Methods with Successive and Parallel Approximations of Inverse Operator for the Nonlinear Least Squares Problem

For solving the nonlinear least-square problem, we propose iterative methods that use successive and parallel approximations of the inverse operator instead of solving a linear system of equations. The convergence order as well as the convergence radius of the proposed methods are studied. Finally, we carry out numerical experiments on a set of test problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Slipping–rolling transitions of a body with two contact points

Nonlinear Dynamics - In this paper, the general kinematics and dynamics of a rigid body is analysed, which is in contact with two rigid surfaces in the presence of dry friction. Due to the rolling...     

IR spectra of photopolymerized C60 films. Experimental and density functional theory study

IR spectra of photopolymerized fullerene films obtained by simultaneous deposition and UV irradiation were measured in the range of 1500-450 cm(-1). The degree of the polymerization of the C60 films was estimated to be about 95%. To assist the assignment of the experimental IR spectra of the films, quantum chemical calculations of the equilibrium structures of the C60 dimers and trimers were performed at the DFT(B3LYP)/3-21G level of theory. Next, IR frequencies and intensities for those structures were calculated. For the five-trimer structures found in the calculations, the relative stabilities were determined at the B3LYP/4-31G and B3LYP/6-31G levels and used to select the lowest-energy trimers, which are Trimer A (angle between monomer centers is 90 degrees ) and Trimer B (angle between monomer centers is 120 degrees). Next, the IR spectra of the polymerized fullerene films were compared with the calculated frequencies of the lowest-energy dimer and the two lowest-energy trimers. On the basis of this analysis and on the comparison of the film spectra with the IR spectra of the C60 dimer and trimer spectra obtained by other methods, it was shown that the main components of the films are C60 dimers and the orthorhombic (O) polymer phase. The tetragonal (T) and rhombohedral (R) polymers, as well as small amounts of monomers, were also found. Although vibrational frequencies of different C60 phases are similar in most cases, we found several unique spectral features of the C60 dimer and other polymers that may be used to determine the composition of the polymerized C60 film.     

Permanganate oxidation of alkenes. Substituent and solvent effects. Difficulties with MP2 calculations

The permanganate oxidation of alkenes has been studied both experimentally and computationally. Transition state structures were located for the reaction of permanganate ion with a variety of monosubstituted alkenes at the B3LYP/6-311++G** level. Although the calculated activation energy for the reaction with ethene was reasonable, the calculated effect of substituents, based on the energies of the reactants, was much larger than that experimentally found. This was shown to be due to the formation of an intermediate charge-dipole complex which led to the transition state. Reaction field calculations found the complex to disappear in a high dielectric constant medium, and the range of activation energies for the reaction in solution became quite small. MP2 calculations were carried out in order to have a comparison with the DFT results. MP2-MP4 gave unusual results for calculations on permanganate ion as well as chromate ion and iron tetraoxide. They also gave markedly unreasonable results for the activation energy of the reaction of permanganate with ethane. CCSD/6-311++G** calculations gave satisfactory results for permanganate ion and chromate ion. At this level of theory, the reaction of permanganate with ethene was found to have a very early transition state, when the bond lengths of the reactants just began to change. The reaction was calculated to be very exothermic (-69 kcal/mol), and this was confirmed via calorimetry. The rates of permanganate oxidation of allyl alcohol and acrylonitrile were determined, and they had similar reactivities. The kinetics and the products of the reaction of permanganate with crotonate ion were examined in some detail.     

Multiple solutions for discrete boundary value problems

A recent multiplicity theorem for the critical points of a functional defined on a finite-dimensional Hilbert space, established by Ricceri, is extended. An application to Dirichlet boundary value problems for difference equations involving the discrete p-Laplacian operator is presented.