Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Beyond linear least squares regression

Regression is a collection of statistical methods that are used to study relationships among predictor and response variables. In addition to the most popular linear model, solved by least squares, several other techniques have found an application in analytical chemistry. Biased methods such as stepwise regression, ridge regression, principal components regression, and partial least squares regression are especially useful in cases of poorly or underdetermined systems with collinearity. When structural and/or distributional assumptions associated with linear least squares are violated, nonlinear regression, robust regression or generalized least squares estimators may offer potential remedies.     

Regulating complex dynamics in firms and economic systems

The decisions of economic firms are often guided by simple routines. We demonstrate that such a routine may generate suboptimal chaotic dynamics, where the suboptimality is due to the fact that decision outcomes are in regions of low performance. We discuss a simple, but effective, method to regulate such dynamics in order to improve the performance. The method works as follows: In a first step, from time series information one has to identify critical starting areas which may lead to a crash or an outcome with low performance. In a second step, one has to perturb the system when it enters these critical areas. Due to sensitive dependence on initial conditions, already minor interventions suffice to prevent harmful events and to obtain better results.     

Can density functional theory (DFT) be used as an aid to a deeper understanding of tandem mass spectrometric fragmentation pathways?

Prediction of tandem mass spectrometric (MS/MS) fragmentation for non‐peptidic molecules based on structure is of immense interest to the mass spectrometrist. If a reliable approach to MS/MS prediction could be achieved its impact within the pharmaceutical industry could be immense. Many publications have stressed that the fragmentation of a molecular ion or protonated molecule is a complex process that depends on many parameters, making prediction difficult. Commercial prediction software relies on a collection of general heuristic rules of fragmentation, which involve cleaving every bond in the structure to produce a list of ‘expected’ masses which can be compared with the experimental data. These approaches do not take into account the thermodynamic or molecular orbital effects that impact on the molecule at the point of protonation which could influence the potential sites of bond cleavage based on the structural motif. A series of compounds have been studied by examining the experimentally derived high‐resolution MS/MS data and comparing it with the in silico modelling of the neutral and protonated structures. The effect that protonation at specific sites can have on the bond lengths has also been determined. We have calculated the thermodynamically most stable protonated species and have observed how that information can help predict the cleavage site for that ion. The data have shown that this use of in silico techniques could be a possible way to predict MS/MS spectra. Copyright © 2009 John Wiley & Sons, Ltd.     

Gold Catalysis: In Situ EXAFS Study of Homogeneous Oxidative Esterification

Gold complexes were prepared and investigated as catalysts for the oxidative esterification of aldehydes. Stabilisation by pyridine ligands gave good conversions and the in situ extended X‐ray absorption fine structure (EXAFS) study of the reactions indicated that the reaction mixtures contained only mononuclear gold species. Thus, this is the first proof for a homogeneous gold‐catalysed oxidation reaction; the presence of nanoparticles could be excluded experimentally.