On the condition number of matrices arising in the tikhonov regularization method

In this work, we consider the regularization method for linear ill-posed problems. For operators and approximating subspaces satisfying certain conditions and for a specific form of the regularization parameter, upper and lower bounds are given for the condition number of the corresponding discrete problem.     

X-ray evidence for the absence of 90°-domains in polycrystalline PbTiO3 thin films with grains below a critical size

3 thin films is systematically studied by using X-ray diffraction (XRD). The PbTiO₃ thin films with different average grain sizes were prepared on various substrates by a sol-gel process. The films on NaCl and fused glass are randomly grain-oriented, while those on (111)Pt/Ti/SiO₂/Si are highly {100} cubic index grain-oriented . It is found from the XRD patterns of the films on NaCl that with decreasing average grain size from 230 to 80 nm, the intensities of high h index (h>l) peaks (hkl), such as (100), (110), (200), (201), (210), (211), etc., decrease rapidly and ultimately disappear, whereas another set of peaks (lkh), including (001), (002), (102), (112), etc., are still intense. This interesting result suggests that at grain size below a certain critical size an increasing number of grains no longer show 90°-domains, which is confirmed by TEM observations. Meanwhile, X-ray evidence of such a grain-size-related absence of 90°-domains is also found for PbTiO₃ films on Pt(111) and fused-glass substrates. The volume fractions of single-domain grains (without 90°-domains) in the films are estimated from their XRD patterns. By combining SEM and TEM investigations, the critical grain size for the formation of 90°-domains is further determined to be near 200 nm. Received: 19 December 1996/Accepted: 24 March 1997     

Optical quenching of the magnetoplastic effect in NaCl crystals

Observations indicate that illuminating NaCl crystals by ultraviolet light (λ=350 nm) suppresses the magnetoplastic effect. The processes induced by illumination take place in a subsystem of point defects and are related to a change in the state of magnetically sensitive dislocation pinning sites. Fiz. Tverd. Tela (St. Petersburg) 39, 1389–1391 (1996)     

The oxidation of 3,3′-dimethoxy benzidine with potassium bromate in acidic solutions

The kinetics and mechanism of the oxidation of 3,3′-dimethoxybenzidine (oda, o-dianisidine) by potassium bromate in aqueous acidic medium were studied by monitoring the formation rate of the reaction product, 3,3′-dimethoxy 4,4′-diphenoquinone at 447 nm. The reaction is, first order with respect to both the substrate and oxidant, and second order with respect to H⁺. The oda: bromate stoichiometric ratio is 1:1. Plausible mechanism and rate laws are proposed accounting the experimental findings. Computer simulations were done using the proposed mechanism.     

Bromination of 4-substituted thiazolylhydrazones

4-Phenyl- and 4-methylthiazolylhydrazones were brominated in chloroform and acetic acid. The use of 1.5 equivalents of bromine led to the exclusive formation of the 5-halo derivatives. Effects of substituents on the thiazole ring and of the hydrazone fragment aromatic ring on the course of the reaction were discovered.Urals State Technical University-UPI, 620002 Yekaterinburg. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 7, pp. 991–995, July, 1995. Original article submitted March 1, 1995.     

Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.