Serdica Mathematical Journal

Volume 29, Number 1, 2003

C O N T E N T S

• Ivanov, V. I. Caracterizations of the solution sets of generalized convex minimization problems ompactness in the first Baire class and Baire-1 operators (pp. 1-10)
• Pang, L.-P. Z.-Q. Xia. A PVT-type algorithm for minimizing a nonsmooth convex function (pp. 11-32)
• Danchev, P. V. Sylow p-subgroups of Abelian group rings (pp. 33-44)
• Geoffroy M., S. Hilout, A. Pietrus. Acceleration of convergence in Dontchev's iterative method for solving variational inclusions (pp. 45-54)
• Cízek P., M. Fabian. Adequate compacta which are Gul'ko or Talagrand (pp. 55-64)
• Kostov Vl. P. On arrangements of real roots of a real polynomial and its derivatives (pp. 65-74)
• Khristov, E. Kh. On the transformations of symplectic expansions and the respective Bäcklund Transformation for the KdV Equation (pp. 75-94)

CHARACTERIZATIONS OF THE SOLUTION SETS OF GENERALIZED CONVEX MINIMIZATION PROBLEMS
Vsevolod I. Ivanov vsevolodivanov@yahoo.com

2000 Mathematics Subject Classification: 90C26, 90C20, 49J52, 47H05, 47J20. Key words: generalized convexity, pseudoconvex function, quasiconvex quadratic function, solution set, variational inequality.

In this paper we obtain some simple characterizations of the solution sets of a pseudoconvex program and a variational inequality. Similar characterizations of the solution set of a quasiconvex quadratic program are derived. Applications of these characterizations are given.

A PVT-TYPE ALGORITHM FOR MINIMIZING A NONSMOOTH CONVEX FUNCTION
Li-Ping Pang lipingpang@163.com, Zun-Quan Xia zqxiazhh@dlut.edu.cn zqxia@mail.dlptt.ln.cn

2000 Mathematics Subject Classification: 90C25, 68W10, 49M37. Key words: Parallel algorithm, synchronous parallel, convex minimization, Moreau-Yosida regularization, strong convexity, descent condition, forced function.

A general framework of the (parallel variable transformation) PVT-type algorithm, called the PVT-MYR algorithm, for minimizing a nonsmooth convex function is proposed, via the Moreau-Yosida regularization. As a particular scheme of this framework an e-scheme is also presented. The global convergence of this algorithm is given under the assumptions of strong convexity of the objective function and an e-descent condition determined by an e-forced function. An appendix stating the proximal point algorithm is recalled in the last section.

SYLOW p-SUBGROUPS OF ABELIAN GROUP RINGS
P. V. Danchev

2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34. Key words: unit groups. direct factors, basic subgroups, direct sums of cyclics.

Let PG be the abelian modular group ring of the abelian group G over the abelian ring P with 1 and prime char P = p. In the present article, the p-primary components U_p(PG) and S(PG) of the groups of units U(PG) and V(PG) are classified for some major classes of abelian groups.

Suppose K is a first kind field with respect to p in char K ¹ p and A is an abelian p-group. In the present work, the p-primary components U_p(KA) and S(KA) of the group of units U(KA) and V(KA) in the semisimple abelian group ring KA are studied when they belong to some central classes of abelian groups. The established criteria extend results obtained by us in Compt. rend. Acad. bulg. Sci. (1993). Moreover, the question for the isomorphic type of the basic subgroup of S(KA) is also settled. As a final result, it is proved that if A is a direct sum of cyclics, the group of all normed p-units S(KA) modulo A, that is, S(KA)/A, is a direct sum of cyclics too. Thus A is a direct factor of S(KA) with a direct sum of cyclics complementary factor provided s_p(K) Ê N. This generalizes a result due to T. Mollov in Pliska Stud. Math. Bulgar. (1986).

ACCELERATION OF CONVERGENCE IN DONTCHEV'S ITERATIVE METHOD FOR SOLVING VARIATIONAL INCLUSIONS
M. Geoffroy michel.geoffroy@univ-ag.fr, S. Hilout, A. Pietrus alain.pietrus@univ-ag.fr

2000 Mathematics Subject Classification: 47H04, 65K10. Key words: Multiapplication, Aubin Continuity, Cubic Convergence.

In this paper we investigate the existence of a sequence (x_k) satisfying 0 Î f(x_k)+ Ñf(x_k)(x_k+1-x_k) +¹/₂Ѳ f(x_k) (x_k+1-x_k)²+G(x_k+1) and converging to a solution x^* of the generalized equation 0 Î f(x)+G(x); where f is a function and G is a set-valued map acting in Banach spaces. We show that the previous sequence is locally cubic convergent to x^* whenever the set-valued map [f(x^*)+ Ñf(x^*)(· -x^*) +¹/₂ Ѳf(\xs)(· - \xs)² +G(·)]^-1 is M-pseudo-Lipschitz around (0,x^*).

ADEQUATE COMPACTA WHICH ARE GUL'KO OR TALAGRAND
Petr Cízek, Marián Fabian fabian@math.cas.cz

2000 Mathematics Subject Classification: 54H05, 03E15, 46B26 Key words: K-analytic space, K-countably determined space, analytic set, coanalytic set, adequate family, ill-founded tree, well-founded tree, Mercourakis space.

We answer positively a question raised by S. Argyros: Given any coanalytic, nonalytic subset S¢ of the irrationals, we construct, in Mercourakis space c₁ (S¢), an adequate compact which is Gul'ko and not Talagrand. Further, given any Borel, non F_s subset S¢ of the irrationals, we construct, in c₁ (S¢), an adequate compact which is Talagrand and not Eberlein.

ON ARRANGEMENTS OF REAL ROOTS OF A REAL POLYNOMIAL AND ITS DERIVATIVES
Vladimir Petrov Kostov kostov@math.unice.fr

2000 Mathematics Subject Classification: 12D10. Key words: arrangement of roots.

We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its s-th derivative are realized by real polynomials.

ON THE TRANSFORMATIONS OF SYMPLECTIC EXPANSIONS AND THE RESPECTIVE BÄCKLUND TRANSFORMATION FOR THE KDV EQUATION
E. Kh. Khristov hristov@fmi.uni-sofia.bg

2000 Mathematics Subject Classification: Primary: 34B40; secondary: 35Q51, 35Q53. Key words: Symplectic expantion, KdV equation, Bäcklund transformation.

By using the Deift-Trubowitz transformations for adding or removing bound states to the spectrum of the Schrödinger operator on the line we construct a simple algorithm allowing one to reduce the problem of deriving symplectic expansions to its simplest case when the spectrum is purely continuous, and vice versa. We also obtain the transformation formulas for the correponding recursion operator. As an illustration of this approach, the Bäcklund transformations for the KdV equation are constructed.