Djeddour, K., A. Mokkadem, M. Pelletier.
On the recursive estimation of the location and of the size of the
mode of a probability density
(pp. 651688)
A B S T R A C T S
OSCILLATION OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL
EQUATIONS
E. M. Elabbasy
emelabbasy@mans.edu.eg,
T. S. Hassan
tshassan@mans.edu.eg
2000 Mathematics Subject Classification:
34K15, 34C10.
Key words:
Oscillation, neutral delay, differential equations.
In this paper, we study the oscillatory behavior of first order
nonlinear neutral delay differential equation
(x(t) − q(t) x(t − σ(t))) ′ +f(t,x( t − τ(t))) = 0, 

where σ, τ ∈ C([t_{0},∞),(0,∞)),
q Î C([t_{0},∞), [0,∞)) and
f ∈ C([t_{0},∞) ×R,R). The obtained results extended and improve
several of the well known previously results in the literature. Our
results are illustrated with an example.
TAYLOR SPECTRUM AND CHARACTERISTIC FUNCTIONS OF
COMMUTING 2CONTRACTIONS
Berrabah Bendoukha
bbendoukha@gmail.com
2000 Mathematics Subject Classification:
47A10, 47A13.
Key words:
Characteritic function, 2contraction, Taylor spectrum.
In this paper, we give a description of Taylor spectrum of commuting
2contractions in terms of characteritic functions of such contractions. The
case of a single contraction obtained by B. Sz. Nagy and C. Foias is
generalied in this work.
GENERALIZED DSYMMETRIC OPERATORS I
S. Bouali
said.bouali@yahoo.fr,
M. Echchad
m.echchad@yahoo.fr
2000 Mathematics Subject Classification:
Primary: 47B47, 47B10; secondary 47A30.
Key words:
Generalized derivation, selfadjoint
derivation ranges, Dsymmetric operators.
Let H be an infinitedimensional complex Hilbert space and let
A, B ∈ L(H), where L(H) is the algebra
of operators on H into itself. Let δ_{AB}:
L(H) → L(H) denote the generalized derivation
δ_{AB}(X) = AX − XB. This note will initiate a study on the class of pairs (A,B) such that [‾(R(δ_{AB}))] = [‾(R(δ_{B*A*}))]; i.e.
[‾(R(δ_{AB}))] is selfadjoint.
PREDEGREE POLYNOMIALS OF PLANE CONFIGURATIONS IN
PROJECTIVE SPACE
Dimitre Tzigantchev
dtzigantchev@maritime.edu
2000 Mathematics Subject Classification:
14N10, 14C17.
Key words:
Planes, hyperplanes, arrangements, configurations.
We work over an algebraically closed field of characteristic zero.
The group PGL(4) acts naturally on P^{N} which
parameterizes surfaces of a given degree in P^{3}. The
orbit of a surface under this action is the image of a rational map
PGL(4) ⊂ P^{15}→P^{N}. The
closure of the orbit is a natural and interesting object to study.
Its predegree is defined as the degree of the orbit closure
multiplied by the degree of the above map restricted to a
general P^{j}, j being the dimension of the orbit. We
find the predegrees and other invariants for all surfaces supported
on unions of planes. The information is encoded in the socalled
predegree polynomials , which possess nice multiplicative
properties allowing us to compute the predegree (polynomials) of
various special plane configurations.
The predegree has both combinatorial and geometric significance. The
results obtained in this paper would be a necessary step in the
solution of the problem of computing predegrees for all surfaces.
WARPED PRODUCT SEMISLANT SUBMANIFOLDS OF A SASAKIAN MANIFOLD
Falleh R. AlSolamy
falleh@hotmail.com,
Viqar Azam Khan
viqar@gmail.com
2000 Mathematics Subject Classification:
53C40, 53C25.
Key words:
Sasakian manifolds, semislant warped
product submanifold, contact CRsubmanifold, warped product
manifold.
In the present note, it is proved that there donot exist warped
product semislant submanifolds in a Sasakian manifold other than
contact CRwarped product submanifolds and thus the results obtained
in [8] are generalized.
FIRSTORDER CONDITIONS FOR OPTIMIZATION
PROBLEMS WITH QUASICONVEX INEQUALITY CONSTRAINTS
Ivan Ginchev
iginchev@eco.uninsubria.it,
Vsevolod I. Ivanov
vsevolodivanov@yahoo.com
2000 Mathematics Subject Classification:
90C46, 90C26, 26B25, 49J52.
Key words:
Nonsmooth optimization, Dini
directional derivatives, quasiconvex functions, pseudoconvex
functions, quasiconvex programming, KuhnTucker conditions.
The constrained optimization problem min f(x), g_{j}(x) ≤ 0
(j = 1,…p) is considered, where f : X → R and
g_{j} : X → R are nonsmooth functions with domain
X ⊂ R^{n}. Firstorder necessary and firstorder
sufficient optimality conditions are obtained when g_{j} are
quasiconvex functions. Two are the main features of the paper: to
treat nonsmooth problems it makes use of Dini derivatives; to obtain
more sensitive conditions, it admits directionally dependent
multipliers. The two cases, where the Lagrange function satisfies a
nonstrict and a strict inequality, are considered. In the case of a
nonstrict inequality pseudoconvex functions are involved and in
their terms some properties of the convex programming problems are
generalized. The efficiency of the obtained conditions is
illustrated on examples.
CONVEXITY AROUND THE UNIT OF A BANACH ALGEBRA
Vladimir Kadets
vova1kadets@yahoo.com,
Olga Katkova
olga.m.katkova@univer.kharkov.ua,
Miguel Martín
mmartins@ugr.es,
Anna Vishnyakova
anna.m.vishnyakova@univer.kharkov.ua
2000 Mathematics Subject Classification:
Primary: 46B20. Secondary: 46H99, 47A12.
Key words:
Unital Banach
algebra, strongly extreme point, midpoint modulus of local
convexity.
We estimate the (midpoint) modulus of convexity at the unit
1 of a Banach algebra A showing that
inf {max_{±}1 ± x − 1 : x ∈ A, x=ε}
≥ (π/4e)ε²+o(ε²)
as ε → 0. We also give a characterization of
twodimensional subspaces of Banach algebras containing the identity
in terms of polynomial inequalities.
OSTROWSKI TYPE INEQUALITIES OVER SPHERICAL SHELLS
George A. Anastassiou
ganastss@memphis.edu
2000 Mathematics Subject Classification:
26D10, 26D15.
Key words:
Ostrowski inequality, sharp
inequality, multivariate inequality, spherical shell.
Here are presented Ostrowski type inequalities
over spherical shells. These regard sharp or close to sharp
estimates to the difference of the average of a multivariate
function from its value at a point.
ON THE RECURSIVE ESTIMATION OF THE LOCATION AND OF THE
SIZE OF THE MODE OF A PROBABILITY DENSITY
Khédidja Djeddour
djeddour@math.uvsq.fr,
Abdelkader Mokkadem
mokkadem@math.uvsq.fr,
Mariane Pelletier
pelletier@math.uvsq.fr
2000 Mathematics Subject Classification:
62G07, 62L20.
Key words:
Location and size of the mode, nonparametric
estimation, stochastic approximation algorithms, averaging
principle.
Tsybakov [31] introduced the method of stochastic approximation to
construct a recursive estimator of the location q of the mode
of a probability density. The aim of this paper is to provide a
companion algorithm to Tsybakov's algorithm, which allows to
simultaneously recursively approximate the size m of the mode.
We provide a precise study of the joint weak convergence rate of
both estimators. Moreover, we introduce the averaging principle of
stochastic approximation algorithms to construct asymptotically
efficient algorithms approximating the couple (q,m).
Back