Vol. 37 No. 3 (202) (2003)

For convolution transforms it has been received inversion formula, when $\varphi(x) \in L^{2}(-\infty, +\infty)$, and inversion functions $E(x)=\prod\limits^{\infty}_{k=1}\Big(1- \dfrac{s^2}{a_k^2}\Big)$  have complex roots satisfying to conditions  $\sum\limits_{k=1}^{\infty}<+\infty, |\arg a_k|\leq\dfrac{π}{4}$.

In $L^{2}(\mathbb{R})$ space an $m\geq 2$, linear order self-adjoint differential operator is observed the coefficients of which have precise behaviour in infinity. The operator’s point spectrum is examined. Particularly the limitation of point spectrum and the non-infinity of boundary points set are proved.

The Dirichlet problem is observed by different authors both for one equation and for systems of equations. In these authors’ articles the boundary function is continuous or has weak singularity (integral singularity). This article observe the case where boundary function may also have not weak singularity. In $M_D(х_1,х_2,\ldots,х_h,\infty; l_1,l_2,\ldots,l_h,l_{h+1})$ class is observed $$\begin{cases} A\dfrac{\partial^2u}{\partial x^2} +2B\dfrac{\partial^2u}{\partial x\partial y}+C\dfrac{\partial^2u}{\partial y^2}=0,\\ u(x,0)=f(x), x\neq х_1,х_2,\ldots,х_h, \end{cases}$$ the boundary problem, where $f(x)\in N_\Gamma (х_1,х_2,\ldots,х_h,\infty; l_1,l_2,\ldots,l_h,l_{h+1})$ . It is proved that the problem has a solution and one solution is found.

In the present paper an extension of the well-known Little's formula in the terms of random variables is established. The interpretation of obtained formula is given. The result allows application to linear-parametrical models in order to prove some conservation laws.

The study is about the $C_i$ periodic word construction of the free Burnsaide group. With the obtained results we can build other however bigger needed $C_i$  periodic words.

The work is devoted to ternary hyperidentities of associativity, which are determined by the equality $((x, y, z), u, v) = (x, y, (z, u v))$. We get the following three hyperidentities: $$ X(Y(x, y, z), u, v) = Y(x, y, X(z, u, v)),  $$ $$X(X(x, y, z), u, v) = Y(x, y, Y(z, u, v)), $$ $$X(Y (x, y, z), u, v) = X (x, y, Y(z, u, v)).$$ The criteria of realization are proved for each of them in the reversible algebras.

The problem of stability of system for second order of non-linear differential equations in critical case has been considered, when the characteristic equation, corresponding to linear approximation of system, has pair imagining roots. Sufficient conditions have been obtained in case of which the trivial solution of the considered system is either asymptotic stable or non-stable for acting force.

Non-steady-state flow of real incompressible liquid in a cylindric pipe with porous lateral surfaces is investigated. Nonstationarity of the problem is conditioned by non-steady-state movement of the fluid which is expressed in differential equations of motion. The problem is solved by the operational calculus method by use of Laplace double integral transformations on spatial and time coordinates. Characteristics of the flow motion are determined as functions of spatial and time coordinates.

The confinement potential and charge distribution in the abrupt contact of two two-dimensional systems is considered theoretically. It is shown that a quasi- one dimensional electron channel can be formed near the interface between two quantum-size semiconductor films with different thickness and doping levels. Using the structure with double such “dimensional” heterojunctions a rectangular potential well confining one-dimensional electrons (quantum wire) can be realized.

In the present paper the domain measures and domain border measures are determined. For this purpose, the connection between the disk-shape domain measures and the phase difference between the initial rays and those passed through domain is used. The influence of salts and other mixtures on domain measures is defined, which is expressed by the refraction index change.

In connection with the problems of supplement in this work the generalization of Ailar’s $\varphi(m)$  functions is introduced and a formula is got for the definition of the value of this generalized function.