Vol. 34 No. 1 (192) (2000)

In contrast to the two well-known methods for introducing classical orthogonal systems of algebraic polynomials (see [1--5], respectively), this paper proposes a completely different method for introducing these same polynomials. At the same time, integral representations for them that are different from the known ones are also given. The latter, in comparison with the Rodrigues formulas known in the literature, are simple and, what is more important, the integrands are not multivalued, but single-valued. In this paper it is proved that the system of functions $$I^{(\alpha,\beta)}_{n,n_1,a}=\frac{{\ae}_{n,n_1,a}}{2\pi_i}\int\limits_C\frac{(1-\frac{x}{a})^{n_1+\zeta}(\frac{x}{a})^{-\zeta}}{\Gamma(n_1+\zeta+\alpha+1)\Gamma(-\zeta+\beta+1)}\cdot\frac{d\zeta}{\prot\limits^m_{v=0}(\zeta+v)},$$ where $n_1\geq n\geq 0$ are integers, $\alpha>-1, \beta>-1, a>0$ are arbitrary numbers, $x\in(0,a)$, the simple contour $C$ encloses neighborhoods of points $0,-1,-2,...,-n, {\ae}{n,n_1,a}=\Gamma(n_1+n+\alpha+\beta=2),$ generates the above-mentioned orthogonal polynomials, namely: for $n_1=n, a=1$ we obtain the Jacobi polynomials, and for $n_1=a\rightarrow\infty$ -- the Laguerre polynomials $L^{(\beta)}_n(x),$ multiplied by $\exp^{-x}$. At the endpoints $[0,a] \mathrm{Y}\limits^{(\alpha, \beta)_{n,n_1,a}(x)} $ is understood in the sense of $x\rightarrow0+, x\rightarrow a–$.

This article proves the existence of semigroup attractors generated by mixed problems for degenerate second-order evolution equations, where degeneration can occur either on the entire boundary or on a part of it. The compactness of the attractors in the space L₂ is established, and a Lyapunov function is constructed to elucidate their structure.

This paper is devoted to the spectral theorem for a Banach representation of a compact group G in a Banach space X generated by Delsarte shifts.

An equality in a semigroup variety is called a superidentity if, substituting any term of the variety into the functional symbols of this equality, we obtain identities. Accordingly, an equality in a semigroup variety is called a presuperidentity if, substituting any term of the variety, excluding projection terms, we obtain identities into the functional symbols of this equality. In this paper, satisfiability criteria for superidentities and presuperidentities of left and right distributivities of various ranks were found.

In this paper, it is proved that the solution of the boundary value problem in an unbounded domain $\Omega$ can be obtained as the limit as $r \rightarrow \infty$ of the solution u_r of the boundary value problem in a bounded domain $\Omega \cap \Omega B_ {r, \mu$, where $B_ {r, \mu}=\left\{x; \big |x \big |_\mu < r\right\}$ is a generalized ball. A similar result is obtained for solutions of problems in a half-space.

This paper presents a solution to a plane elasticity problem for a cylindrically orthotropic rectangular plate with displacements specified on one edge and stresses on the other three. A small physical parameter is introduced, and the stress function is considered in the problem as a power of this parameter. In recurrent boundary value problems, the stress function is represented as a Fourier series. Infinite systems of linear algebraic equations are obtained to determine the expansion coefficients. The solution to the problem for specific parameter values is reduced to numerical results, allowing one to study the nature of the stress distribution near the embedment.

We study the problem of optimally stabilizing the vibrations of a rectangular plate, one side of which is rigidly fixed, while the others are free. The plate is stabilized using a control force applied to its upper surface. The problem is solved using beam functions. An optimal control action is obtained that minimizes the objective functional, which represents the total energy.

Using the formulas of the theory of radio emission of pulsars given in the review [1], based on the available observational data, the radiances of pulsars and the magnetic moments of 491 neutron stars were calculated.

This paper examines the transmission and reflection of light under normal incidence on a layer of a medium with a spiral periodic structure, the axis of which is perpendicular to the boundary surfaces. The influence of changes in various medium parameters (layer thickness, real and imaginary parts of the average permittivity, and dielectric anisotropy) on previously discovered effects is studied. The characteristics of anomalously strong absorption of radiation in periodic media near the diffraction reflection region are investigated. The physical mechanisms underlying the observed patterns are discussed.