Serdica Mathematical Journal

Volume 33, Numbers 2-3, 2007

C O N T E N T S

• Sixty years of Institute of Mathematics and Informatics (pp. i-ii)

  Survey

• Dwilewicz, R. Z. An analytic point of view at toric varieties (pp. 163-240)

  Articles

• El Borai, M. M., M. I. Abbas. Solvability of an infinite system of singular integral equations (pp. 241-252)
• Hassan, A. A. M., S. M. Amer. Existence of solutions for a class of quasi-linear singular integro-differential equations (pp. 253-266)
• Hatamleh, R. Linear colligations and dynamic system corresponding to operators in the Banach space (pp. 267-286)
• Ganyushkin, O. V. Mazorchuk. On nilpotent subsemigroups in some matrix semigroups (pp. 287-300)
• Öztürk, Ö. On Thom polynomials for A₄(−) via Schur functions (pp. 301-320)
• Zapalowski, P. Invariant functions on Neil parabola in Cⁿ (pp. 321-338)
• Gala, S. A note on Div-Curl lemma (pp. 339-350)
• Al-Mahameed, M. M. Extended maximum principles (pp. 351-362)
• Arnaoudov, I., V. Georgiev, G. Venkov. Does Atkinson-Wilcox expansion converges for any convex domain? (pp. 363-376)
• Ganchev, G., O. Kassabov. Hermitian manifolds of pointwise constant antiholomorphic sectional curvatures (pp. 377-386)

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  A B S T R A C T S

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  AN ANALYTIC POINT OF VIEW AT TORIC VARIETIES
  Roman J. Dwilewicz romand@umr.edu

  2000 Mathematics Subject Classification: Primary 14M25, 32Q99; Secondary 14L30. Key words: Toric varieties, convex polytopes, polyhedral cones.

  The goal of this paper is to give an elementary and analytic introduction to toric varieties, a first view at these objects that will prepare the readers, especially analysts, for a more advanced study. These notes could serve a wider audience from many, mainly analytic fields, who want to learn basics about toric varieties. Also they can be used to introduce this subject in graduate or advanced undergraduate courses.

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  SOLVABILITY OF AN INFINITE SYSTEM OF SINGULAR INTEGRAL EQUATIONS
  Mahmoud M. El Borai m_m_elborai@yahoo.com,
  Mohamed I. Abbas m_i_abbas77@yahoo.com

  2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15. Key words: Infinite system of singular integral equations, Banach sequence space, Differential equations of fractional orders.

  Schauder's fixed point theorem is used to establish an existence result for an infinite system of singular integral equations in the form:

  (1) x_i(t) = a_i(t)+ ∫^t₀ (t − s)^{− α} (s, x₁(s), x₂(s), …) ds,

  where i = 1,2,…, α ∈ (0,1) and t ∈ I = [0,T].

  The result obtained is applied to show the solvability of an infinite system of differential equation of fractional orders.

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  EXISTENCE OF SOLUTIONS FOR A CLASS OF QUASI-LINEAR SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS
  A. A. M. Hassan,
  S. M. Amer amrsammer@hotmail.com

  2000 Mathematics Subject Classification: 45F15, 45G10, 46B38. Key words: Quasi-linear singular integro-differential equations, Schauder's fixed point theorem, Sobolev space.

  An existence theorem is proved for a class of quasi-linear singular integro-differential equations with Cauchy kernel.

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  LINEAR COLLOGATIONS AND DYNAMIC SYSTEM CORRESPONDING TO OPERATORS IN THE BANACH SPACE
  Raed Hatamleh raedhat@yahoo.com

  2000 Mathematics Subject Classification: Primary 47A48, 93B28, 47A65; Secondary 34C94. Key words: Banach space, Dynamic system, Colligations, Coupling.

  New concepts of linear colligations and dynamic systems, corresponding to the linear operators, acting in the Banach spaces, are introduced. The main properties of the transfer function and its relation to the dual transfer function are established.

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  ON NILPOTENT SUBSEMIGROUPS IN SOME MATRIX SEMIGROUPS
  Olexandr Ganyushkin ganiyshk@univ.kiev.ua,
  Volodymyr Mazorchuk mazor@math.uu.se

  2000 Mathematics Subject Classification: 20M20, 20M10. Key words: Nilpotent semigroup, stochastic matrix, partial order, block, polyhedron.

  We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup W_n of all n×n real matrices with non-negative coefficients and the semigroup D_n of all doubly stochastic real matrices.

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  ON THOM POLYNOMIALS FOR A₄(−) VIA SCHUR FUNCTIONS
  Özer Öztürk ozero@metu.edu.tr

  2000 Mathematics Subject Classification: 05E05, 14N10, 57R45. Key words: Thom polynomials, singularities, global singularity theory, classes of degeneracy loci, Schur functions, resultants.

  We study the structure of the Thom polynomials for A₄(−) singularities. We analyze the Schur function expansions of these polynomials. We show that partitions indexing the Schur function expansions of Thom polynomials for A₄(−) singularities have at most four parts. We simplify the system of equations that determines these polynomials and give a recursive description of Thom polynomials for A₄(−) singularities. We also give Thom polynomials for A₄(3) and A₄(4) singularities.

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  INVARIANT FUNCTIONS ON NEIL PARABOLA IN Cⁿ
  Pawel Zapalowski Pawel.Zapalowski@im.uj.edu.pl

  2000 Mathematics Subject Classification: Primary 32F45. Key words: generalized Neil parabola, Carathéodory pseudodistance, Kobayashi pseudodistance, Carathéodory-Reiffen pseudometric, Kobayashi-Royden pseudometric.

  We present the Carathéodory and the inner Caratheodory distances and the Carathéodory-Reiffen metric on generalized Neil parabolas in Cⁿ. It is a generalization of the results from [4] and [5].

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  A NOTE ON DIV-CURL LEMMA
  Sadek Gala sadek.gala@gmail.com

  2000 Mathematics Subject Classification: 42B30, 46E35, 35B65. Key words: Compactness compensated, Hardy space, Sobolev space.

  We prove two results concerning the We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and div (u^→v) ∈ H¹(R^d) which include as a particular case, the result of [3].

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  EXTENDED MAXIMUM PRINCIPLES
  M. M. Al-Mahameed al_mahameed2000@yahoo.com

  2000 Mathematics Subject Classification: 35B50, 35L15. Key words: Maximum principles, elliptic equations.

  In this paper we introduce some new results concerning the maximum principles for second order linear elliptic partial differential equations defined on a noncompact Riemannian manifold.

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  DOES ATKINSON-WILCOX EXPANSION CONVERGES FOR ANY CONVEX DOMAIN?
  I. Arnaoudov jna@tu-sofia.bg,
  V. Georgiev georgiev@dm.unipi.it,
  G. Venkov gvenkov@tu-sofia.bg

  2000 Mathematics Subject Classification: 35C10, 35C20, 35P25, 47A40, 58D30, 81U40. Key words: Atkinson-Wilcox expansion theorem, Helmholtz equation, far field pattern, convex domain, second-order recurrence relations.

  The Atkinson-Wilcox theorem claims that any scattered field in the exterior of a sphere can be expanded into a uniformly and absolutely convergent series in inverse powers of the radial variable and that once the leading coefficient of the expansion is known the full series can be recovered uniquely through a recurrence relation. The leading coefficient of the series is known as the scattering amplitude or the far field pattern of the radiating field. In this work we give a simple characterization of the strictly convex domains, such that a reasonable generalization of the Atkinson-Wilcox expansion converges uniformly in the corresponding exterior domain. All these strictly convex domains are spheres.

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  HERMITIAN MANIFOLDS OF POINTWISE CONSTANT ANTIHOLOMORPHIC SECTIONAL CURVATURES
  Georgi Ganchev ganchev@math.bas.bg,
  Ognian Kassabov okassabov@vtu.bg

  2000 Mathematics Subject Classification: Primary 53B35, Secondary 53C50. Key words: Hermitian manifolds, antiholomorphic sectional curvatures.

  In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.

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