Serdica Mathematical Journal
Volume 22, Number 4, 1996
C O N T E N T S
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Nikola Obreshkoff - Biographical data
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Nikola Obreshkoff - Bibliography
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Klebanov, L., S.T. Rachev.
Sums of a random number of random variables and their
approximations with \nu-accompanying infinitely
divisible laws
(pp. 471-496)
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Ivanov, K.G., A.Sharma.
Quadratic mean radius of a polynomial in C(Z)
(pp. 497-514)
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Craven Th., G. Csordas.
Problems and theorems in the theory of multiplier sequences
(pp. 515-524)
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Kalashnikov V.
Calculation of reliability characteristics for
regenerative models
(pp. 525-546)
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Bakan A., Th. Craven, G. Csordas, A. Golub
Weakly increasing zero-diminishing sequences
(pp. 547-570)
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Bakalov B., E. Horozov, M. Yakimov.
Backlund-Darboux transformations in Sato's Grassmannian
(pp. 571-586)
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Schinzel A.
Triples of positive integers
with the same sum and the same product
(pp. 587-588)
A B S T R A C T S
SUMS OF A RANDOM NUMBER OF RANDOM
VARIABLES AND THEIR APPROXIMATIONS
WITH \nu-ACCOMPANYING INFINITELY
DIVISIBLE LAWS
Lev B. Klebanov, Svetlozar T. Rachev
1991 Mathematics Subject Classification:
60F05, 60E07, 60E10.
Key words: infinitely divisible laws,
geometric sums, rate of convergence,
probability metrics.
In this paper a general theory of a random number of
random variables is constructed. A description of all
random variables \nu admitting an analog of the
Gaussian distribution under \nu-summation, that is,
the summation of a random number \nu of random
terms, is given. The \nu-infinitely divisible
distributions are described for these \nu-summations
and finite estimates of the approximation of \nu-sum
distributions with the help of \nu-accompanying infinitely
divisible distributions are given. The results include, in
particular, the description of geometrically infinitely
divisible and geometrically stable distributions as well
as their domains of attraction.
QUADRATIC MEAN RADIUS OF A POLYNOMIAL IN C(Z)
K.G. Ivanov, A. Sharma
1991 Mathematics Subject Classification: 30C15, 26C10.
Key words: algebraic polynomials,
location of zeros, Schoenberg conjecture.
A Schoenberg conjecture connecting quadratic mean radii
of a polynomial and its derivative is verified for some
kinds of polynomials, including fourth degree ones.
PROBLEMS AND THEOREMS IN THE THEORY OF MULTIPLIER SEQUENCES
Thomas Craven, George Csordas
1991 Mathematics Subject Classification:
primary 26C10, 30D15; secondary 30D10.
Key words: Laguerre-P\'olya class,
multiplier sequences, distribution of zeros of
entire functions, \lambda-sequences.
The purpose of this paper is (1) to highlight some
recent and heretofore unpublished results in the theory
of multiplier sequences and (2) to survey some open
problems in this area of research. For the sake of clarity
of exposition, we have grouped the problems in three
subsections, although several of the problems are
interrelated. For the reader's convenience, we have
included the pertinent definitions, cited references
and related results, and in several instances,
elucidated the problems by examples.
CALCULATION OF RELIABILITY CHARACTERISTICS FOR
REGENERATIVE MODELS
Vladimir Kalashnikov
1991 Mathematics Subject Classification: 60K15.
Key words: regenerative process, semi-regenerative process,
geometric sum, two-sided bounds.
If a regenerative process is represented as semi-regenerative,
we derive formulae enabling us to calculate basic
characteristics associated with the first occurrence time
starting from corresponding characteristics for the
semi-regenerative process. Recursive equations, integral
equations, and Monte-Carlo algorithms are proposed for
practical solving of the problem.
WEAKLY INCREASING ZERO-DIMINISHING SEQUENCES
Andrew Bakan, Thomas Craven, George Csordas, Anatoly Golub
1991 Mathematics Subject Classification:
Primary 30D15; Secondary 26C10, 30D10, 65D05.
Key words: weakly increasing sequences,
zero-diminishing sequences, zeros of entire functions,
interpolation
The following problem, suggested by Laguerre's Theorem (1884),
remains open: Characterize all real sequences
\{\mu_k\}_{k=0}^\infty which have the zero-diminishing
property; that is, if p(x)=\sum_{k=0}^n a_k x^k is
any real polynomial, then \sum_{k=0}^n \mu_k a_k x^k
has no more real zeros than p(x).
In this paper this problem is solved under the additional
assumption of a weak growth condition on the sequence
\{\mu_k\}_{k=0}^\infty, namely
\lim_{\overline{n\to\infty}}|\mu_n|^{1/n}< \infty.
More precisely, it is established that the real sequence
\{\mu_k\}_{k\geq 0} is a weakly increasing
zero-diminishing sequence if and only if there exists
\sigma\in\{+1,-1\} and an entire function
\Phi(z)=be^{az}\prod\limits_{n\geq 1}
\left (1+{x\over{ \alpha_n}}\right ),\;
a,b\in R^1,\;b\neq 0,\;\alpha_n>0\;
\forall n\geq 1,\;\sum\limits_{n\geq 1}
{1\over{\alpha_n}}<\infty,
such that
\mu_k={{\sigma^k}\over{\Phi(k)}},\; \forall k\geq 0.
BACKLUND-DARBOUX TRANSFORMATIONS IN SATO'S GRASSMANNIAN
B. Bakalov, E. Horozov, M.Yakimov
1991 Mathematics Subject Classification:
Primary 58F07; secondary 58F37.
Key words:
Backlund-Darboux transformation,
Sato's Grassmannian, tau-function, spectral algebra.
We define Backlund-Darboux transformations in Sato's
Grassmannian. They can be regarded as Darboux transformations
on maximal algebras of commuting ordinary differential
operators. We describe the action of these transformations
on related objects: wave functions, tau-functions and
spectral algebras.
TRIPLES OF POSITIVE INTEGERS WITH THE SAME SUM AND THE
SAME PRODUCT
A. Schinzel
1991 Mathematics Subject Classification:
11D25, 11G05.
Key words: rational solutions
It is proved that for every k there exist
k triples of positive integers
with the same sum and the same product.
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