Serdica Mathematical Journal
Volume 29, Number 2, 2003
C O N T E N T S
-
Kalenda, O. F. K.
A characterization of weakly Lindelöf determined Banach spaces
(pp. 95-108)
-
Borisova, G. S.
The operators Ag = gA+[`(g)]A*
for a class of nondissipative operators A with a
limit of the corresponding correlation function
(pp. 109-140)
-
Johanis, M.
Approximation of Lipschitz mappings
(pp. 141-148)
-
Azevedo, S. S.
A basis for Z-graded identities of matrices over infinite fields
(pp. 149-158)
-
Chbani, Z., H. Riahi.
Variational principles for monotone and maximal bifunctions
(pp. 159-166)
-
Laghdir, M., N. Benkenza.
Duality in constrained DC-optimization via Toland's duality
approach
(pp. 167-176)
-
Kostov Vl. P.
On a stratification defined by real roots of polynomials
(pp. 177-186)
-
Molina, M., M. Mota, A. Ramos.
On the extinction probability for bisexual
branching processes in varying environments
(pp. 187-194)
A B S T R A C T S
A CHARACTERIZATION OF WEAKLY LINDELÖF DETERMINED BANACH SPACES
Ondrej F. K. Kalenda
kalenda@karlin.mff.cuni.cz
2000 Mathematics Subject Classification:
46B26, 46B03, 46B04.
Key words:
Weakly Linde\"of determined Banach space, projectional
resolution of the identity, complemented subspace, Corson compact
space, Valdivia compact space.
We prove that a Banach space X is weakly Lindelöf determined
if (and only if) each non-separable Banach space isomorphic to a
complemented subspace of X has a projectional resolution of the
identity. This answers a question posed by S. Mercourakis and
S. Negrepontis and yields a converse of Amir-Lindenstrauss'
theorem. We also prove that a Banach space of the form C(K)
where K is a continuous image of a Valdivia compactum is weakly
Lindelöf determined if (and only if) each non-separable Banach space
isometric to a subspace of C(K) has a projectional resolution of
the identity.
THE OPERATORS Ag = gA+[`(g)]A*
FOR A CLASS OF NONDISSIPATIVE OPERATORS A WITH A
LIMIT OF THE CORRESPONDING CORRELATION FUNCTION
Galina S. Borisova
g.borisova@fmi.shu-bg.net
2000 Mathematics Subject Classification:
Primary 47A48, Secondary 60G12.
Key words:
Operator colligation, nondissipative
curve, correlation function, wave operator, scattering operator.
In this work we present the operators Ag = gA+[`(g)]A* with
Âg = 1/2 in the case of an
operator A from the class of nondissipative operators generating
nonselfadjoint curves, whose correlation functions have a limit as
t®±¥. The asympthotics of the stationary curves
eitAgf as t®±¥ onto the absolutely
continuous subspace of Ag are obtained. These
asymptotics and the obtained asymptotics in [9] of the
nondissipative curves eitAf allow to construct the scattering
theory for the couples (Ag,A) and (A,Ag). We
consider the basic terms from the scattering theory - wave
operators, a scattering operator and the question of a similarity
of A and Ag. We obtain explicitly the wave operators,
the scattering operator and the similarity of A and
Ag.
APPROXIMATION OF LIPSCHITZ MAPPINGS
Michal Johanis
johanis@karlin.mff.cuni.cz
2000 Mathematics Subject Classification:
46B03.
Key words:
Approximation, uniform Gâteaux differentiability,
Lipschitz mappings.
We prove that any Lipschitz mapping from a separable Banach space into any
Banach space can be approximated by uniformly Gâteaux differentiable
Lipschitz mapping.
A BASIS FOR Z-GRADED IDENTITIES OF MATRICES OVER INFINITE FIELDS
Sergio S. Azevedo
sergios@ime.unicamp.br
2000 Mathematics Subject Classification:
16R10, 16R20, 16R50.
Key words:
Matrix algebra, variety of algebras,
polynomial identities, graded identities.
The algebra Mn(K) of the matrices n×n over a field K can
be regarded as a Z-graded algebra. In this paper, it is
proved that if K is an infinite field, all the Z-graded
polynomial identities of Mn(K) follow from the identities:
|
|
|
| |
|
| |
|
zyx,
a(x) = -a(y) =
a(z), |
|
| |
|
where a is the degree of the corresponding variable.
This is a generalization of a result of Vasilovsky about the Z-graded
identities of the algebra Mn(K) over fields of characteristic 0.
VARIATIONAL PRINCIPLES FOR MONOTONE AND MAXIMAL BIFUNCTIONS
Zaki Chbani
chbaniz@ucam.ac.ma,
Hassan Riahi
h-riahi@ucam.ac.ma
2000 Mathematics Subject Classification:
49J40, 49J35, 58E30, 47H05.
Key words:
Brø ndsted-Rockafellar principle, equilibrium problem.
We establish variational principles for monotone and maximal bifunctions of
Brø ndsted-Rockafellar type by using our characterization of bifunction's
maximality in reflexive Banach spaces. As applications, we give an existence
result of saddle point for convex-concave function and solve an approximate
inclusion governed by a maximal monotone operator.
DUALITY IN CONSTRAINED DC-OPTIMIZATION VIA TOLAND'S DUALITY
APPROACH
M. Laghdir
laghdir@ucd.ac.ma,
N. Benkenza
2000 Mathematics Subject Classification:
90C48, 49N15, 90C25.
Key words:
Conjugate, difference convex, Toland-Singer
duality, DC-constraint.
In this paper we reconsider a nonconvex duality
theory established by B. Lemaire and M. Volle (see [4]),
related to a primal problem of minimizing the difference of two
convex functions subject to a DC-constraint. The purpose of this
note is to present a new method based on Toland-Singer duality
principle. Applications to the case when the constraints are
vector-valued are provided.
ON A STRATIFICATION DEFINED BY REAL ROOTS OF POLYNOMIALS
Vladimir Petrov Kostov
kostov@math.unice.fr
2000 Mathematics Subject Classification:
12D10.
Key words:
Multiplicity vector; multiplicity surplus.
We consider the family of polynomials
P(x,a) = xn+a1xn-1+¼+
an,x,ai Î R,
and the stratification of
Rn @
{ (a1,¼,an)
|ai Î R}
defined by the
multiplicity vector of the real roots of P. We prove smoothness
of the strata and a transversality property of their
tangent spaces.
ON THE EXTINCTION PROBABILITY FOR BISEXUAL BRANCHING PROCESSES IN
VARYING ENVIRONMENTS
Manuel Molina
mmolina@unex.es,
Manuel Mota
mota@unex.es,
Alfonso Ramos
aramos@unex.es
2000 Mathematics Subject Classification:
60J80.
Key words:
Bisexual branching processes. Branching processes
in varying environments. Extinction probability.
In this paper, the bisexual branching process in varying
environments introduced in [9] is considered and some sufficient
conditions for the existence of positive probability of
non-extinction are established.
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