Necessary conditions for the stability of difference schemes

For difference equations of the form uⁿ⁺=Auⁿ+f ⁿh, n ≤ T/h the necessary condition for stability due to von Neumann is well known; this condition is expressed in terms of the spectrum of the operator A: r(A)≤1+ch. In this note, for a certain class of difference equations, we express this condition in terms of the spectral radius of the symbol of the operator A.     

Reactions with electron transfer to N-acylheteroaromatic cations

The behavior of some benzoquinolines and imidazoles in reactions with acylating agents in the presence of zinc dust was investigated. The possibility of free-radical hetarylation under these conditions of organic compounds by the intermediately formed N-acylheteroaromatic radicals was established. The structure of 2,2-diacetyl-1,1,2,2-tetrahydro-1,1-diisoquinolyl was determined by means of x-ray diffraction analysis.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 2, pp. 229–237, February, 1977.     

Separation of water-soluble vitamins by reversed-phase high performance liquid chromatography with ultra-violet detection: application to polyvitaminated premixes

Nine water-soluble vitamins: [thiamine (B1), ascorbic acid (C), nicotinamide (PP), pyridoxine (B6), calcium pantothenate (B5), folic acid (B9), cyanocobalamin (B12), riboflavin (B2) and biotin (B8)] were separated on a YMC-Pack Pro C18 column (250 mm x 4.6 mm, 5 microm particle size) in a single run with a gradient elution of mobile phase consisting of 0.025% trifluoroacetic acid pH 2.6 (solvent A) and acetonitrile (solvent B). The separation was achieved within 17 min with a flow rate of 0.8 ml min(-1) and the detection was performed at two wavelengths (210 and 275 nm). The calibration graphs plotted with six concentrations of each vitamin were linear with a regression coefficient R2 > 0.995. The method was applied for the quantification of vitamins B1, C, PP, B6, B5, B9 B2 and B8 in polyvitaminated premixes (premixes) used for the fortification of infant nutrition products. The sample preparation involves an aqueous extraction of vitamins and two different samples dilution were used prior the LC-analysis. The specificity of the method was demonstrated by the retention characteristics, UV spectra and by comparing the peak purity with the standard of each vitamin. The repeatability of the method was evaluated at different level of concentrations on 12 premixes and the coefficients of variation (CVr) were below 6.5%. The values of the intermediate precision (CV1) were below 9.6% (n = 6). The concentrations of vitamins found in premixes with our method were comparable to the declared values, since no bias was found between the two sets of results at 95% confidence. The simplicity of the procedure should make it highly desirable for quality control of premixes in the food industry.     

Theoretical investigation of the structure of X-ray photoelectron spectra of crystals

In the one-electron approximation general relations are obtained which determine the X-ray photoemission intensity from mono- and poly-crystalline solids. To describe the excited electron states plane waves and orthogonalized plane waves have been used. It is shown that consideration of orthogonalization terms is important when photoionization cross-sections of valence electrons are to be calculated. Results are discussed using the examples of transition metals, noble metals, semiconductors and dielectrics possessing the diamond and zinc-blende structure. The spectra from poly-crystals and single crystals of transition metals can differ markedly, but in the case of dielectrics the difference is small. Calculated X-ray photoelectron spectra of crystals are in good agreement with experimental data. To obtain detailed information of the spectrum structure, especially when transition metals are considered, the many-electron effects must be taken into account.     

Scattering at a strongly singular potential

Hubert's equation for the resolvent and the maximum principle for the equation u + V(x)u=0 is used to construct a self-adjoint extension of the operator + V(x) with a positive singular potential V(x). The properties of this operator are investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 217–228, August, 1970.In conclusion I wish to thank sincerely A. N. Tikhonov for his interest in the work, and V. A. Il'in, V. P. Maclov, A. A. Samarskii, A. G. Sveshinkov, I. A. Shishmarev, and especially A. A. Arsen'ev for the interesting and informative discussions we had concerning this work.