Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.     

Asymptotics of the transition probabilities of the simple random walk on self-similar graphs

It is shown explicitly how self-similar graphs can be obtained as `blow-up' constructions of finite cell graphs . This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals.

For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph , starting at a vertex of the boundary of . It is proved that the expected number of returns to before hitting another vertex in the boundary coincides with the resistance scaling factor.

Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the -step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpinski graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the -step transition probabilities.

     

A nondestructive method for estimation of the fracture toughness of CrMoV rotor steels based on ultrasonic nonlinearity

The objective of this paper is to develop a nondestructive method for estimating the fracture toughness (K(IC)) of CrMoV steels used as the rotor material of steam turbines in power plants. To achieve this objective, a number of CrMoV steel samples were heat-treated, and the fracture appearance transition temperature (FATT) was determined as a function of aging time. Nonlinear ultrasonics was employed as the theoretical basis to explain the harmonic generation in a damaged material, and the nonlinearity parameter of the second harmonic wave was the experimental measure used to be correlated to the fracture toughness of the rotor steel. The nondestructive procedure for estimating the K(IC) consists of two steps. First, the correlations between the nonlinearity parameter and the FATT are sought. The FATT values are then used to estimate K(IC) using the K(IC) versus excess temperature (i.e., T-FATT) correlation that is available in the literature for CrMoV rotor steel.     

Nuclear hyperfine coupling interactions in the rotational spectra of the linear and bent isomers of HF-N2O

The rotational spectra of six isotopomers of the linear and bent isomers of HF-N₂O have been collected in the 7-18 GHz region with a Fourier transform microwave spectrometer. The nuclear hyperfine structure in the spectra produced by HF spin-spin coupling interaction and nuclear quadrupole coupling interactions due to the D nucleus of DF and the nuclei of N₂O have been resolved and analyzed. In the linear isomer, H in HF is bonded to the terminal N in N₂O. The NF bond lengths are 2.9808(2) Å for the HF-containing isotopomers and 2.9732(2) Å for the DF-containing isotopomers. The zero point angles are 23.1° for HF and 31-34° for N₂O. The hyperfine constants suggest that the HF bond is lengthened by 0.0105 Å upon complexation and that the electric field gradients of the two nitrogen nuclei in N₂O are perturbed differently in the complex. In the bent isomer, the hydrogen bond is formed between HF and O in N₂O. The intermolecular distances are 3.4942(2) Å for the HF-containing isotopomers and 3.4436(2) Å for the DF-containing isotopomers, with HF and N₂O forming angles of 34° and 46°, respectively, with the intermolecular axis. The nuclear quadrupole coupling constants of the two nitrogen nuclei do not indicate electric field gradient perturbation in this isomer.     

Phase transition and analysis of AC conductivity data of the (Cs)0.26(Rb)0.74H(SO4)0.89(SeO4)0.11 compound

The X-ray diffraction, vibrational and impedance spectroscopy studies of (Cs)_0.26(Rb)_0.74H(SO₄)_0.89(SeO₄)_0.11 (CsRbHSSe) new solid solution are presented. The title compound undergo a superionic phase transition (SPT) at This transition was confirmed by an abrupt increase of conductivity. The bulk impedance parameters of CsRbHSSe, RbH(SO₄)_0.81(SeO₄)_0.19 (RbHSSe) and CsH(SO₄)_0.76(SeO₄)_0.24 (CsHSSe) were determined from an analysis of AC conductivity data measured in a wide temperature range. The charge carriers concentration in the samples investigated has been evaluated using the Almond-West formalism and shown to be independent of temperature.     

The Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as spin chains, with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey tree. The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C [ ln² ]^–1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.     

Thioformaldehyde S-methylide and thioacetone S-methylide: an ab initio MO study of structure and cycloaddition reactivity

The mechanisms of cycloaddition of thioformaldehyde S-methylide and thioacetone S-methylide, as models for an alkyl-substituted ylide, to thioformaldehyde and thioacetone, as well as to ethene as a model for a C=C double bond have been studied by ab initio calculations. Restricted and unrestricted B3LYP/6-31G* calculations were performed for the geometries of ground states, transition structures, and intermediates. Although basis sets with more polarization functions were tested, the 6-31G* basis set was applied throughout. Single-point CASPT2 calculations are reported for analysis of the unsubstituted system. The stabilities of structures with high biradical character seem to be overestimated by DFT methods in comparison to CASPT2. The general trends of the results are independent of the level of theory. Thioformaldehyde adds to thioformaldehyde S-methylide without activation energy, and the activation energies for two-step biradical pathways to 1,3-dithiolane are low. C,S biradicals are more stable than C,C biradicals. The two-step cycloaddition is not competitive with the concerted cycloaddition. Methyl substitution in the 1,3-dipole and the dipolarophile does not change the mechanistic relationships. TSs for the concerted formation of the regioisomeric cycloadducts of thioacetone Smethylide and thioacetone were located. Concerted addition remains the preferred reaction. The reactivity of the C=S double bond is high relative to that of the C=C double bond.     

Relationship between the hydration degree of poly-N-isopropylacrylamide gel and activity of immobilized α-chymotrypsin

The influence of the pH, temperature, and dimethyl sulfoxide concentration on the hydration degree of the poly-N-isopropylacrylamide gel and the activity of -chymotrypsin immobilized into the polymer was studied. The behavior of more hydrophilic preparations based on polyacrylamide and copolymer of acrylamide and acrylic acid was studied for comparison. An increase in both the temperature and dimethyl sulfoxide content decreases the hydration of the poly-N-isopropylacrylamide, which correlates with a decrease in the activity of the immobilized enzyme. The use of substrates with different structures and an irreversible inhibitor proves that the change in the properties of -chymotrypsin immobilized into the poly-N-isopropylacrylamide gel is related to the change in the rate constants of enzymatic reactions. Comparison of all experimental data obtained suggested an opportunity of local interactions between the protein globule and polymeric chains with a change in the hydration degree of poly-N-isopropylacrylamide during its phase transition.