Serdica Mathematical Journal

Volume 25, Number 1, 1999

C O N T E N T S

• Kalamidas, N. Some properties of g- and P-spaces (pp. 1-4)
• Martín-Gutiérrez, E. On a conditional Cauchy-type functional equation involving powers (pp. 5-10)
• Sianesi, F. Relative compactness for hyperspaces (pp. 11-18)
• Mitov, K. V. Limit theorems for regenerative excursion processes (pp. 19-40)
• Pyrih, P. On the sense preserving mappings in the Helm topology in the plane (pp. 41-46)
• Kostov, Vl. P. On the hyperbolicity domain of the polynomial xⁿ+a₁x^n-1+...+ a_n (pp. 47-70)
• Lucente, S. Nonlinear wave equation with vanishing potential (pp. 71-82)
• Erdem, S. Some para-Hermitian related complex structures and non-existence of semi-Riemannian metric on some spheres (pp. 83-90)

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  A B S T R A C T S

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  SOME PROPERTIES OF g- AND P-SPACES
  N. Kalamidas

  1991 Mathematics Subject Classification: 54C35, 54A25. Key words: pointwise convergence, C_p(x), g-space, Frechet-Urysohn property, strictly positive measure, P-space.

  A g-space with a strictly positive measure is separable. An example of a non-separable g-space with c.c.c. is given. A P-space with c.c.c. is countable and discrete.

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  ON A CONDITIONAL CAUCHY-TYPE FUNCTIONAL EQUATION INVOLVING POWERS
  Emma Martín-Gutiérrez

  1991 Mathematics Subject Classification: 39B52. Key words: functional equation.

  We solve the functional equation f(x^m+y) = f(x)^m+f(y) in the realm of polynomials with integer coefficients.

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  RELATIVE COMPACTNESS FOR HYPERSPACES
  Francesca Sianesi sianesi@vmimat.mat.unimi.it

  1991 Mathematics Subject Classification: 54B20. Key words: hyperspaces topologies, relative compactness, sup measures, gap functional.

  The present paper contains results characterizing relatively compact subsets of the space of the closed subsets of a metrizable space, equipped with various hypertopologies.

  We investigate the hyperspace topologies that admit a representation as weak topologies generated by families of gap functionals defined on closed sets, as well as hit-and-miss topologies and proximal-hit and-miss topologies.

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  LIMIT THEOREMS FOR REGENERATIVE EXCURSION PROCESSES
  Kosto V. Mitov kmitov@af-acad.bg

  1991 Mathematics Subject Classification: 60F05; 60K05; 60J80. Key words: Alternating renewal processes, regenerative processes, limit theorems, branching processes, state-dependent immigration.

  The regenerative excursion process Z(t), t = 0,1,2,... is constructed by two independent sequences X = {X_i, i ³ 1 } and Z = {T_i, (Z_i(t), 0 £ t < T_i), i ³ 1 }.

  For the embedded alternating renewal process, with interarrival times X_i - the time for the installation and T_i - the time for the work, are proved some limit theorems for the spent worktime and the residual worktime, when at least one of the means of X_i and T_i is infinite.

  Limit theorems for the process Z(t) are proved, too.

  Finally, some applications to the branching processes with state-dependent immigration are given.

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  ON THE SENSE PRESERVING MAPPINGS IN THE HELM TOPOLOGY IN THE PLANE
  Pavel Pyrih pyrih@karlin.mff.cuni.cz

  1991 Mathematics Subject Classification: Primary 54A10; Secondary 26A34,31C40. Key words: fine topology, density topology.

  We introduce the helm topology in the plane. We show that (assuming the helm local injectivity and the Euclidean continuity) each mapping which is oriented at all points of a helm domain U is oriented at U.

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  ON THE HYPERBOLICITY DOMAIN OF THE POLINOMIAL xⁿ+a₁x^n-1+¼+ a_n
  Vladimir Petrov Kostov kostov@hera.unice.fr

  1991 Mathematics Subject Classification: Primary: 14P10; secondary: 14B05. Key words: hyperbolicity domain, stratum, multiplicity vector.

  We consider the polynomial P_n = xⁿ+ a₁x^n-1+¼+a_n, a_i Î R. We represent by figures the projections on Oa₁... a_k, k £ 6, of its hyperbolicity domain P = {a Î Rⁿ  |  all roots of P_n are real}. The set P and its projections P^k in the spaces Oa₁... a_k, k £ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection P^k® P^k-1 is a segment or a point. We prove that this is also true for the strata of P of dimension ³ k. This implies that for any two adjacent strata there always exist a space Oa₁... a_k, k £ n, such that from the projections of the strata in it one is ``above'' the other w.r.t. the axis Oa_k. We show

  1) how to find this k and which stratum is ``above'' just by looking at the multiplicity vectors of the strata;

  2) how to obtain the relative position of a stratum of dimension l and of all strata of dimension l+1 and l+2 to which it is adjacent.

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  NONLINEAR WAVE EQUATION WITH VANISHING POTENTIAL
  Sandra Lucente lucente@mail.dm.unipi.it

  1991 Mathematics Subject Classification: 35L05, 35B65. Key words: nonlinear wave equation, large data.

  We study the Cauchy problem for u_tt- Du+V(x)u⁵ = 0 in 3-dimensional case. The function V(x) is positive and regular, in particular we are interested in the case V(x) = 0 in some points. We look for the global classical solution of this equation under a suitable hypothesis on the initial energy.

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  SOME PARA-HERMITIAN RELATED COMPLEX STRUCTURES AND NON-EXISTENCE OF SEMI-RIEMANNIAN METRIC ON SOME SPHERES
  Sadettin Erdem serdem@arf.math.metu.edu.tr

  1991 Mathematics Subject Classification: 53C15, 53C50, 53C55. Key words: Hermitian, para-Hermitian and indefinite metric.

  It is shown that the spheres S²ⁿ (resp: S^k with k º 1 mod 4) can be given neither an indefinite metric of any signature (resp: of signature (r, k - r) with 2 £ r £ k-2) nor an almost paracomplex structure. Further for every given Riemannian metric on an almost para-Hermitian manifold with the associated 2-form f one can construct an almost Hermitian structure (under certain conditions, two different almost Hermitian structures) whose associated 2-form(s) is f.

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