Serdica Mathematical Journal

Volume 25, Number 3, 1999

C O N T E N T S

• Buyuklieva, S. New binary extremal self-dual codes of lengths 50 and 52 (pp. 185-190)
• Vodev, G. On the uniform decay of the local energy (pp. 191-206)
• Ibrahim, Sobhy El-sayed. The general differential operators generated by a quasi-differential expressions with their interior singular points (pp. 207-240)
• Kozubowski, T. J., K. Podgórski. Geometric stable laws through series representations (pp. 241-256)
• Hájek, P. A remark on S.M. Bates' theorem (pp. 257-258)

A B S T R A C T S

NEW BINARY EXTREMAL SELF-DUAL CODES OF LENGTHS 50 AND 52
Stefka Buyuklieva

1991 Mathematics Subject Classification: 05A05. Key words: self-dual codes.

New extremal binary self-dual codes of lengths 50 and 52 are constructed. Some of them are the first known codes with such weight enumerators. The structure of their automorphisms groups are shown.

ON THE UNIFORM DECAY OF THE LOCAL ENERGY
Georgi Vodev vodev@math.univ-nantes.fr

1991 Mathematics Subject Classification: 35P25, 35J15, 47F05. Key words: cutoff resolvent, local energy decay.

It is proved in [1], [2] that in odd dimensional spaces any uniform decay of the local energy implies that it must decay exponentially. We extend this to even dimensional spaces and to more general perturbations (including the transmission problem) showing that any uniform decay of the local energy implies that it must decay like O(t^-2n), t >> 1 being the time and n being the space dimension.

THE GENERAL DIFFERENTlAL OPERATORS GENERATED BY A QUASI-DIFFERENTIAL EXPRESSIONS WITH THEIR INTERIOR SINGULAR POINTS
Sobhy El-sayed Ibrahim

1991 Mathematics Subject Classification: 34A05, 34B25, 34C11, 34E15, 34G10, 47E05. Key words: quasi-differential expressions, regular and singular end-points, regularly solvable operators, Hilbert space, boundary conditions.

The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M⁺ are considered over a regoin (a,b) on the real line, -¥ £ a < b £ ¥, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L²_w (a,b) which are regularly solvable with respect to the minimal operators T₀(M) and T₀(M⁺). In addition to direct sums of regularly solvable operators defined on the separate subintervals, there are other regularly solvable restrications of the maximal operator which involve linking the various intervals together in interface like style.

GEOMETRIC STABLE LAWS THROUGH SERIES REPRESENTATIONS
Tomasz J. Kozubowski tkozubow@cecasun.utc.edu
Krzysztof Podgórski kpodgorski@math.iupui.edu

1991 Mathematics Subject Classification: 60E07, 60F05, 60F15, 60F17, 60G50, 60H05. Key words: Geometric compound, invariance principle, Linnik distribution, Mittag-Leffler distribution, random sum, stable distribution, stochastic integral.

Let (X_i) be a sequence of i.i.d. random variables, and let N be a geometric random variable independent of (X_i). Geometric stable distributions are weak limits of (normalized) geometric compounds, S_N = X₁+¼+X_N, when the mean of N converges to infinity. By an appropriate representation of the individual summands in S_N we obtain series representation of the limiting geometric stable distribution. In addition, we study the asymptotic behavior of the partial sum process S_N(t) = å_{i = 1}^[Nt] X_i, and derive series representations of the limiting geometric stable process and the corresponding stochastic integral. We also obtain strong invariance principles for stable and geometric stable laws.

A REMARK ON S.M. BATES' THEOREM
Petr Hájek

1991 Mathematics Subject Classification: 58C25. Key words: Separable Banach space, smooth surjection, homogeneous polynomial surjection, noncompact operator.

In his paper [1], Bates investigates the existence of nonlinear, but highly smooth, surjective operators between various classes of Banach spaces. Modifying his basic method, he obtains the following striking results.