Serdica Mathematical Journal
Volume 34, Number 2, 2008
C O N T E N T S

Sarkar, A. K.
A unified grouptheoretic method on
improper partial semibilateral generating functions
(pp. 373382)

Bensalem, N.
Régularité Lipschitzienne des géodésiques
minimisantes pour quelques distributions affines
(pp. 383394)

Elabbasy, E. M., W. W. Elhaddad.
Oscillation criteria
for nonlinear differential equations of second order with damping
term
(pp. 395414)

Dzhumadil'daev, A. S.
qLeibniz algebras
(pp. 415440)

Marinov, R. Ts.
An iterative procedure for solving
nonsmooth generalized equation
(pp. 441454)

Argyros, I. K., S. Hilout.
Steffensen methods for
solving generalized equations
(pp. 455466)

Kovacheva, R. K.
Zeros of sequences of partial sums and
overconvergence
(pp. 467482)

Kurbanov, S.
Limit theorems for noncritical branching
processes with continuous state space
(pp. 483488)

Gourdin, D., H. Kamoun, O. Ben Khalifa.
Solutions globales régulières pour quelques équations
lineaires d'évolution du type pseudodifférentiel singulier
(pp. 489508)

Veleva, E.
Test for independence of the variables with
missing elements in one and the same column of the empirical
correlation matrix
(pp. 509530)
A B S T R A C T S
A UNIFIED GROUPTHEORETIC METHOD ON IMPROPER PARTIAL
SEMIBILATERAL GENERATING FUNCTIONS
Asit Kumar Sarkar
asit_kumar_sarkar@yahoo.com
2000 Mathematics Subject Classification:
33A65, 33C20.
Key words:
Generating relation, Improper partial
semibilateral generating function.
A unifed grouptheoretic method of obtaining more general class of
generating functions from a given class of improper partial
semibilateral generating functions involving Laguerre and
Gegenbauer polynomials are discussed.
RÉGULARITÉ LIPSCHITZIENNE DES GÉODÉSIQUES
MINIMISANTES POUR QUELQUES DISTRIBUTIONS AFFINES
Naceurdine Bensalem
naceurdine_bensalem@yahoo.fr
2000 Mathematics Subject Classification:
49J15, 49J30, 53B50.
Key words:
Contrôle optimal, Régularité Lipschitzienne,
Distribution affine, Géodésique.
In the context of subRiemannian geometry and the Lipschitzian
regularity of minimizers in control theory, we investigate some
properties of minimizing geodesics for certain affine distributions.
In particular, we consider the case of a generalized H2strong
affine distribution and the case of an affine Plaff system of
maximal class.
OSCILLATION CRITERIA FOR NONLINEAR DIFFERENTIAL EQUATIONS
OF SECOND ORDER WITH DAMPING TERM
E. M. Elabbasy
emelabbasy@mans.edu.eg
W. W. Elhaddad
2000 Mathematics Subject Classification:
34C10, 34C15.
Key words:
Oscillation, second order nonlinear
differential equation.
Some new criteria for the oscillation of all solutions of second
order
differential equations of the form
(d/dt)(r(t)ψ(x)dx/dt^{α−2}(dx/dt))+
p(t)φ(x^{α−2}x,r(t)
ψ(x)dx/dt^{α−2}(dx/dt))+q(t)x^{α−2}
x=0,
and the more general equation
(d/dt)(r(t)ψ(x)dx/dt^{α−2}(dx/dt))+p(t)φ(g(x),r(t)
ψ(x)dx/dt^{α−2} (dx/dt))+q(t)g(x)=0,
are established. our results generalize and extend some known
oscillation criterain in the literature.
qLEIBNIZ ALGEBRAS
A. S. Dzhumadil'daev
askar@math.kz,
askar56@hotmail.com
2000 Mathematics Subject Classification:
Primary 17A32, Secondary 17D25.
Key words:
Leibniz algebras, Zinbiel algebras, OmniLie algebras,
polynomial identities, qcommutators
An algebra (A,ο) is called Leibniz if
aο(bοc) = (a ο b)ο c(a ο c) ο b for all a,b,c ∈ A. We
study identities for the algebras A^{(q)} = (A,ο_{q}), where
a ο_{q} b = a ο b+q b ο a is the qcommutator. Let
Char K ≠ 2,3. We show that the class of qLeibniz algebras is defined
by one identity of degree 3 if q^{2} ≠ 1, q ≠−2, by two
identities of degree 3 if q = −2, and by the commutativity
identity and one identity of degree 4 if q = 1. In the case of
q = −1 we construct two identities of degree 5 that form a base
of identities of degree 5 for −1Leibniz algebras. Any identity
of degree < 5 for −1Leibniz algebras follows from the
anticommutativity identity.
AN ITERATIVE PROCEDURE FOR SOLVING NONSMOOTH GENERALIZED
EQUATION
Rumen Tsanev Marinov
marinov_r@yahoo.com
2000 Mathematics Subject Classification:
47H04, 65K10.
Key words:
Setvalued maps, generalized equation, linear
convergence, Aubin continuity.
In this article, we study a general iterative procedure of the
following form
0 ∈ f(x_{k})+F(x_{k+1}),
where f is a function and F is a set valued map acting from a
Banach space X to a linear normed space Y, for solving
generalized equations in the nonsmooth framework.
We prove that this method is locally Qlinearly convergent to x^{*}
a solution of the generalized equation
0 ∈ f(x)+F(x)
if the setvalued map
[f(x^{*})+g(·)−g(x^{*})+F(·)]^{−1}
is Aubin continuous at (0,x^{*}), where g:X→ Y is a
function, whose Fréchet derivative is LLipschitz.
STEFFENSEN METHODS FOR SOLVING GENERALIZED EQUATIONS
Ioannis K. Argyros
ioannisa@cameron.edu,
Saïd Hilout
said_hilout@yahoo.fr
2000 Mathematics Subject Classification:
65G99, 65K10, 47H04.
Key words:
Steffensen's method, Banach space,
setvalued mapping, generalized equations, Aubin continuity,
divided difference, Newton's method.
We provide a local convergence analysis for Steffensen's method in
order to solve a generalized equation in a Banach space setting.
Using well known fixed point theorems for setvalued maps
[13] and Hölder type conditions introduced by us in
[2] for nonlinear equations, we obtain the superlinear
local convergence of Steffensen's method. Our results compare
favorably with related ones obtained in [11].
ZEROS OF SEQUENCES OF PARTIAL SUMS AND OVERCONVERGENCE
Ralitza K. Kovacheva
rkovach@math.bas.bg
2000 Mathematics Subject Classification:
30B40, 30B10, 30C15, 31A15.
Key words:
Overconvergence, HadamardOstrowski gaps, equilibrium measure.
We are
concerned with overconvergent power series. The main idea is to
relate the distribution of the zeros of subsequences of partial sums
and the phenomenon of overconvergence. Sufficient conditions for a
power series to be overconvergent in terms of the distribution of
the zeros of a subsequence are provided, and results of
JentzschSzegö type about the asymptotic distribution of the
zeros of overconvergent subsequences are stated.
LIMIT THEOREMS FOR NONCRITICAL BRANCHING PROCESSES WITH
CONTINUOUS STATE SPACE
S. Kurbanov
kurbanov@kma.zcu.cz
2000 Mathematics Subject Classification:
Primary 60J80, Secondary 60G99.
Key words:
Random variable, branching process, decreasing immigration, independent
increment, factorial moment.
In the paper a modification of the branching stochastic process with
immigration and with continuous states introduced by Adke S. R. and
Gadag V. G. (1995) is considered. Limit theorems for the noncritical
processes with or without nonstationary immigration and finite
variance are proved. The subcritical case is illustrated with
examples.
SOLUTIONS GLOBALES RÉGULIÈRES POUR QUELQUES
ÉQUATIONS LINEAIRES D'ÉVOLUTION DU TYPE
PSEUDODIFFÉRENTIEL SINGULIER
D. Gourdin
gourdin@math.jussieu.fr,
H. Kamoun,
O. Ben Khalifa
2000 Mathematics Subject Classification:
35C15, 35D05, 35D10, 35S10, 35S99.
Key words:
Linear Cauchy problem,
global solution, Sobolev spaces, Integral Kirchhof solutions.
We give here examples of equations of type (1)
∂_{tt}^{2} y p(t, D_{x}) y = 0, where p is a singular pseudodifferential
operator with regular global solutions when the Cauchy data are
regular, t ∈ R, x ∈ R^{5} .
TEST FOR INDEPENDENCE OF THE VARIABLES WITH MISSING
ELEMENTS IN ONE AND THE SAME COLUMN OF THE EMPIRICAL CORRELATION
MATRIX
Evelina Veleva
eveleva@abv.bg
2000 Mathematics Subject Classification:
62H15, 62H12.
Key words:
Multivariate normal distribution, Wishart
distribution, correlation matrix completion,
maximum likelihood ratio test.
We consider variables with joint multivariate normal distribution
and suppose that the sample correlation matrix has missing elements,
located in one and the same column. Under these assumptions we
derive the maximum likelihood ratio test for independence of the
variables. We obtain also the maximum likelihood estimations for the
missing values.
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