Vol. 43 No. 2 (219) (2009)

In the present paper the class of semi-analytical functions in the polydisk $U^n\subset\mathbb{C}^n$ is introduced. This class is an extension of the set of holomorphic functions. For $n=1$ the concept of semi-analyticity coincides with analyticity. The Dirichlet problem with values given on the distinguished boundary of the polydisk always has a solution in the set of real parts of semi-analytical functions. Therefore, to investigate semi-analytical functions one can apply the potential theory methods, like one does it for the one-dimensional case. In the present paper the Schwarz type integral representation for the above-mentioned functions is obtained.

The paper discusses a homogeneous one-dimensional pseudodifferential equation with a symbol of the form

$$A(x, \xi)=A_0(\xi)+\sum\limits_{k=1}^N th\frac{\pi}{\alpha}\left (x-\lambda_k+i\frac{\alpha\beta}{2}\right )A_k(\xi)$$

$$(x, \xi, \lambda \in \mathbb{R},  \alpha>0,  -1<\beta<1,  k=1,2,…,N ), $$

where  $A_k(\xi)  ( k=0,1,...,N)$  are locally integrable functions from class of symbols of non-negative order r . The method of bringing the pseudodifferential equation to a system of onedimensional singular integral equations with Cauchy’s kernel is proposed.

The present paper studies existence and uniqueness of solution of initial boundary value problems for the non-linear degenerate pseudoparabolic equations. It is proved that if operators L and M satisfy certain conditions, the problem has a unique solution in corresponding functional spaces.

In this paper we consider the correctness of Lagrange bivariate interpolation problem, where interpolation parameters are integrals over certain bounded plane regions. Here we study the case of bivariate polynomials of degree not exceeding one.

The bivariate interpolation problem, where the interpolation parameters are integrals over bounded regions, is considered in this paper. H. Hakopian posed a hypothesis for this problem in the case, when regions are obtained from intersection of lines in general position [2]. Till now the hypothesis is proved for polynomials of degree ≤1 . In this paper we bring a new proof. Meanwhile we solve the problem in more general setting – in the case of arbitrary regions.

In the present paper generalizations of classical Pleijel identities are obtained. We refer these identities as Pleijel type identities. Particular cases of these identities are proved in [1], [3] and [5].

In this paper we construct infinite order automorphisms of free periodic groups B (m, n)  of sufficiently large period n with m ≥ 2 generators. From the obtained results it follows that the quotient group of the group       Aut(B (m, n)) with respect to normal subgroup of inner automorphisms is infinite.

In this work we discuss level placement (numeration, arrangement) by height optimal algorithms for transitive oriented and bipartite oriented graphs. There are described three definitions of the oriented graph, and for those three definitions it is solved the level placement problem for transitive oriented graph. The problem of level placement of bipartite oriented graph is solved by the linear complexity algorithm, whereas the problems of level placement of transitive oriented graph are solved by the quadratic complexity algorithms.

An imaging technique has been created (using a focused He-Ne laser beam as a probing signal) capable of imaging the grain structure of HTS thin films with 2–3 μ spatial resolution. It is based on detection of an inductance change of a single-layer flat pick-up coil, placed at the face of the specimen. This leads to frequency changes of a stable tunnel diode oscillator. Test device enabled 2D-mapping of the grain structure of the bridge-shaped YBaCuO film. Basically, the method is capable of imaging fine peculiarities of normal-metallic to superconductive phase transition and 2D-current distribution, as well as may identify localized defects in thin HTS-materials with sub-μ spatial resolution, using non-bolometric response. However, the achieved 2–3 μ resolution of a bolometric nature (in a given device with ∼3 mm-size coil) is limited and depends on how narrow is possible to focus the probing beam, while the own spatial resolution of the tested flat-coil technique is better than 0.1 μ, and can be improved by 1–2 orders of the value by reducing pick-up coil size.

UV-VIS spectrophotometry and circular dichroism (CD) methods were applied to study the binding of DNA with three novel Ag-containing porphyrins meso-tetra(4N-allylpyridyl) (AgTAlPyP4), meso-tetra(4N-butylpyridyl) (AgTButPyP4) and meso-tetra(4N-oxyethylpyridyl) (AgTOEtPyP4) at high ionic strength. The binding constant K_b and stoichiometry n were determined from binding isotherms for each ligand–DNA complex. The Gibbs free energy, enthalpy and entropy of binding also were calculated using the values of K_b . Obtained results show that these porphyrins associate with the duplex DNA via outside binding mode.

In this work the isotherms of particle adsorption on membranes are evaluated theoretically. The changes of adsorption center structure and the process of particle adsorption are considered in a consistent way. It is shown that the account of potential well deformation at adsorption process may result in qualitatively different regimes of adsorption. Dependence on the parameters of particle interaction with the adsorption center allows to obtain either Longmuir or S-shaped isotherms of adsorption.

In the present paper the asymptotic variance of the classical Von-Neiman–Fuglede Theorem for elements of the complex Banach algebra is extended.