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

DOI: https://doi.org/10.46991/PYSUA.2000.34.1
Published: 2000-04-26

Mathematics

  • Mathematics

    ON A SYSTEM OF FUNCTIONS GENERATING CLASSICAL ORTHOGONAL SYSTEMS OF ALGEBRAIC POLYNOMIALS

    H. V. Badalyan
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    Abstract

    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 $x\rightarrow0+, x\rightarrow a–$.

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  • Mathematics

    ATTRACTORS OF THE DEGENERATING EVOLUTIONARY EQUATIONS

    A. V. Tsutsulian
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    Abstract

    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.

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  • Mathematics

    ON THE SPECTRAL THEOREM FOR A BANACH REPRESENTATION OF A COMPACT GROUP GENERATED BY DELSARTE SHIFTS

    M. I. Karakhanyan
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    Abstract

    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.

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  • Mathematics

    HYPERIDENTITIES OF LEFT AND RIGHT DISTRIBUTIVITIES IN VARIETIES OF SEMI­GROUPS

    Yu. M. Movsisyan
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    Abstract

    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.

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  • Mathematics

    ON APPROXIMATION OF THE SOLUTIONS OF BOUNDARY-VALUE PROBLEMS IN UNBOUNDED DOMAINS

    G. V. Dallakyan
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    Abstract

    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 ur 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.

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