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. 689-714)

  Articles

• Hassan, T. S. Interval oscillation for second order nonlinear differential equations with a damping term (pp. 715-732)
• Dhagat, V. B., A. Sharma. Weak and semi compatible maps in probabilistic metric space using implicit relation (pp. 733-742)
• Ezzaldine, H., V. P. Kostov. Even and old overdetermined strata for degree 6 hyperbolic polynomials (pp. 743-770)
• Komeda, J., A. Ohbuchi. Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve II (pp. 771-782)
• Kolev, N. New upper bound for the edge Folkman number F_e(3,5;13) (pp. 783-790)

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.uni-giessen.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 ≥ t0,

where α ≥ 1, a ∈ C¹([t₀,∞);(0,∞)) and p,q ∈ C([t₀,∞);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 [t₀,∞), rather than on the whole half-line 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, Semi-compatible 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 self-maps 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 NON-GAP FOUR ON A DOUBLE COVERING OF A HYPERELLIPTIC CURVE II
Jiryo Komeda komeda@gen.kanagawa-it.ac.jp,
Akira Ohbuchi ohbuchi@ias.tokushima-u.ac.jp

2000 Mathematics Subject Classification: Primary 14H55; Secondary 14H30, 14J26.
Key words: Weierstrass semigroup of a point, double covering of a hyperelliptic curve, 4-semigroup.

A 4-semigroup means a numerical semigroup whose minimum positive integer is 4. In [7] we showed that a 4-semigroup 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 4-semigroup.

NEW UPPER BOUND FOR THE EDGE FOLKMAN NUMBER F_e(3,5;13)
Nikolay Kolev nickyxy@fmi.uni-sofia.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

G e →   (a1, …, ar)

means that in every r-coloring 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

Fe(a1, …, ar; q) = min{|V(G)| : G e →   (a1, …, ar; 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

Fe(3,5;13) ≤ 21.

This improves the bound

Fe(3,5;13) ≤ 24,

proved by Kolev and Nenov.