Serdica Mathematical Journal
Volume 30, Number 1, 2004
C O N T E N T S

Mihajlova, A. D.
Estimate for the number of zeros of Abelian integrals
on elliptic curves
(pp. 116)

Gadhi, N., A. Metrane.
Optimality conditions for D.C. vector optimization problems
under D.C. constraints
(pp. 1732)

Finta, Z.
Direct and converse theorems for generalized
Bernsteintype operators
(pp. 3342)

Komeda, J., A. Ohbuchi.
Weierstrass points with first nongap four on a double
covering of a hyperelliptic curve
(pp. 4354)

Mahdi, S.
Conditional confidence interval for the scale parameter of
a Weibull distribution
(pp. 5570)

Elabbasy, E. M., T. S. Hassan.
Oscillation criteria for first order delay differential equations
(pp. 7186)

Karim, D.
Zerodimensionality and Serre rings
(pp. 8794)

Khadzhiivanov, N., N. Nenov.
Sequences of maximal degree vertices in graphs
(pp. 95102)
A B S T R A C T S
ESTIMATE FOR THE NUMBER OF ZEROS OF ABELIAN INTEGRALS ON ELLIPTIC CURVES
Ana Dimitrova Mihajlova
a.mihajlova@fmi.shubg.net
2000 Mathematics Subject Classification:
Primary 34C07, secondary 34C08.
Key words:
Abelian intergrals, limit cycles.
For h Î (0,^{1}/_{6}) we obtain an upper bound for the
number of zeros of the Abelian integral h ® I(h) = ò_{d(h)}[g(x,y)dxf(x,y)dy],
where d(h) is the closed connected component of
the level curve
{[(x^{2}+y^{2})/2][(x^{3})/3]+axy^{2} = h}, a
Î (0,1). The bound depends explicitly
on the maximum of degrees of the polynomials f and g.
OPTIMALITY CONDITIONS FOR D.C. VECTOR OPTIMIZATION PROBLEMS UNDER D.C. CONSTRAINTS
N. Gadhi
n.gadhi@ucam.ac.ma
A. Metrane
metrane@ucam.ac.ma
2000 Mathematics Subject Classification:
Primary 90C29; Secondary 49K30.
Key words:
Convex mapping, D.C. mapping,
LagrangeFritzJohn multipliers, Local weak minimal solution, Optimality
condition, Subdifferential.
In this paper, we establish necessary optimality conditions and sufficient
optimality conditions for D.C. vector optimization problems under D.C.
constraints. Under additional conditions, some results of [9] and
[15] are also recovered.
DIRECT AND CONVERSE THEOREMS FOR GENERALIZED BERNSTEINTYPE OPERATORS
Zoltán Finta
fzoltan@math.ubbcluj.ro
2000 Mathematics Subject Classification:
41A25, 41A27, 41A36.
Key words:
GoodmanSharma operator,
direct and converse approximation theorems, $K$functional.
We establish direct and converse theorems for
generalized parameter dependent Bernsteintype operators. The
direct estimate is given using a Kfunctional and the inverse
result is a strong converse inequality of type A, in the
terminology of [2].
WEIERSTRASS POINTS WITH FIRST NONGAP FOUR ON A DOUBLE
COVERING OF A HYPERELLIPTIC CURVE
Jiryo Komeda
komeda@gen.kanagawait.ac.jp
Akira Ohbuchi
ohbuchi@ias.tokushimau.ac.jp
2000 Mathematics Subject Classification:
Primary 14H55; Secondary 14H30, 14H40, 20M14.
Key words:
Weierstrass semigroup of a point, double covering of a
hyperelliptic curve, 4semigroup.
Let H be a 4semigroup, i.e., a numerical
semigroup whose minimum positive element is four.
We denote by 4r(H)+2 the minimum element of H
which is congruent to 2 modulo 4. If the genus g of H is larger
than 3r(H)1, then there is a cyclic covering
p:C® P^{1} of curves
with degree 4 and its ramification point P
such that the Weierstrass semigroup H(P) of P is H (Komeda [1]).
In this paper it is showed that we can construct a double covering of
a hyperelliptic curve and its ramification point P such that H(P)
is equal to H even if g £ 3r(H)1.
CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF
A WEIBULL DISTRIBUTION
Smail Mahdi
smahdi@uwichill.edu.bb
2000 Mathematics Subject Classification:
62F25, 62F03.
Key words:
Weibull distribution, rejection of a
preliminary hypothesis, conditional and
unconditional interval estimator, likelihood
ratio interval, coverage probability, average length, simulation.
A twosided conditional confidence interval
for the scale parameter q of a Weibull
distribution is constructed. The construction
follows the rejection of a preliminary test for
the null hypothesis:
q = q_{0} where q_{0}
is a given value. The confidence bounds are derived according
to the method set forth by Meeks and D'Agostino (1983) and
subsequently used by Arabatzis et al. (1989) in Gaussian models
and more recently by Chiou and Han (1994, 1995) in
exponential models. The derived conditional confidence
interval also suits non large samples since it is based
on the modified pivot statistic advocated in Bain and
Engelhardt (1981, 1991). The average length and the coverage
probability of this conditional interval are compared with
whose of the corresponding optimal unconditional interval
through simulations. The study has shown that both intervals are
similar when the population scale parameter is far enough from
q_{0}. However, when q is in the vicinity of
q_{0}, the conditional interval outperforms the
unconditional one in terms of length and also maintains a reasonably
high coverage probability. Our results agree with the findings of
Chiou and Han and Arabatzis et al. which contrast with whose
of Meeks and D'Agostino stating that the unconditional interval is
always shorter than the conditional one. Furthermore, we derived
the likelihood ratio confidence interval for q and compared
numerically its performance with the two other interval estimators.
OSCILLATION CRITERIA FOR FIRST ORDER DELAY DIFFERENTIAL EQUATIONS
E. M. Elabbasy
emelabbasy@mans.edu.eg
T. S. Hassan
tshassan@mans.edu.eg
2000 Mathematics Subject Classification:
34K15.
Key words:
Oscillation, Delay differential equations.
This paper is concerned with the oscillatory behavior of firstorder delay
differential equation of the form
 .
x

( t) +p( t) x( t(t) ) = 0, 

where p, t Î C[ [ t_{0},¥) ,R^{+}],
R^{+} = [ 0,¥), t( t) is nondecreasing, t( t) < t for t
³ t_{0} and lim_{t®
¥}t( t) = ¥.
Let the numbers k and l be defined by
k = 
lim
t® ¥


inf
 
ó õ

t
t(t)

p(s)ds and l = 
lim
t® ¥


sup
 
ó õ

t
t(t)

p(s)ds. 

It is proved here that when l < 1 and 0 < k
£ ^{1}/_{e} all solutions of
this equation oscillate in several cases in which the condition
l > 
e1
e2


æ ç
è

k+ 
1
l_{1}


ö ÷
ø

 
1
e2

, 

holds, where l_{1} is the smaller root of the equation
l = e^{kl}.
ZERODIMENSIONALITY AND SERRE RINGS
D. Karim
ikarim@ucam.ac.ma
2000 Mathematics Subject Classification:
Primary 13A99; Secondary 13A15, 13B02, 13E05.
Key words:
Zerodimensional ring, semiquasilocal ring, Nagata
ring, von Neumann regular ring, directed union of Artinian
subrings, Serre ring.
This paper deals with zerodimensionality. We investigate the problem of
whether a Serre ring R < X > is expressible as a directed union of
Artinian subrings. In particular, we show that
Õ_{a Î A}(R_{a} < X_{a} > ) is not a directed union of Artinian
subrings, where {R_{a}}_{a Î A} is an infinite
family of zerodimensional rings and each X_{a} is an
indeterminate over R_{a}.
SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS
Nickolay Khadzhiivanov
hadji@fmi.unisofia.bg
Nedyalko Nenov
nenov@fmi.unisofia.bg
2000 Mathematics Subject Classification:
05C35.
Key words:
Maximal degree vertex, complete $s$partite graph, Turan's graph.
Let G(M) where M Ì V(G) be the set of all
vertices of the graph G adjacent to any vertex of M.
If v_{1},...,v_{r} is a vertex sequence in G such
that G(v_{1},...,v_{r}) = Æ and
v_{i} is a maximal degree vertex in G(v_{1},...,v_{i1}),
we prove that e(G) £ e(K(p_{1},...,p_{r})) where K(p_{1},...,p_{r})
is the complete rpartite graph
with p_{i} = G(v_{1},...,v_{i1})\G(v_{i}).
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