On the discretization of differential and Volterra integral equations with variable delay

In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss (-Legendre) points will no longer yield the optimal (local) orderO(h ^2m ). This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406).     

The multipole expansion far from an isolated source

The static field equations of general relativity are examined as deviations from a Schwarzschild solution in isotropic form. The multipole expansion is recovered with hypergeometric radial dependence.     

Synthesis and Characterization of Novel Aminophosphine Ligands Based on Ferrocenodecaline Backbones

Summary.  Novel aminophosphine ligands for enantioselective transition metal catalysts based on different ferroceno cis- and trans-decaline backbones were synthesized and structurally characterized. Their palladium dichloride complexes were tested in the asymmetric Grignard cross coupling reaction of vinyl bromide and phenylethyl magnesium chloride, but only very low enantioselectivities were obtained. Steric strain in the aminophosphine ligands causes a severe backbone deformation and in addition leads to a slowed rotation of the respective dimethylamino group as was detected by variable temperature NMR spectroscopy. Received January 4, 2002. Accepted January 16, 2002     

Some Observations Concerning 1H-NMR. Spectra of 5-hydroxy- and 5-acetoxy-steroid compounds

It was found that analysis of the ¹H-NMR. signals (at 360 (preferably) and/or 100 MHz) of the protons at C (4) and of the equatorial α-proton at C (6), particularly when these are located in the 2–3 ppm region and therefore convenient for detection and identification, may be of valuable aid in the structural and configurational characterization of 5-hydroxy- and 5-acetoxy-steroids (unsubstituted or containing a hydroxy or acyloxy substituent at C (3)).     

Intramolecular Alkylation of Aromatic Compounds XXXVI [1]. Stereoselective Synthesis of C/D- cis -Configured Ergolines

 The stereoselective synthesis of cis-ergoline is presented. Starting from rac-N-benzoyl tryptophan methyl ester, the key compound indolinylmethylpyridin-3-one was prepared via a seven-step reaction in good yield. Since its cyclization to the desired ergolinone failed, the key compound was reduced to yield the two diastereomeric pyridin-3-ols; only one of them cyclized in trifluoromethanesulfonic acid, affording cis-ergoline. Catalytic hydrogenation of the latter gave N,N′-dimethyldihydroergoline, the X-ray crystallography of which revealed both the correct structure and identical relative configurations at C-5a and C-6a (SS or RR). Hydroboration and subsequent perruthenate oxidation of the Δ⁹-ergoline provided access to the regioisomeric ergolinols and ergolinones.     

A new synthesis of aryl hetaryl ketones via SRN1 reactions of halogenated heterocycles with potassiophenylacetonitrile followed by phase-transfer catalyzed decyanation

Reaction of halogenated pyridines, quinolines, pyrimidines, and pyrazines with potassiophenylacetonitrile ( 5 ) in liquid ammonia under near-uv irradiation affords secondary nitriles, which then undergo oxidative decyanation under phase-transfer catalytic conditions to afford aryl hetaryl ketones in excellent yields.     

Desulfurization of thiophenes by radio-frequency plasma processing

Thiophene and several alkylthiophenes are desulfurized by glow discharges to 50–80%, the main products being ethine, ethene, and low-boiling hydrocarbons. Addition of oxygen increases the desulfurization, augments the production of C₂-hydrocarbons, and reduces polymer formation.     

Complete density matrix measured for Na(32 P) excited by electron impact

Anisotropies observed in collisional deexcitation of laser excited Na atoms are analysed in terms of the density matrix describing the Na(3P) atom after the process Na(3² S)+e ^?(E)→Na(3² P)+e ^?(E?δE). Collision energiesE range from 5.1 to 22.1 eV, and scattering angles fromθ _col=0 to 25°. The results are presented with reference to a coordinate frame in which the angular momentum transferred during collision is parallel to the quantization axis. This choice allows a convenient exploitation of reflection symmetry and offers most direct relation of measurable quantities to the sublevel excitation amplitudes as e.g. computed by close coupling and distorted wave methods. Comparison between experiment and theory provides a sensitive probe of the strengths and shortcomings of the theory when predicting these alignment and orientation parameters. Since we discuss a complete set of data containing all accessible information on the atomic orbital the present measurements allow the detection of a deviation from coherence in the excitation for small scattering anglesθ _col≈5–10° atE=5.1 eV which we attribute to electron exchange.