Estimates from Below for Lebesgue Constants

Estimates from below for the norms of linear means of multiple Fourier series are obtained. These means are given by some function λ and generalize the well-known Bochner-Riesz means. Sharpness of these estimates is established. The assumptions on λ are rather weak and of local character. Our results contain as particular cases a number of earlier published results. Proofs are based on the authors' new results on asymptotics of the Fourier transform of piecewise-smooth functions. Some applications of the results obtained are given, namely, orders of growth of the Lebesgue constants for "ovals" and "hyperbolic crosses" are evaluated and sharp conditions on the modulus of smoothness of a function are given, for this function to be approximated by the linear means.     

Investigation of complexes of DNA with synthetic peptide fragments of the N-end of histone H2B

The nature of the interaction of high- and low-molecular-weight DNAs (6×10⁶ and 3×10⁵ daltons) with synthetic oligopeptides of the N-end of histone H2B having the sequences 1–21, 1–10, 1–13, 11–21, and 14–21, differing in molecular weight and amino acid composition, as a function of the amount of peptide component in the complex and the ionic strength of the solution has been studied by the methods of UV and CD spectroscopy and the spectrophotometric analysis of melting curves. It has been shown that of all the peptides studied only the 1–21 peptide possesses the capacity of condensing DNA. This capacity depends on the amount of peptide component in the complex, the molecular weight of the DNA, and the ionic strength of the solution. The interaction with peptides under all the conditions studied, without changing the conformational parameters of the DNA, stabilizes its secondary structure in relation to the action of the temperature, which depends on the number of lysine residues in the peptides.     

Electromagnetic wave scattering by many small particles

Scattering of electromagnetic waves by many small particles of arbitrary shapes is reduced rigorously to solving linear algebraic system of equations bypassing the usual usage of integral equations. The matrix elements of this linear algebraic system have physical meaning. They are expressed in terms of the electric and magnetic polarizability tensors. Analytical formulas are given for calculation of these tensors with any desired accuracy for homogeneous bodies of arbitrary shapes. An idea to create a “smart” material by embedding many small particles in a given region is formulated.     

Acceleration of Convergence of A Continuous Analog of the Newton Method

Continuous Newton-type method is constructed which converges faster than a given exponential.     

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.     

Conformational features of peptide fragments of the C-end of histone H1

The conformational possibilities of four synthesized oligopeptides with the amino acid sequences 165–172, 173–184, 152–172, and 152–184 of the C-end of histone H1 of calf thymus above been studied in solution under various conditions by the method of circular dichroism.V. I. Nikitin Institute of Chemistry, Academy of Sciences of the Tadchik SSR, Dushanbe. Institute of Cytology, Academy of Sciences of the USSR, Leningrad. Translated from Khimiya Prirodnykh Soedinenii, No. 6, pp. 793–798, November–December, 1990.     

A new approach to the inverse scattering and spectral problems for the Sturm-Liouville equation

New proofs of the known uniqueness theorems for the one-dimensional inverse spectral and scattering problems are given. Proof of the invertibility of all of the steps in the inversion procedures of Gelfand-Levitan and Marchenko is given. The proposed method of investigation yields some new results, for example, a Marchenko-type equation at x = 0 which holds on the whole axis, rather than on a half-axis, as usual for the scattering theory on half-axis. It also yields a new method, shorter and simpler than earlier published, for proving that the potential in the class L_1,1, obtained by the Marchenko reconstruction procedure, generates the scattering data from which it was reconstructed.