Serdica Mathematical Journal
Volume 21, Number 2, 1995
C O N T E N T S

Christov, O., D. Dragnev.
Perturbations of Systems Describing the Motion of a
Particle in Central Fields.
(pp. 91108)

Stamov, G.
Integral Manifolds and Perturbations of the
Nonlinear Part of Systems of Autonomous
Differential Equations with Impulses at Fixed Moments.
(pp. 109122)

Dion, J.P., G. Gauthier, A. Latour.
Branching Processes with Immigration and Integervalued Time Series.
(pp. 123136)

Gonska, H., DingXuan Zhou.
On an Extremal Problem concerning Bernstein Operators.
(pp. 137150)

Aboussoror, A., P. Loridan.
Strongweak Stackelberg Problems in Finite Dimensional Spaces.
(pp. 151170)
A B S T R A C T S
PERTURBATIONS OF SYSTEMS DESCRIBING THE MOTION
OF A PARTICLE IN CENTRAL FIELDS
Ognyan Christov, Dragomir Dragnev
1991 Mathematics Subject Classification: 70D10, 70H05, 70K89, 58F07, 34D10.
Key words:
particle in central field, KAMtheory, Abelian integrals,
PicardFuchs equations.
The present paper deals with the KAMtheory conditions
for systems describing the motion of a particle in central field.
INTEGRAL MANIFOLDS AND PERTURBATIONS OF THE
NONLINEAR PART OF SYSTEMS OF AUTONOMOUS
DIFFERENTIAL EQUATIONS WITH IMPULSES AT FIXED
MOMENTS
G. T. Stamov
1991 Mathematics Subject Classification: 34A37.
Key words: integral manifolds, impulsive differential equations.
Sufficient conditions are obtained for the existence of local integral
manifolds of autonomous systems of differential equations with impulses at
fixed moments. In case of perturbations of the nonlinear part an estimate
of the difference between the manifolds is obtained.
BRANCHING PROCESSES WITH IMMIGRATION AND INTEGERVALUED TIME SERIES
J.P. Dion, G. Gauthier and A. Latour
1991 Mathematics Subject Classification:62M10, 60J80.
Key words: integervalued time series, branching processes with
immigration, estimation, consistency, asymptotic normality.
In this paper, we indicate how integervalued autoregressive time
series Ginar(d) of ordre d, d ³ 1, are simple functionals of
multitype branching processes with immigration. This allows the
derivation of a simple criteria for the existence of a stationary
distribution of the time series, thus proving and extending some
results by AlOsh and Alzaid [1], Du and Li [9]
and Gauthier and Latour [11].
One can then transfer results on estimation in
subcritical multitype branching processes to stationary
Ginar(d) and
get consistency and asymptotic normality for the corresponding
estimators. The technique covers autoregressive moving average time
series as well.
ON AN EXTREMAL PROBLEM CONCERNING BERNSTEIN OPERATORS
Heinz H. Gonska and DingXuan Zhou
1991 Mathematics Subject Classification: 41A15, 41A17.
Key words: Bernstein operators, best constant,
second modulus of smoothness, Kfunctional.
The best constant problem for Bernstein operators with respect to the
second modulus of smoothness is considered.
We show that for any
^{1}/_{2} £ a < 1,
there is an N(a) Î N such
that for n ³ N(a),

sup
1a £ ^{k}/_{n} £ a


ê ê
ê

B_{n} 
æ ç
è

f, 
k
n


ö ÷
ø

f 
æ ç
è


k
n


ö ÷
ø


ê ê
ê

£ c w_{2} 
æ ç
è

f, 
1
Ön


ö ÷
ø

, 

where c is a constant, 0 < c < 1.
STRONGWEAK STACKELBERG PROBLEMS IN FINITE DIMENSIONAL SPACES
Abdelmalek Aboussoror and Pierre Loridan
1991 Mathematics Subject Classification: 90D40, 90C31, 90C25.
Key words: Marginal functions, twolevel optimization,
limits of sets, stability, convex analysis.
We are concerned with twolevel optimization problems called strongweak
Stackelberg problems, generalizing the class of Stackelberg problems
in the strong and weak sense. In order to handle the fact that the
considered twolevel optimization problems may fail to have a solution
under mild assumptions, we consider a regularization involving
eapproximate optimal solutions in the lower
level problems. We prove the existence of optimal solutions for such
regularized problems and present some approximation results when the
parameter e goes to zero.
Finally, as an example, we consider an optimization
problem associated to a best bound given in [2]
for a system of nondifferentiable convex inequalities.
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