Vol. 44 No. 1 (221) (2010)

In the present paper the Weierstrass multipliers are generalized and the theorem of convergence of corresponding infinite products is proved. Representations of Blaschke type functions are established through Weierstrass type functions and Blaschke function. A way construction of new Blaschke type products is shown, and a method of proof of their convergence is developed. Some relations between Blaschke type products are established.

The paper suggests a factorization construction method for a class of second order matrix-functions. The left lower element of these matrix-functions may be represented as combinations of other three elements of matrix-function and two functions, meromorphic respectively in interior and exterior regions of the unit disk.

In the present paper the boundary value problem for the Sobolev type equation $$
\begin{cases}
\dfrac{\partial}{\partial t}L(u(t,x))+M(u(t,x))=f(t,x), \\ \qquad t>0,~~~x=(x_1,\ldots,x_n)\in \Omega\subset\mathbb{R}^n,\\
u\big|_{\partial\Omega}=0,\\
(Lu)(0,x)=g(z),\quad x\in\Omega,\end{cases}
$$

is considered, where L and M are second-order differential operators. It is proved that under some conditions this problem in the corresponding space has the unique solution.

In the present paper the Neumann problem for the equation $Lu \equiv (t^\alpha u^\alpha)''+  au +f,$ where $0\leq \alpha\leq 4,~t\in[0, b],~f\in L_2(0, b)$ , is considered. Firstly, the weighted Sobolev space $W^2_\alpha$ and generalized solution for the abovementioned equation are defined. Then, the existence and uniqueness of the generalized solution is studied, as well as the spectrum and the domain of corresponding operator are described.

Regression model generated by two-parametric distribution of moderate growth and arising in bioinformatics is considered. Consistency (in a weak sense) of least square estimates for parameters is proved. Distributions of least square estimates for parameters and of Gaussian noise variation estimate are obtained. Results may be used for statistical hypothesis testing with regard to parameters of model.

The paper presents buckling loads of isotropic rectangular plates with two simply supported opposite edges and the other two edges elastically supported against rotation. An analytical method that uses the Lévy solution method is employed to determine the buckling loads of the mentioned rectangular plates. The convergence and comparison of the results with those available in the literature indicate the accuracy and the validity of the proposed technique. Effects of the elastic restraint parameters on the mode shapes are illustrated in graphic forms.

In this paper the polymorphic lambda terms are considered, where no type information is provided for the variables. The aim of this work is to prove that presented typification algorithm [1] typifies such terms in most common way.

We study quantum effects of scalar fields in cosmological models of Friedman–Robertson–Walker with a power-low scale factor and spatial topology $R^2\time (S^l)^q$. Recurrent formulae are obtained for positive-frequency Wightman function, vacuum expectation values of the field squared and energy density.

Studying the spectral peculiarities of nonlinear-dispersive similariton, generated in single-mode optical fiber without gain (passive fiber), we reveal that the bandwidth of such a similariton is conditioned only by the initial pulse power. This property of nonlinear-dispersive similariton can be used for the pulse duration measurements at the femtosecond time scale, alternatively to the autocorrelation technique.

In this note two-parametric model, generated by distribution of moderate growth, is considered. The parametric function of unknown parameters, for which there is unbiased, effective estimate, is obtained. The estimate itself is built too.

In the paper a notion of ordinary theory is considered. It is proved that some systems of first order predicate calculus are ordinary. This property is used for a proof complexity comparison in the considered systems.

The paper is devoted to the experimental research of bilayer lipid membranes stability (BLM) at joint action of an electrostatic field and hydrostatic pressure. It is shown, that with increase in a potential difference on BLM the average time of a life of a membrane decreases. Presence of hydrostatic pressure increasing leads to additional reduction of average time of the membrane life.