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.     

Erratum: Preparation of size‐controlled In2O3 nanoparticles

The article referenced above was first published online on 7 August 2007 with incorrect pagination; the pagination has now been corrected online and in print. Copyright © 2007 John Wiley & Sons, Ltd.     

Electrooxidation of carbo/thiocarbohydrazide and their hydrazone derivatives at a glassy carbon electrode

Electrochemical oxidation of thio/carbohydrazide and their hydrazone derivatives Benzaldehyde thiocarbohydrazone [BTCH] diacetylene thiocarbohydrazone [DATCH] have been studied in Brit-ton Robinson buffer in aqueous and nonaqueous media at a glassy carbon electrode. The effects of pH, sweep rate, concentration, temperature and surfactants have been studied. The complex bis (carbo/thiocabohydrazide) Zn(II) chloride was also subjected to voltammetric analysis in order to understand the reactivity both in free and metal bound states. The reaction conditions were optimized for the determination of above compounds in micrograms quantities by differential pulse voltammetry, analytical utility of this investigation is also highlighted.     

Tuning size and catalytic activity of nano-clusters of cobalt oxide

Cobalt oxides were prepared by three different methods: (1) by reacting cobalt nitrate with oxalic acid, (2) co-precipitating cobalt nitrate with sodium carbonate, and (3) using sodium dodecyl sulphate as organic surfactant. All three samples were characterized before and after calcination by solvent extraction and the resulting products examined by IR spectroscopy. In the case of method 3, the removal of surfactant was followed by TGA studies. Products from all three methods were identified by XRD. Peaks in low angle XRD indicate the porous nature of the oxides. The morphology of the pores was studied by transmission electron microscopy. Some irregular pore structures were obtained for samples from methods 1 and 2, with an average size of 4–6 nm. Only the product from method 3 using SDS as template showed ordered structure and optimum size, and Brunauer-Emmet-Teller surface areas of the as-prepared, as well as the treated samples, exhibited H3 type hysteresis. The samples from the three methods were used as catalysts in the oxidation reaction of cyclohexane under mild conditions and the catalytic efficiency of the cobalt oxide was comparable with mesoporous cobalt oxides.     

Protonation constant of salicylidene (N-benzoyl)glycyl hydrazone and its coordination behaviour towards some bivalent metal ions

Protonation constant of an unsymmetrical Schiff base, salicylidene(N-benzoyl)glycyl hydrazone (SalBzGH), and formation constants of its complexes have been determined potentiometrically at different temperatures in aqueous dioxane medium. Complexes of SalBzGH with VO(IV), Mn(II), Co(II), Ni(II), Cu(II), Zn(II), Cd(II) and Hg(II) have been prepared. Elemental analyses, pH-metric, molar conductance, magnetic susceptibility, electronic, IR, ESR, XRD (powder) and NMR studies have been carried out to study the coordination behaviour of SalBzGH toward these metal ions. pH-metric and 1H NMR studies show the presence of two dissociable protons in the ligand. IR and NMR spectra suggest the tridentate nature of the ligand, coordinating as a uninegative species in the Mn(II) complex and as a dinegative species in all the other complexes. Presence of two different conformers of the ligand at room temperature and stabilization of a single conformer upon complex formation have been established from¹H NMR spectra of the metal-free ligand, Zn(II) and Hg(II) complexes recorded at 296 K. Electronic and ESR spectra indicate highly distorted tetragonal geometry for VO(IV) and Cu(II) complexes. XRD powder patterns of the Zn(II) complexes are indexed for an orthorhombic crystal system.     

Investigation of surface topography, morphology and structure of amorphous carbon films by AFM and TEM

Morphology and structure of amorphous carbon films deposited with a pulsed arc source (LASER-ARC) have been studied using microscopical methods (SEM, TEM and AFM), electron diffraction and spectroscopical investigation (EELS). The parameters of the arc source and the deposition conditions (substrate temperature) influence morphology and structure of deposited amorphous carbon films. Especially the incorporation and growth of particles, embedded in the film have been investigated. By particle analysis using an optical microscope a majority of particles that is smaller than 500 nm has been determined. The morphology has been also demonstrated similar by AFM and TEM images. Their number and size of particles is strongly influenced by the deposition temperature. The structure of amorphous film is characterized by the EELS-spectra, but the particle structure was not detectable.     

Spectroscopic investigation of Pt—methanol interface in perchloric acid medium by sum-frequency generation

We recorded vibrational spectra of the CO poison resulting from the electrochemical decomposition of CH₃OH on platinum by sum-frequency generation using the infrared free electron laser CLIO. At high CH₃OH concentration (0.1 M) in the electrolyte, both multi-bonded and linearly bonded CO are equally present. At lower CH₃OH concentration (0.004 M), the multi-bonded species is predominant. The CO poison still remains on the surface at potentials (0.05 V/NHE) close to the hydrogen evolution onset.     

Thin-layer chromatography of radioactively labelled cholesterol and precursors from biological material

Summary The investigation methods of the action of xenobiotics on sterol biosynthesis from ¹⁴C-acetate in rat hepatocyte cultures can be developed, with regard to extraction using Extrelut and the separation of the sterol pattern by thin-layer chromatography, in such a way that they are suitable for wider application, e.g., screening. Good visualisation and recognition of changes in the sterol pattern are possible using autoradiography of the thin-layer chromatogram.

---

Dünnschicht-Chromatographie von radioaktiv markiertem Cholesterin und Vorstufen aus biologischem Material. Eine einfache und empfindliche Methode zur Untersuchung einer Beeinflussung des Sterolbiosynthesewegs

---

Dedicated to Dr. E. Roesch on occasion of his 65th birthday