Serdica Mathematical Journal

Volume 24, Number 3-4, 1998

C O N T E N T S

• Iliev, A. I. Generalization of Ehrlich-Kjurkchiev method for multiple roots of algebraic equations (pp. 215-224)
• Yordanov, R. G. Dubrovin type equations for completely integrable systems associated with a polynomial pencil (pp. 225-256)
• Zagorodniuk, S. On a five-diagonal Jacobi matrices and orthogonal polynomials on rays in the complex plane (pp. 257-282)
• Teleiko, A. On extension of functors onto the Kleisli category of the inclusion hyperspace monad (pp. 283-288)
• Amara, Ch. Directions de majoration d'une fonction quasiconvexe et applications (pp. 289-306)
• Cauty, R. La classe borélienne ne détermine pas le type topologique de C_p(X) (pp. 307-318)
• Trifonov, O. Integer points close to a smooth curve (pp. 319-338)
• Dimkov, G., J. Dziok. Generalized problem of sratlikeness for products of p-valent starlike functions (pp. 339-344)

A B S T R A C T S

GENERALIZATION OF EHRLICH-KJURKCHIEV METHOD FOR MULTIPLE ROOTS OF ALGEGRAIC EQUATIONS
Anton I. Iliev aii@ulcc.uni-plovdiv.bg

1991 Mathematics Subject Classification: 61H05. Key words: simultaneous roots finding, quaternary convergence.

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.

DUBROVIN TYPE EQUATIONS FOR COMPLETELY INTEGRABLE SYSTEMS ASSOCIATED WITH A POLYNOMIAL PENCIL
Russi G. Yordanov

1991 Mathematics Subject Classification: 35Q53, 35Q58. Key words: N-soliton solution, N-gap solution, KdV, polynomial pencil.

Dubrovin type equations for the N-gap solution of a completely integrable system associated with a polynomial pencil is constructed and then integrated to a system of functional equations. The approach used to derive those results is a generalization of the familiar process of finding the 1-soliton (1-gap) solution by integrating the ODE obtained from the soliton equation via the substitution u = u(x+ lt).

ON A FIVE-DIAGONAL JACOBI MATRICES AND ORTHOGONAL POLYNOMIALS ON RAYS IN THE COMPLEX PLANE
S. Zagorodniuk

1991 Mathematics Subject Classification: 47A57, 34L05. Key words: moments problem, Jacobi matrix, orthogonal polynomials.

Systems of orthogonal polynomials on the real line play an important role in the theory of special functions [1]. They find applications in numerous problems of mathematical physics and classical analysis. It is known, that classical polynomials have a number of properties, which uniquely define them. Particularly, one can deduce, that they satisfy the following recurrence: Jp = lp, where

p = æ ç ç ç ç è p0 p1 . ö ÷ ÷ ÷ ÷ ø

vector of polynomials p_k(l), k = 0,...,¥, J - symmetric, semi-infinite matrix:

J = æ ç ç ç ç ç è b0 a0 0 0 . a0 b1 a1 0 . 0 a1 b2 a2 . . . . . . ö ÷ ÷ ÷ ÷ ÷ ø ,

b_k - real, a_k > 0, k = 0,...,¥, which have a three diagonal structure [2].

In Chapter 1 of our work we consider the class of polynomials with described above recurrence, where J - is not necessarily symmetric one. The more wide conditions are proposed for J. The subject of orthogonality for such systems of polynomials is investigated. The corresponding theorems of orthogonality are obtained.

Chapter 2 of the work is devoted to the solution of some symmetric moments problem. The statement of the problem is a generalization of a well-known moments problem on the real line [3]. The criterion for the real case is known [3]. The generalized problem is found to be closely connected with a semi-infinite fivediagonal matrices (we also call them Jacobi). The criterion of solvability is proved.

ON EXTENSION OF FUNCTION ONTO THE KLEISLI CATEGORY OF THE INCLUSION HYPERSPACE MONAD
Andrii Teleiko topos@mf.franko.lviv.ua

1991 Mathematics Subject Classification: 54B30, 18C20. Key words: monad, Kleisli category, weakly normal functor, inclusion hyperspace.

It is proved that there exists no extension of any non-trivial weakly normal functor of finite degree onto the Kleisli category of the inclusion hyperspace monad.

DIRECTIONS DE MAJORATION D'UNE FONCTION QUASICONVEXE ET APPLICATIONS
Charki Amara

1991 Mathematics Subject Classification: 52A20, 90C26, 90C31. Key words: recession cones, quasiconvex functions, sensitivity analysis, epiconvergence, marginal functions.

We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces.

LA CLASSE BORÉLIENNE NE DÉTERMINE PAS LE TYPE TOPOLOGIQUE DE C_p(X)
Robert Cauty cauty@math.jussieu.fr

1991 Mathematics Subject Classification: 54C35, 54H05. Key words: espaces de fonctions C_p(X), classe borélienne.

We prove that, for every countable ordinal a ³ 3, there exists countable completely regular spaces X_a and Y_a such that the spaces C_p(X _a) and C_p(Y_a) are borelian of class exactly M_a, but are not homeomorphic.

INTEGER POINTS CLOSE TO A SMOOTH CURVE
Ognian Trifonov mailto:trifonov@math.sc.edu

1991 Mathematics Subject Classification: Primary 11N25, secondary 11N64. Key words: Lattice points, elementary methods in number theory.

We review the existing estimates for the number of integer points close to a smooth curve and improve on some of these.

GENERALIZED PROBLEM OF STARLIKENESS FOR PRODUCTS OF p-VALENT STARLIKE FUNCTIONS
Georgi Dimkov gdimkov@math.bas.bg
Jacek Dziok

1991 Mathematics Subject Classification: 30C45. Key words: univalent, p-valent, starlike of order a.

We consider functions of the type

F(z) = zp n Õ j = 1  é ë fj(z)/zp ù û aj

where f_j are p-valent functions starlike of order a_j and a_j are complex numbers. The problem we solve is to find conditions for the centre and the radius of the disc {z:|z-w| < r}, contained in the unit disc {z:|z| < 1} and containing the origin, so that its transformation by the function F be a domain starlike with respect to the origin.