Minimal basis sets in calculations of intermolecular interaction energies

Several minimal (7, 3/3) Gaussian basis sets have been used to calculate the energies and some other properties of CH₄ and H₂O. Improved basis sets developed for these molecules have been extended to NH₃ and HF and employed to H₂CO and CH₃OH. Interaction energies between XH_n molecules have been calculated using the old and the new minimal basis sets. The results obtained with the new basis sets are comparable in accuracy to those calculated with significantly more extended basis sets involving polarization functions. Binding energies calculated using the counterpoise method are not much different for the new and the old minimal basis sets, and are likely to be more accurate than the results of much more extended calculations.     

Effect of oxidant on the reaction of [RhCl(cyclooctadiene)(phosphine)] with 1-hexene

Summary A promoting role of an oxidant, present in commercial 1-hexene, in the substitution of phosphine in the complex [RhCl(COD)(phosphine)] (1) where the phosphine is PPh₃ or 1/2 BPS-2 [bis(diphenylphosphinoethyl)tetra-methyldisiloxane] and COD=cycloocta-1,5-diene, has been detected and explained. When [oxidant]>[(1)] two reaction steps are distinguished: an oxidation of phosphine to phosphine oxide with generation of [RhCl(COD)], followed by its fast dimerization, and an oxidation of the dimer to Rh^III species. When [oxidant]<[(1)] the latter step is not observed and the reaction of [RhCl(COD)] with 1-hexene is favoured, particularly when an excess of phosphine (even at high oxidant concentration) is present. Most rate constants of the individual steps were evaluated.     

Quadratic harnesses, -commutations, and orthogonal martingale polynomials

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a -commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

     

Classical Versions of q-Gaussian Processes:¶Conditional Moments and Bell's Inequality

We show that classical processes corresponding to operators which satisfy a q-commutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell's inequality for their covariances can be extended from q=−1 to the entire range −1≤q < 1. Received: 18 September 2000 / Accepted: 17 November 2000     

Stretched exponential survival statistics for microorganisms in radiation field

The so-called concave or tailed survival curves are reported both for multi and single species bacterial populations. Taking as an example Bacillus pumilus, frequently encountered in decontamination studies, it is shown that the tailed survival curves are adequately described by stretched (0<<1) exponential SF=exp[−(D/D₀)], where SF denotes the fraction of species surviving the irradiation dose D, D₀ is the effective dose, and is the dispersion parameter interpreted phenomenologically in terms of radiation sensivity distribution for single species population under the given experimental conditions.     

Efficient optimization of the reward-risk ratio with polyhedral risk measures

In problems of portfolio selection the reward-risk ratio criterion is optimized to search for a risky portfolio offering the maximum increase of the mean return, compared to the risk-free investment opportunities. In the classical model, following Markowitz, the risk is measured by the variance thus representing the Sharpe ratio optimization and leading to the quadratic optimization problems. Several polyhedral risk measures, being linear programming (LP) computable in the case of discrete random variables represented by their realizations under specified scenarios, have been introduced and applied in portfolio optimization. The reward-risk ratio optimization with polyhedral risk measures can be transformed into LP formulations. The LP models typically contain the number of constraints proportional to the number of scenarios while the number of variables (matrix columns) proportional to the total of the number of scenarios and the number of instruments. Real-life financial decisions are usually based on more advanced simulation models employed for scenario generation where one may get several thousands scenarios. This may lead to the LP models with huge number of variables and constraints thus decreasing their computational efficiency and making them hardly solvable by general LP tools. We show that the computational efficiency can be then dramatically improved by alternative models based on the inverse ratio minimization and taking advantages of the LP duality. In the introduced models the number of structural constraints (matrix rows) is proportional to the number of instruments thus not affecting seriously the simplex method efficiency by the number of scenarios and therefore guaranteeing easy solvability.     

Carathéodory balls and norm balls of the domainsH n ={z ∈ ℂ n : |z 1|+…+|z n |<1}

In this short note we prove that the only Carathéodory balls in domains given in the title (forn≥2) which are the norm balls are the ones with the center at 0. It is a generalization of the result of B. Schwarz, who proved this theorem in casen=2.