Serdica Mathematical Journal

Volume 30, Number 4, 2004

C O N T E N T S

• Zagorodnyuk, S. M. Direct and inverse spectral problems for (2N+1)-diagonal, complex, symmetric, non-hermitian matrices (pp. 471-482)
• González, M., R. Martínez, M. Mota. A note on the asymptotic behaviour of a periodic multitype Galton-Watson branching process (pp. 483-494)
• Galanova, N. Yu. Symmetric and asymmetric gaps in some fields of formal power series (pp. 495-504)
• Andruszkiewicz, R. R. Essential cover and closure (pp. 505-512)
• Aryal, G., S. Nadarajah. Information matrix for beta distributions (pp. 513-526)
• Argyros, S. A., A. D. Arvanitakis. Regular averaging and regular extension operators in weakly compact subsets of Hilbert spaces (pp. 527-548)
• Khoury, J. A note on elementary derivations (pp. 549-570)

• Fourth Popov Prize Awarded to Sergei Denissov

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  A B S T R A C T S

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  DIRECT AND INVERSE SPECTRAL PROBLEMS FOR (2N+1)-DIAGONAL, COMPLEX, SYMMETRIC, NON-HERMITIAN MATRICES
  S. M. Zagorodnyuk Sergey.M.Zagorodnyuk@univer.kharkov.ua

  2000 Mathematics Subject Classification: 42C05. Key words: inverse problems, difference equation.

  We consider the following linear (2N)-th order difference equation:

  å_{i = 1}^N ( a_k-i,i y_k-i + a_k,iy_k+i ) + a_k,0 y_k = l^N y_k, k = 0,1,2,... ,

  where a_i,j Î C: a_i,N ¹ 0, l is a complex parameter, (y₀, y₁, ..., y_k, ...) = [y^®]^T is a vector solution, N is a fixed integer. It can be written in the following matrix form: J [y^®] = l^N [y^®], where J is a (2N+1)-diagonal, symmetric matrix. We give an easy procedure for solving of the direct and the inverse spectral problems for the equation. Guseynov used a procedure of the Gelfand-Levitan type for the case N = 1. We use another procedure and this procedure is more easy and transparent.

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  A NOTE ON THE ASYMPTOTIC BEHAVIOUR OF A PERIODIC MULTITYPE GALTON-WATSON BRANCHING PROCESS
  M. González mvelasco@unex.es
  R. Martínez rmartinez@unex.es
  M. Mota mota@unex.es

  2000 Mathematics Subject Classification: 60J80. Key words: Periodic Multitype Galton-Watson Branching Process, almost sure convergence, cycle.

  In this work, the problem of the limiting behaviour of an irreducible Multitype Galton-Watson Branching Process with period d greater than 1 is considered. More specifically, almost sure convergence of some linear functionals depending on d consecutive generations is studied under hypothesis of non extinction. As consequence the main parameters of the model are given a convenient interpretation from a practical point of view. For a better understanding of the theoretical results, an illustrative example is provided.

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  SYMMETRIC AND ASYMMETRIC GAPS IN SOME FIELDS OF FORMAL POWER SERIES
  N. Yu. Galanova natagyi@mail2000.ru

  2000 Mathematics Subject Classification: 03E04, 12J15, 12J25. Key words: Non-archimedean real closed fields, super-real fields, h₁-fields, semi-h₁-fields, fields of formal power series, symmetric gaps.

  We consider non-archimedean real closed fields of cardinality À₁ that have special type of symmetric gaps and compare these fields with well known h₁-fields (Hausdorff), semi-h₁-fields, and some super-real fields (Dales, Woodin). All these fields are realized as fields of formal power series. We describe all symmetric Dedekind and non-Dedekind gaps of semi-h₁-fields (in particular, for a nonstandard real line). We consider a construction of fields with symmetric gaps that are not semi-h₁. By this construction we give examples of fields with different asymmetric gaps.

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  ESSENTIAL COVER AND CLOSURE
  R. R. Andruszkiewicz randrusz@math.uwb.edu.pl

  2000 Mathematics Subject Classification: 16N80, 16S70, 16D25, 13G05. Key words: essential extension, essential cover, essential closure, accessible subring.

  We construct some new examples showing that Heyman and Roos construction of the essential closure in the class of associative rings can terminate at any finite or the first infinite ordinal.

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  INFORMATION MATRIX FOR BETA DISTRIBUTIONS
  Gokarna Aryal
  Saralees Nadarajah snadaraj@math.iupui.edu

  2000 Mathematics Subject Classification: 33C90, 62E99. Key words: Beta distributions, Beta function, Hypergeometric function.

  The Fisher information matrix for three generalized beta distributions are derived.

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  REGULAR AVERAGING AND REGULAR EXTENSION OPERATORS IN WEAKLY COMPACT SUBSETS OF HILBERT SPACES
  Spiros A. Argyros sargyros@math.ntua.gr
  Alexander D. Arvanitakis aarva@math.ntua.gr

  2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20. Key words: C(K) spaces, Weakly compact sets, Regular averaging operators, Regular extension operators.

  For weakly compact subsets of Hilbert spaces K, we study the existence of totally disconnected spaces L, such that C(K) is isomorphic to C(L).
  We prove that the space C(B_H) admits a Peczynski decomposition and we provide a starshaped weakly compact K, subset of B_H with non-empty interior in the norm topology, and such that C(K) @ C(L) with L totally disconnected.

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  A NOTE ON ELEMENTARY DERIVATIONS
  Joseph Khoury jkhoury@matrix.cc.uottawa.ca

  2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15. Key words: Derivations, Hilbert fourteenth problem.

  Let R be a UFD containing a field of characteristic 0, and B_m = R[Y₁,...,Y_m] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B₃ such that kerD is a finitely generated R-algebra then the generators of kerD can be chosen to be linear in the Y_i's. In this paper, we prove that this does not hold for B₄. We also investigate R-elementary derivations D of B_m satisfying one or the other of the following conditions:

  1. D is standard.
  2. kerD is generated over R by linear constants.
  3. D is fix-point-free.
  4. kerD is finitely generated as an R-algebra.
  5. D is surjective.
  6. The rank of D is strictely less than m.

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