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.     

Effect of pressure on the proton transfer rate from a photoacid to a solvent. 4. Photoacids in methanol

The pressure dependence of the excited-state proton dissociation rate constant of four photoacids, 2-naphthol-6,8-disulfonate (2N68DS), 10-hydroxycamptothecin (10-CPT), 5-cyano-2-naphthol (5CN2), and 5,8-dicyano-2-naphthol (DCN2), are studied in methanol. The results are compared with the results of the pressure dependence study we recently conducted for several photoacids in water, ethanol, and propanol. The pressure dependence is explained using an approximate stepwise two-coordinate proton transfer model. The increase in rate, as a function of pressure, manifests a strong dependence of proton tunneling on the distance which decreases with an increase of pressure between the two oxygen atoms involved in the process. The decrease in the proton transfer rate with increasing pressure reflects the dependence of the reaction on the solvent relaxation rate. We found that, for the relatively weak photoacids 2N68DS, 10-CPT, and 5CN2, the proton transfer rate constant increases by a factor of about 5-8 at a pressure of about 1.5 GPa. For a strong photoacid like DCN2, the rate increase was only by a factor of 2.     

Spectres infrarouges et raman du complexe solide MgBr22Et2O: Changement de phase et isomérie de rotation des molécules d'éther coordonnées

Infrared and Raman spectra of solid magnesium bromide dietherate MgBr₂2Et₂O have been studied at different temperatures between 300 and 90 °K in the 4000 to 30 cm^? range. Infrared spectra of this compound at various pressures up to 10 kbar have also been investigated. The MgBr₂2Et₂O crystal has two phases and there is a reversible transition between them. This phase transition concerns primarily the conformational change of (C₂Hg₅)₂O molecules and much less the arrangement of the ligands around the central magnesium atom. At room temperature or low pressure the GG conformation appears to be predominant while at low temperature or high pressure there is only TG (or TT) conformation of ether molecules. The enthalpy difference and transition temperature between the phases have been determined calorimetrically. An assignment of the spectra of both phases is given. The analysis of intramolecular vibrations gives information about the relative contribution of conformational change and coordination effect to the frequency shifts of (C₂H₅)₂O vibrations. The assignment of magnesium-ligand vibrations on the other hand allows us to estimate the Mg-Br and Mg-O force constants.     

On the nature of the surprisingly small (red) shift in the halothane...acetone complex.

The halothane???acetone and fluoroform???acetone complexes are studied using the second‐order Møller–Plesset (MP2) method with a cc‐pVTZ basis set and the density functional theory (DFT) method with a TZVP basis set. Whereas halothane exhibits a small red shift upon complexation, fluoroform shows a pronounced blue shift. To explain this difference in behavior, we perform symmetry‐adapted perturbation theory (SAPT) and natural bond orbital (NBO) analyses. Although the composition of the total stabilization energy of each complex is different, that alone does not provide a satisfactory explanation for the difference in the spectral shifts. This difference is interpreted as a result of the interplay of the hyperconjugation and rehybridization mechanisms. The small and surprising red shift of the C? H stretching frequency of halothane, which resulted from the complexation of this species with acetone,is explained by the compensation of the two above‐mentioned mechanisms. On the other hand, the fluoroform???acetone complex exhibits a blue shift of the C? H stretching frequency upon complexation, the most likely reason for this shift being a concerted occurrence of the hyperconjugation and rehybridization mechanisms. The calculated shift of the C? H stretching vibration frequencies of halothane (+27 cm^?1) agree with the experimental value of +5 cm^?1.     

Complexes of iron(III) salen and saloph Schiff bases with bridging dicarboxylic and tricarboxylic acids

Six new dinuclear or trinuclear FeIII complexes involving tetradentate Schiff bases N,N′-bis(salicylidene)ethylenediamine (salenH2) or bis(salicylidene)-o-phenylenediamine (salophH2) with 2,5-pyridinedicarboxylic acid, acetylenedicarboxylic acid or 1,3,5-benzenetricarboxylic acid have been synthesized and characterized by means of elemental analysis, i.r. spectroscopy, thermal analyses, conductivity measurements and variable-temperature magnetochemical measurements to the temperature of liquid nitrogen. The complexes can be characterized as high-spin distorted octahedral FeIII bridged by carboxylic acids. The dicarboxylic or tricarboxylic acids play a role as bridges for weak antiferromagnetic intramolecular exchange. The antiferromagnetic coupling parameters J vary in the -1.99 to -5.47cm-1 range for the dimers, whilst the values are -2.35 and -1.42cm-1 for the salen and saloph trimers, respectively. One complex, namely [{Fe(saloph)}2(2,5-dicarpy)]middot H2O, obeys the Curie-Weiss law. This revised version was published online in June 2006 with corrections to the Cover Date.     

Iminodiacetic acid functionalised monolithic silica chelating ion exchanger for rapid determination of alkaline earth metal ions in high ionic strength samples

Iminodiacetic acid has been covalently bonded to a bare silica monolith to produce the first reported high-performance monolithic chelating ion exchange column. Using the new column, separation and determination of traces of alkaline earth metal ions (low ppm) in high ionic strength samples (up to 2 M NaCl and KCl brines), could be achieved in under 40 s. At an eluent flow rate of 4 mL min(-1) retention time precision was < 1.2% (n = 9) for Mg(II) and Ca(II), with detector linearity (n = 5) over the range 2-10 mg L(-1) of between R2 = 0.985 and R2 = 0.995. In 1 M KCl and NaCl brine samples, detection limits of 0.2 mg L(-1) were possible.