Entgegnung zu den vorstehenden Bemerkungen von Herrn W. Kochanowsky

Ohne Zusammenfassung     

The small peptide-catalyzed direct asymmetric aldol reaction in water

Simple modular di- and tripeptides with a primary amine at the N-terminus catalyze the aqueous asymmetric aldol reaction between unmodified ketones and aldehydes to furnish the corresponding beta-hydroxy ketones with up to 86% ee in water and 99% ee in aqueous media.     

Experimental and computational studies of alkali-metal coinage-metal clusters

Coinage and alkali metal mixed clusters, M4Na- (M = Cu, Au) have been investigated experimentally using photoelectron spectroscopy and computationally at correlated ab initio levels. The related Cu4Li-, Ag4Li-, Ag4Na-, and Au4Li- clusters as well as the neutral Cu4Li2 and Cu4Na2 clusters have also been studied computationally. The calculations show that the two lowest isomers of the negatively charged clusters include a pyramidal C4v structure and a planar C2v species. For Cu4Li- and Cu4Na-, the C4v structure is calculated at correlated ab initio level to be 30.9 and 16.9 kJ/mol below the planar C2v isomer, whereas the planar isomers of Au4Li- and Au4Na- are found to be 29.7 and 49.4 kJ/mol below the pyramidal ones. For Ag4Li- and Ag4Na-, the pyramidal isomers are the lowest ones. Comparison of the calculated and measured photoelectron spectra of Cu4Na- and Au4Na- shows that the pyramidal Cu4Na- cluster of C4v symmetry and the planar Au4Na- of C2v symmetry are detected experimentally. Calculations of the magnetically induced current density in Cu4Li- and Cu4Li2 using the Gauge-Including Magnetically Induced Current (GIMIC) method show that strong ring currents are sustained mainly by the highest-occupied molecular orbital primarily derived from the Cu 4s. The GIMIC calculations thus show that the Cu4(2-) ring is -aromatic and that the d orbitals do not play any significant role for the electron delocalization effects. The present study does not support the notion that the square-planar Cu4(2-) is the first example of d-orbital aromatic molecules.     

Complexes of iron and cobalt with new tripodal amido-polyphosphine hybrid ligands

Divalent complexes of iron and cobalt with new, monoanionic tripodal amido-polyphosphine ligands have been thoroughly characterized, and XRD analysis reveals geometries that are distinct for this class of ligand.     

Evolutionary game theory in growing populations

Existing theoretical models of evolution focus on the relative fitness advantages of different mutants in a population while the dynamic behavior of the population size is mostly left unconsidered. We present here a generic stochastic model which combines the growth dynamics of the population and its internal evolution. Our model thereby accounts for the fact that both evolutionary and growth dynamics are based on individual reproduction events and hence are highly coupled and stochastic in nature. We exemplify our approach by studying the dilemma of cooperation in growing populations and show that genuinely stochastic events can ease the dilemma by leading to a transient but robust increase in cooperation.     

Expectation of the truncated randomly weighted sums with dominatedly varying summands

We consider the asymptotic behavior of the values P(S > x), E(S 1_{S>x}), and E(S | S > x). Here S = θ₁X₁ + θ₂X₂ + · · · + θ_nX_n is a randomly weighted sum of the basic random variables X₁,X₂, . . . , X_n, which follow some special dependence structure, and {θ₁, θ₂, . . . , θ_n} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X₁,X₂, . . .,X_n} and {θ₁, θ₂, . . . , θ_n} are supposed to be independent. We derive asymptotic formulas in the case where the number of summands n is fixed and the distributions of the basic random variables are dominatedly varying.We apply them to some values related to the risk measures of certain weighted sums.     

Bentley’s conjecture on popularity toplist turnover under random copying

Bentley et al. studied the turnover rate in popularity toplists in a ‘random copying’ model of cultural evolution. Based on simulations of a model with population size N, list length ? and invention rate μ, they conjectured a remarkably simple formula for the turnover rate: $\ell \sqrt{\mu}$ . Here we study an overlapping generations version of the random copying model, which can be interpreted as a random walk on the integer partitions of the population size. In this model we show that the conjectured formula, after a slight correction, holds asymptotically.