A B S T R A C T S

GROUPS OF ORDER 32 AS GALOIS GROUPS
Ivo M. Michailov i.michailov@fmi.shu-bg.net

2000 Mathematics Subject Classification: 12F12.
Key words: embedding problem, Galois extension, quaternion algebra, obstruction.

We find the obstructions to realizability of groups of order 32 as Galois groups over arbitrary field of characteristic not 2. We discuss explicit extensions and automatic realizations as well.

OLIGOPOLY MODEL OF A DEBIT CARD NETWORK
Peter Manchev manchev.p@bnbank.org

JEL Classification: G21, L13.
Key words: debit card, payment networks, switch fees, access pricing.

The paper builds an oligopoly model of a debit card network. It examines the competition between debit card issuers. We show that there is an optimal pricing for the debit card network, which maximizes all issuer's revenues. The paper also shows that establishing a link between debit card networks averages the costs provided that there is no growth in the customer's usage of the networks, resulting from the link.

THE ASYMPTOTIC BEHAVIOUR OF THE FIRST EIGENVALUE OF LINEAR SECOND-ORDER ELLIPTIC EQUATIONS IN DIVERGENCE FORM
Alexander Fabricant
Nikolai Kutev
Tsviatko Rangelov rangelov@math.bas.bg

2000 Mathematics Subject Classification: 35J70, 35P15.
Key words: linear elliptic equations, eigenvalue problem, asymptotic behavior, dynamical systems.

The asymptotic of the first eigenvalue for linear second order elliptic equations in divergence form with large drift is studied. A necessary and a sufficient condition for the maximum possible rate of the first eigenvalue is proved.

A NEW CHARACTERIZATION OF WEIGHTED PEETRE K-FUNCTIONALS
Borislav R. Draganov bdraganov@fmi.uni-sofia.bg
Kamen G. Ivanov kamen@math.bas.bg

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10.
Key words: K-functional, modulus of smoothness, linear operator, fractional integral.

Certain types of weighted Peetre K-functionals are characterized by means of the classical moduli of smoothness taken on a proper linear transforms of the function. The weights with power-type asymptotic at the ends of the interval with arbitrary real exponents are considered. This paper extends the method and results presented in [3].

ON SOME EXTREMAL PROBLEMS OF LANDAU
Szilárd Révész revesz@renyi.hu, revesz@ihp.jussieu.fr

2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.
Key words: Prime number formula, positive trigonometric polynomials, positive definite functions, extremal problems, Borel measures, convexity, duality.

The prime number theorem with error term presents itself as &pi'(x) = ∫₂^x [dt/ logt] + O ( x e^{- K logL x}). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial ²/₃ (1+cos x)² = 1+⁴/₃ cosx +¹/₃ cos2x. Landau considered more general positive definite nonnegative cosine polynomials 1+a₁cos x+… + a_ncos nx ≥ 0, with a₁> 1,a_k ≥ 0 (k = 1,…,n), and deduced the above error term with L = 1/2 and any K< 1/(2V(a))^½, where V(a): = (a₁+a₂+…+ a_n)/(( (a₁)^½-1)²). Thus the extremal problem of finding V: = minV(a) over all admissible coefficients, i.e. polynomials, arises.

The question was further studied by Landau and later on by many other eminent mathematicians. The present work surveys these works as well as current questions and ramifications of the theme, starting with a long unnoticed, but rather valuable Bulgarian publication of Lubomir Chakalov.