The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

Synthesis of hydrocarbons and alcohols from CO and H2 in the presence of CO-Rh catalysts

Catalysts of 4.5% Co- 0.1% Rh- 5:10% Cu/TiO ₂ display high activity in the synthesis from CO and H₂ of a mixture of hydrocarbons and alcohols at 250–300°C and pressures of 0.1–6.0 MPa; these catalysts are more selective than traditional Co catalysts with respect to alcohol formation.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 8, pp. 1707–1709, August, 1990.     

A top-down approach to protein structure studies using chemical cross-linking and Fourier transform mass spectrometry

In a preliminary communication we described a top-down approach to the determination of chemical cross-link location in proteins using Fourier transform mass spectrometry (FT-MS). We have since extended the approach to use a series of homobifunctional cross-linkers with the same reactive functional groups, but different cross-linker arm lengths. Correlating cross-linking data across a series of related linkers allows the distance constraint derived from a cross-link between two reactive side chains to be determined more accurately and increases the confidence in the assignment of the cross-links. In ubiquitin, there are seven lysines with primary amino groups and the amino terminus. Disuccinimidyl suberate (DSS, cross-linker arm length = 11.4 A), disuccinimidyl glutarate (DSG, cross-linker arm length = 7.5 A) and disuccinimidyl tartrate (DST, cross- linker arm length = 5.8 A) are homobifunctional cross-linking reagents that react specifically with primary amines. Using tandem mass spectrometry (MS/MS) on the singly, internally cross-linked precursor ion of ubiquitin, we found cross-links with DSS and DSG between the amino terminus and Lys 6, between Lys 6 and Lys 11, and between Lys 63 and Lys 48. Using disuccinimidyl tartrate (DST), the shortest cross-linker in the series, only the cross-links between the amino terminus and Lys 6, and between Lys 6 and Lys 11 were observed. The observed cross-links are consistent with the crystal structure of ubiquitin, if the lysine side chains and the amino terminus are assumed to have considerable flexibility. In a separate study, we probed the reactivity of the primary amino groups in ubiquitin using the amino acetylating reagent, N-hydroxy succinimidyl acetate (NHSAc), and a top-down approach to localize the acetylated lysine residues. The reactivity order obtained in that study (M1 approximate, equals K6 approximate, equals K48 approximate, equals K63) > K33 > K11 > (K27, K29), shows that the cross-link first formed in ubiquitin by reaction with DSS and DSG occurs between the most reactive residues.     

Halogens in competition: electronic structure of mixed dihalobenzenes

The electronic structure of all isomeric dihalobenzenes C6H4XY (X, Y = Cl, Br, I) has been investigated by HeI/HeII photoelectron spectroscopy, Green's functions calculations, and comparison with the spectra of related dihalobenzenes C6H4X2 (X = Cl, Br, I). The careful analysis of measured pi orbital and halogen lone pair ionization energies enabled us to describe substituent effects in terms of resonance, inductive, steric, and spin-orbit coupling interactions.     

The effect of fluctuations on emission spectra: practical distinctions between raman and fluorescence spectra

By the use of stochastic theory it is clearly shown that two types of emission should to fluctuations such as would occur during interactions with their environment are resonantly driven by light. In our approach the electromagnetic field is treated in a classical manner and decay is introduced by a Wigner-Weisskopfansatz. The two types of emission should occur for a three-level system and relaxation to other levels is not involved in the mechanism. Also included is a discussion of operational definitions of fluorescence (and other spontaneous emissions) and of the relationship between absorption and scattering in terms of dephasing or T₂ processes, especially as this relates to conventional emission phenomena.     

Amide bond formation using an air-stable source of AlMe3

Synthesis of amides from coupling esters with a range of primary amines can be conveniently achieved in moderate to excellent yields (69-99%) using an air-stable adduct of trimethylaluminium (AlMe₃)₂·DABCO (DABCO is 1,4-diazobicyclo[2.2.2]octane), referred to as DABAL-Me₃. Reactions can be run without requiring the exclusion of atmospheric oxygen or the drying of solvents.     

Combined use of liquid chromatography-nuclear magnetic resonance spectroscopy and liquid chromatography-mass spectrometry for the characterization of an acarbose degradation product

Directly coupled LC-MS and LC-NMR were applied to identify and structurally characterize an acarbose degradation product A in acidic media. A comparative analysis of the stop-flow LC-NMR (1H and TOCSY) and LC-MS data provided evidence that A is structurally related to acarbose, differing from the parent compound in a number of subunits present in the molecule. Spectral analysis revealed that A was the alpha-glucosidase inhibitor amylostatin XG. Complementary information obtained from the two methods led to the structural elucidation of A which was later corroborated by high-resolution NMR spectroscopy of the isolated molecule.     

Thermal investigation of PEG 4000 — oxazepam binary system

A thermal study using DSC and Hot Stage Microscopy (HSM) was carried out to investigate the interaction in solid state of the binary system PEG 4000 — oxazepam, and to establish their phase diagram. The eutectic composition, which melting occurs at lower temperature as compared with the pure components, has been determined. The results obtained by DSC and HSM have indicated that PEG 4000 — oxazepam mixtures displays no obvious incompatibilities, and that the system shows a typical eutectic behaviour. However because of the closeness of the melting of PEG 4000 to the eutectic temperature, it was difficult to determine precisely the eutectic composition and temperature on the basis of DSC measurements alone. The use of heats of fusion corresponding to physical mixtures allowed an estimation of the eutectic composition at 6% w/w oxazepam. Additional information of temperature (57.6C) and composition (5–10% w/w oxazepam) of the eutectic was obtained by HSM using the contact method. This low melting temperature in this range of compositions offers advantages in terms of drug stability and easy manufacture.     

Spectres de vibration et structure d'un cristal d'amidogallate de sodium

Raman spectra (4000-150 cm^?1) of a single crystal of NaGa(NH₂)₄ and infrared spectra (4000-200 cm^?1 ) of a polycrystalline sample have been studied at different temperatures. An assignment of the bands is given. The spectra are discussed assuming S₄ and T_d point group symmetry of the Ga(NH₂)^?₄ ion at low temperature and at room temperature respectively. Metal-ligand and N-H stretching frequencies are compared to those of some other amido metalates.