fully deuterium labelled samples. The latter have been used as model compounds with perfect 1H decoupling and provide a measure of the efficiency of the 1H decoupling sequence. Overall, the effectiveness of 1H decoupling depends strongly on the decoupling sequence utilized, the experimental conditions and the sample studied. Of the decoupling sequences studied, the XiX sequence consistently yielded the best results, although any of the advanced decoupling sequences strongly outperformed the CW sequence and provided improvements over no 1H decoupling. Experiments performed at 7.05T demonstrate that the XiX decoupling sequence is the least sensitive to changes in the 1H transmitter frequency and may explain the superior performance of this decoupling sequence. Overall, the most important factor in the effectiveness of 1H decoupling was the carbon type studied, with the methylene carbon of Cu(ambut)(2) being substantially more sensitive to 1H decoupling than the methine carbon of Cu(ala)(2).H2O. An analysis of the various broadening mechanisms contributing to 13C linewidths has been performed in order to rationalize the different sensitivities of the two carbon sites under the four experimental conditions.     

An unusual [4 + 4] cycloadduct from an o-quinodimethane: Characterisation and computational studies

Trapping of a pyrimidinone-derived o-quinodimethane with tetraphenylcyclopentadienone gives a mixture of the [4 + 2] and formal [4 + 4] adducts. Computational studies show that the latter arises by a sequence of [4 + 2] cycloaddition, homolysis, radical recombination to a spiro-tetrahydrofuran and Claisen rearrangement under the thermolytic conditions.     

Studies on the synthesis of the lasubine alkaloids

Formal syntheses of lasubine II and subcosine II have been completed by the synthesis of epi-lasubine II. The synthesis involves diastereoselective allylation of a methoxy isoxazolidine and a tandem hydrogenation process leading stereoselectively to a trisubstituted piperidine.     

Efficient synthesis of (3R,3aS,6aR)- hexahydrofuro[2,3-b]furan-3-ol from glycolaldehyde

A one-step diastereoselective (up to 98:2) synthesis of the bis-furan alcohol of Darunavir and other HIV drug candidates has been achieved utilizing the novel cyclization of glycolaldehyde and 2,3-dihydrofuran. The cycloaddition was catalyzed by a variety of catalysts including those formed from tin(II) triflate and common chiral ligands such as BINAP and Evans's box ligands. An efficient and unique enzymatic process enhanced the enantiomeric purity to provide the target in optically pure form.     

Determination of phosphate using a highly sensitive paired emitter-detector diode photometric flow detector

The use of a novel inexpensive photometric device, a paired emitter-detector diode (PEDD) has been applied to the colorimetric determination of phosphate using the malachite green spectrophotometric method. The novel miniaturized flow detector applied within this manifold is a highly sensitive, low cost, miniaturized light emitting diode (LED) based detector. The optical flow cell was constructed from two LEDs, whereby one is the light source and the second is the light detector, with the LED light source forward biased and the LED detector reversed biased. The photocurrent generated by the LED light source discharges the junction capacitance of the detector diode from 5 V (logic 1) to 1.7 V (logic 0) and the time taken for this process to occur is measured using a simple timer circuit.The malachite green (MG) method employed for phosphate determination is based on the formation of a green molybdophosphoric acid complex, the intensity of which is directly related to phosphate concentration. Optimum analytical parameters such as reaction kinetics, reagent to sample concentration ratio and emitter wavelength intensity were investigated for the spectrophotometric method. Linear calibration plots that obeyed the Beer-Lambert law were obtained for phosphate in the range of 0.02-2 μM. The dynamic range, sensitivity and limits of detection are reported.