Counterexamples in optimal quadrature

It is widely believed that order of exactness is a good measure of the quality of an algorithm for numerical quadrature. We show that this is not the case, by exhibiting a situation in which the optimal algorithm does not even integrate constants exactly. We also show that there are situations in which the penalty for using equidistant nodes is unbounded. Finally, we show that the complexity of obtaining an -approximation can be an arbitrary function of , i.e., there is no hardest quadrature problem.     

Mass spectra of C-nucleosides II. An unusual fragmentation reaction of the heterocyclic moiety of pyrazomycin and some closely related compounds

The mass spectrum of pyrazomycin has been obtained and reveals a fragmentation pattern with several important peaks which are not normally found in the mass spectra of C-nucleosides. It has now been established, using model compounds, that these unusual fragment ions are a direct result of the juxtaposition of the exocyclic hydroxy and carboxamido groups of the aglycon. It appears that a facile elimination of ammonia and ethanol from o-hydroxycarboxamides and o-hydroxyethylesters, respectively, may be a general fragmentation reaction for aromatic heterocycles. A mass spectrum of the TMS derivative of pyrazomycin has also been obtained and factors which may result in exceptions to the empirical B+30 (M-103) rule for C-nucleosides are discussed.     

Objective function approximations in mathematical programming

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the best approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained.This paper was partially supported by the National Science Foundation and by the Office of Naval Research, and was the basis for a plenary lecture delivered at the IX International Symposium on Mathematical Programming in Budapest, Hungary, August 1976.     

Determination of standard pH values for sodium hydrogen diglycolate buffer (0.05m) from 5–65°C

Values of pa _H ^o for 0.05 mole-kg^?1 aqueous solutions of sodium hydrogen diglycolate in the temperature range 5–65°C have been obtained from cells without transport, and can be fitted to the equation $$\begin{gathered} pa^\circ _H = 3.5098 + 2.222 \times 10^{ - 3} ({T \mathord{\left/ {\vphantom {T {K - 298.15}}} \right. \kern-\nulldelimiterspace} {K - 298.15}}) \hfill \\ + 2.628 \times 10^{ - 5} ({T \mathord{\left/ {\vphantom {T {K - 298.15}}} \right. \kern-\nulldelimiterspace} {K - 298.15}})^2 \hfill \\ \end{gathered} $$ The analysis has been carried out by a multilinear regression procedure using a form of the Clarke and Glew equation. This buffer standard may be a useful alternative to the saturated potassium hydrogen tartrate buffer.     

Quantum mechanical description of electrode reactions: Part II. Treatment of compact layer structure at platinum electrodes by means of the extended Hückel molecular orbital method. Pt(111) surfaces

The Iteraltive Extended Hückel Molecular Orbital method has been adapted to calculation of the properties of an electrode and compact layer. Predictions of the stablest orientations, on the Pt(111) surface of species such as H₂O, Pt, OH^?, H, and the halides, F^?, Cl^?, Br^? and I^?, based upon calculation of the total energy corresponding to various internuclear distances, are reported. The calculations correctly predict self-adsorption of Pt on the Pt(111) surface at the face-centered cubic closest-packing position. The H₂O molecule is predicted to locate itself above three adjoining Pt atoms, with the O atom closest to the surface and the H atoms opposite the O. Similar results were obtained for OH^? and the halides. Atomic H, however, is predicted to drop into the plane of centers of the Pt surface atoms, where it would lie between, three adjacent Pt atoms. Application of the method to electrode studies requires only modest amounts of computer time but produces surprisingly reliable qualitative predictions. Compulation of electrochemical quantities such as charge, differential capacitance, surface tension and potential energy as a function of electrode potential will be described in future work.     

Progress in the Development of Methods for Asymmetric Organic Synthesis

Methods for the enantioselective synthesis of chiral cyclohexane derivatives from aromatic carboxylic acids are presented. Birch reduction-alkylation of chiral benzamides of type 1 occur with high diastereose-lectivities to give 1,4-cyclohexadiene derivatives with quaternary centers located at C(6). Early applications of this chemistry provided target structures with a quaternary center derived from C(1) of the starting benzoic acid derivative. Herein are described 1) the development of a more versatile Birch reduction-alkylation, 2) a practical “asymmetric linkage” between aromatic carboxylic acids and chiral acyclic structures, and 3) asymmetric syntheses of unnatural alkaloids; applications to opiate receptor pharmacology.     

Level decay and orientational kinetics of the rhodamine B monomer and dimer

The population kinetics and the rotational diffusion of the rhodamine B monomer and dimer were measured by using picosecond pulses from a mode-locked Nd : YAG laser to induce and time resolve the concentration-dependent transient absorption saturation of various aqueous solutions of this organic dye.     

Synthesis of an Azasteroid Using an Acyl Iminium Ion-Initiated Tandem Cyclization

An acyl iminium ion-initiated tandem cyclization gave an unexpected dienone product, a seco-azasteroid (2). The factors governing the formation of 2 were investigated in an attempt to optimize its formation. The reaction was applied to a more elaborate system, resulting in the synthesis of the full steroid skeleton of 13-azaandrosta-1,4-diene-3,17-dione (3), which contains the unusual substitution of a chlorine atom for the axial 19-methyl.     

Approximate solutions of the Bloch-McConnell equations for two-site chemical exchange.

Approximate formulas for the transverse-relaxation rate constant and resonance frequency have been derived for nuclear spins subject to two-site chemical exchange. The new results are more accurate than previous approximations when chemical exchange does not approach the fast-exchange limit or when site populations are not highly asymmetric. Results are derived from an effective average evolution operator that contains contributions from both exact eigenvalues of the Bloch-McConnell equations. These expressions are useful for interpretation of chemical exchange processes in NMR spectroscopy.     

New cationic aryldiazo,aryldiimine and arylhydrazine complexes of platinum

The preparation of some new cationic aryldiazo complexes of platinum of formula trans-[Pt(N₂Ar)(PEt₃)₂L]⁺, where N₂Ar = N₂C₆H₄F-m or -p and L = NH₃, Py, Et₃P or EtNC, is described. Protonation of these complexes gives the corresponding aryldiimide complexes trans-[Pt(NHNAr)(PEt₃)₂L]⁺, and reduction of the protonated complexes with molecular hydrogen in the presence of a catalyst gives the arylhydrazine complexes trans-[Pt(NH₂NHAr)(PEt₃)₂L]⁺. Some of the spectroscopic properties of these new complexes are reported and discussed.