Serdica Mathematical Journal
Volume 34, Number 4, 2008
C O N T E N T S
Survey

De Beule, J., A. Klein, K. Metsch, L. Storme.
Partial ovoids and partial spreads of classical finite polar spaces
(pp. 689714)
Articles

Hassan, T. S.
Interval oscillation for second order
nonlinear differential equations with a damping term
(pp. 715732)

Dhagat, V. B., A. Sharma.
Weak and semi compatible maps
in probabilistic metric space using implicit relation
(pp. 733742)

Ezzaldine, H., V. P. Kostov.
Even and old overdetermined strata for degree 6 hyperbolic polynomials
(pp. 743770)

Komeda, J., A. Ohbuchi.
Weierstrass points with first nongap four on a double covering of a
hyperelliptic curve
II
(pp. 771782)

Kolev, N.
New upper bound for the edge Folkman number
F_{e}(3,5;13)
(pp. 783790)
A B S T R A C T S
PARTIAL OVOIDS AND PARTIAL SPREADS OF
CLASSICAL FINITE POLAR SPACES
J. De Beule
jdebeule@cage.ugent.be,
K. Metsch
Klaus.Metsch@math.unigiessen.de,
A. Klein
klein@cage.ugent.be,
L. Storme
ls@cage.ugent.be
2000 Mathematics Subject Classification:
05B25, 51E20.
Key words:
Partial ovoids, partial spreads, classical finite polar spaces.
We survey the main results on ovoids and spreads, large maximal
partial ovoids and large maximal partial spreads, and on small
maximal partial ovoids and small maximal partial spreads in
classical finite polar spaces. We also discuss the main results on
the spectrum problem on maximal partial ovoids and maximal partial
spreads in classical finite polar spaces.
INTERVAL OSCILLATION FOR SECOND ORDER
NONLINEAR DIFFERENTIAL EQUATIONS WITH A DAMPING TERM
Taher S. Hassan
tshassan@mans.edu.eg
2000 Mathematics Subject Classification:
34C10, 34C15.
Key words:
Interval oscillation, Second order, Nonlinear differential
equations.
It is the purpose of this paper to give oscillation criteria for the
second
order nonlinear differential equation with a damping term
(a(t) y′(t))′ + p(t)y′(t) + q(t) y(t) ^{α−1} y(t) = 0, t ≥ t_{0}, 

where α ≥ 1,
a ∈ C^{1}([t_{0},∞);(0,∞)) and p,q ∈
C([t_{0},∞);R). Our results here are different, generalize and improve some known results for oscillation of second order nonlinear
differential equations that are different from most known ones in
the sencse they are based on the information only on a sequence of
subintervals of [t_{0},∞), rather than on the
whole halfline and can be applied to extreme cases such as
∫_{t0}^{∞} q(t) dt = − ∞. Our results
are illustrated with an example.
WEAK AND SEMI COMPATIBLE MAPS IN PROBABILISTIC
METRIC SPACE USING IMPLICIT RELATION
Vanita Ben Dhagat
vanita1_dhagat@yahoo.co.in,
Akshay Sharma
2000 Mathematics Subject Classification:
54H25, 47H10.
Key words:
Menger space, Weak compatible mapping,
Semicompatible mapping, Implicit function, common fixed point.
The concept of semi compatibility is given in probabilistic metric
space and it has been applied to prove the existence of unique
common fixed point of four selfmaps with weak compatibility
satisfying an implicit relation. At the end we provide examples in
support of the result.
EVEN AND OLD OVERDETERMINED STRATA FOR
DEGREE 6 HYPERBOLIC POLYNOMIALS
Hayssam Ezzaldine
haysam82@hotmail.com,
Vladimir Petrov Kostov
kostov@math.unice.fr
2000 Mathematics Subject Classification:
12D10.
Key words:
(strictly) hyperbolic polynomial; Gegenbauer
polynomial; overdetermined stratum.
In the present paper we consider degree 6 hyperbolic polynomials (HPs)
in one variable (i.e. real and with all roots real). We are interested in such
HPs whose number of equalities between roots of the
polynomial and/or its derivatives is higher than expected. We give the
complete study of the four families of such degree 6 even HPs and also
of HPs which are primitives of degree 5 HPs.
WEIERSTRASS POINTS WITH FIRST NONGAP
FOUR ON A DOUBLE COVERING OF A HYPERELLIPTIC CURVE II
Jiryo Komeda
komeda@gen.kanagawait.ac.jp,
Akira Ohbuchi
ohbuchi@ias.tokushimau.ac.jp
2000 Mathematics Subject Classification:
Primary 14H55; Secondary 14H30, 14J26.
Key words:
Weierstrass semigroup of a point, double covering of a hyperelliptic curve,
4semigroup.
A 4semigroup means a numerical semigroup whose minimum positive
integer is 4. In [7] we showed that a 4semigroup with
some conditions is the Weierstrass semigroup of a ramification point
on a double covering of a hyperelliptic curve. In this paper we
prove that the above statement holds for every 4semigroup.
NEW UPPER BOUND FOR THE EDGE FOLKMAN NUMBER
F_{e}(3,5;13)
Nikolay Kolev
nickyxy@fmi.unisofia.bg
2000 Mathematics Subject Classification:
05C55.
Key words:
Folkman graph, Folkman number.
For a given graph G let V(G) and E(G) denote the vertex and
the edge set of G respevtively. The symbol
means that in every rcoloring of E(G)
there exists a monochromatic a_{i}clique of color i for some
i ∈ {1,…,r}. The edge Folkman numbers are defined by the
equality
F_{e}(a_{1}, …, a_{r}; q) = min{V(G) : G 
e
→

(a_{1}, …, a_{r}; q) and cl(G) < q}. 

In this paper we prove a new upper bound on the edge Folkman number
F_{e}(3,5;13), namely
This improves the bound
proved by Kolev and Nenov.
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