Serdica Mathematical Journal

Volume 26, Number 3, 2000

C O N T E N T S

• Brohe M., P. Tossings. Perturbed proximal point algorithm with nonquadratic kernel (pp. 177-206)
• Moudafi, A. A recession notion for a class of monotone bivariate functions (pp. 207-220)
• Rusev, P. Complete systems of Hermite associated functions (pp. 221-228)
• Jankowski, T. Differential equations in abstract cones (pp. 229-244)
• Giambruno, A., M. Zaicev. A characterization of varieties of associative algebras of exponent two (pp. 245-252)
• Kostov, V. P. Some examples of rigid representations (pp. 253-276)

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  A B S T R A C T S

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  PERTURBED PROXIMAL POINT ALGORITHM WITH NONQUADRATIC KERNEL
  M. Brohe, M.Brohe@ulg.ac.be , P. Tossings Patricia.Tossings@ulg.ac.be

  2000 Mathematics Subject Classification: 47H04, 47H05, 47H09, 47H10. Key words: Proximal point algorithm, Bregman functions, generalized resolvent operator, variational convergence.

  Let H be a real Hilbert space and T be a maximal monotone operator on H.

  A well-known algorithm, developed by R. T. Rockafellar [16], for solving the problem
  

  (P)                      ``To find [`x] Î H such that 0 Î T[`x]''
  

  is the proximal point algorithm.

  Several generalizations have been considered by several authors: introduction of a perturbation, introduction of a variable metric in the perturbed algorithm, introduction of a pseudo-metric in place of the classical regularization,...

  We summarize some of these extensions by taking simultaneously into account a pseudo-metric as regularization and a perturbation in an inexact version of the algorithm.

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  A RECESSION NOTION FOR A CLASS OF MONOTONE BIVARIATE FUNCTIONS
  A. Moudafi amoudafi@martinique.univ-ag.fr

  2000 Mathematics Subject Classification: 49M45, 49M10, 65K10, 90C25. Key words: Bivariate function, recession notion, Yosida approximate, variational convergence, convex optimization, maximal monotone operators.

  Using monotone bifunctions, we introduce a recession concept for general equilibrium problems relying on a variational convergence notion. The interesting purpose is to extend some results of P. L. Lions on variational problems. In the process we generalize some results by H. Brézis and H. Attouch relative to the convergence of the resolvents associated with maximal monotone operators.

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  COMPLETE SYSTEMS OF HERMITE ASSOCIATED FUNCTIONS
  Peter Rusev pkrusev@math.bas.bg

  2000 Mathematics Subject Classification: 30B60, 33C45. Key words: Hermite polynomials, Hermite associated functions, completeness.

  It is proved that if the increasing sequence {k_n}_{n = 0}^¥ of nonnegative integers has density greater than 1/2 and D is an arbitrary simply connected subregion of C\R then the system of Hermite associated functions {G_kn(z)}_{n = 0}^¥ is complete in the space H(D) of complex functions holomorphic in D.

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  DIFFERENTIAL EQUATIONS IN ABSTRACT CONES
  Tadeusz Jankowski tjank@mif.pg.gda.pl

  2000 Mathematics Subject Classification: 34A45, 34K99. Key words: Quasilinearization, monotone iterations, superlinear and semi--superlinear convergence.

  We extend the method of quasilinearization to differential equations in abstract normal cones. Under some assumptions, corresponding monotone iterations converge to the unique solution of our problem and this convergence is superlinear or semi--superlinear.

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  A CHARACTERIZATION OF VARIETIES OF ASSOCIATIVE ALGEBRAS OF EXPONENT TWO
  A. Giambruno a.giambruno@unipa.it, M. Zaicev zaicev@mech.math.msu.su

  2000 Mathematics Subject Classification: Primary 16R10, 16P90. Key words: variety of algebras, polynomial identity.

  It was recently proved that any variety of associative algebras over a field of characteristic zero has an integral exponential growth. It is known that a variety V has polynomial growth if and only if V does not contain the Grassmann algebra and the algebra of 2 ×2 upper triangular matrices. It follows that any variety with overpolynomial growth has exponent at least 2. In this note we characterize varieties of exponent 2 by exhibiting a finite list of algebras playing a role similar to the one played by the two algebras above.

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  SOME EXAMPLES OF RIGID REPRESENTATION
  Vladimir Petrov Kostov kostov@math.unice.fr

  2000 Mathematics Subject Classification: 15A24. Key words: monodromy group, rigid representation.

  Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes C_j Ì GL(n,C) (resp. c_j Ì gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices M_j Î C_j (resp. A_j Î c_j) satisfying the equality M₁¼M_p+1 = I (resp. A₁+¼+A_p+1 = 0). The matrices M_j and A_j are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere.

  We give new examples of existence of such (p+1)-tuples of matrices M_j (resp. A_j) which are rigid, i.e. unique up to conjugacy once the classes C_j (resp. c_j) are fixed. For rigid representations the sum of the dimensions of the classes C_j (resp. c_j) equals 2n²-2.

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