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.     

Ultraviolet emission spectra of sunbeds

It is well known that UV radiation contributes to the development of skin cancer. Exposure to solar radiation is predominantly responsible for the high incidence rate of skin cancer, but there are also indications that sunbeds are involved. The aim of the present investigation was to determine the UV emission spectra of sunbeds. It included the most common sunbed models, which cover more than 50% of the Swiss market. The UV emission spectra of sunbeds have special characteristics and are different from the sun spectrum, which can be seen in high-resolution spectral measurements. Sunbed emission spectra are similar to the sun spectrum in the UVB (280-320 nm) range but reach values 10 to 15 times higher in the UVA (320-400 nm) range. An average erythema-effective irradiance of 0.33 W/m2 was determined for sunbeds. This corresponds to a UV index of 13, which is significantly higher than the UV index of 8.5 of the high summer sun at noon at intermediate latitudes. The measurements were spread over the whole effective area of the sunbeds, and an inhomogeneous distribution of the irradiances with variations of up to 30% from the average value was found.     

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.     

A DFT study of pyramidalized alkenes: 7-oxasesquinorbornenes and 7,7′-dioxasesquinorbornenes

  DFT calculations of 7′–oxasesquinorbornenes and 7,7′-dioxasesquinorbornenes using the B3LYP/6–31G* method are reported. All the investigated structures (syn- and anti- derivatives) showed significant non-planarity of the central double bond, with the exception of those anti-derivatives possessing symmetrical structures. The influence of the replacement of the methylene groups at position 7- of the norbornene fragment with oxygen and the introduction of second and third (peripheral) double bonds and benzene rings on the molecular and electronic structures of these molecules have also been investigated. Received: 11 November 2002 / Accepted: 6 June 2002 / Published online: 29 April 2003