Multiple local solutions to nonlinear control processes

We use the topological tool of Nielsen fixed-point theory to study the controllability of perturbed linear control processes whose control space can be reduced to a finite-dimensional one. These methods produce a lower bound on the number of controls that achieve a given target. We concentrate particularly on the case where the perturbation has sublinear growth, and we exhibit examples of multiple controls of various types that are detected by the Nielsen theory method.This research was partly supported by the Research Grant Teoria del Controllo dei Sistemi Dinamici, Ministero della Pubblica Istruzione, Roma, Italy.     

Are there more than five linearly-independent collision invariants for the Boltzmann equation?

The problem of finding the summational collision invariants for the Boltzmann equation is tackled with the aim of proving that the most general solution of the problem is not different from the standard one even when the equation defining a collision invariant is only satisfied almost everywhere inR ³×R ³×S ². The collision invariant is assumed to be in the Hilbert spaceH of the functions which are square integrable with respect to a Maxwellian weight.     

Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

A remark on the KAM theorem applied to a four-vortex system

We consider a planar four-vortex system with unit intensities and apply the KAM theorem for two-dimensional tori with fixed frequency. We obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number=(5—1)/2 and compare our result with numerical experiments.     

The velocity correlation function for the Lorentz gas

The results of variational solutions of the repeated ring and self-consistent repeated ring equations for the two-and three-dimensional overlapping Lorentz gas (LG), as formulated in a previous report, are presented. Calculations of the full velocity correlation function (VCF) for the 2D LG, including long-time tails, are compared with those from molecular dynamics. The trial functions chosen lead to predictions for the long-time tails that improve as the density of the scatterers is increased. At a value of 0.24 for* (= ², where is the density and the radius of scatterers), the self-consistent amplitudes of the long-time tail are within 40% of the molecular dynamics. A limited number of 3D results for the short-time behavior of the repeated ring VCF are presented. The 3D solutions agree with the molecular dynamics to within 10%.     

Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

Theoretical study of the reaction of Sc with SCO in gas phase

In order to elucidate the mechanism of reaction M⁺ + SCO, both triplet and singlet potential energy surfaces (PESs) for the reaction of Sc⁺ + SCO have been theoretically investigated using the DFT (B3LYP/6-311+G*) level of theory. The geometries for reactants, intermediates, transition states and products were completely optimized. All the transition states were verified by the vibrational analysis and the intrinsic reaction coordinate calculations. The involving potential energy curve-crossing dramatically affects reaction mechanism, reaction rate has been discussed, and the crossing points (CPs) have been localized by the approach suggested by Yoshizawa et al. The present results show that the reaction mechanism are insertion–elimination mechanism both along the C–S and C–O bond activation branches, but the C–S bond activation is much more favorable in energy than the C–O bond activation. All theoretical results not only support the existing conclusions inferred from early experiment, but also complement the pathway and mechanism for this reaction.     

General application of jäntti's method for the fast calculation of sorption equilibrium

On the basis of a molecular model for adsorption kinetics Jäntti introduced a method to calculate equilibriums shortly after a change of the pressure of the sorptive gas. In the present paper we show that this method is useful in many more situations than those intended originally.     

Do intramolecular halogen bonds exist? Ab initio calculations and crystal structures’ evidences

Cl···O intramolecular contacts are analysed here. Hence the Cambridge Structural Database (CSD) search was performed to find five-membered Cl–C=C–C=O pseudo-rings closed through Cl···O intramolecular contacts. Such rings were analysed and it was found that some of Cl···O distances are smaller than the corresponding sum of van der Waals radii. Additionally ab initio MP2/6-311++G(d,p) calculations and AIM analyses were carried out here on 3-chloropropenal and its fluoro derivatives to deepen the nature of intramolecular Cl···O interactions, some of findings may indicate that they are attractive and stabilizing ones. This paper is dedicated to Prof. T. M. Krygowski on the occasion of his 70th birthday