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. 1-16)
• Gadhi, N., A. Metrane. Optimality conditions for D.C. vector optimization problems under D.C. constraints (pp. 17-32)
• Finta, Z. Direct and converse theorems for generalized Bernstein-type operators (pp. 33-42)
• Komeda, J., A. Ohbuchi. Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve (pp. 43-54)
• Mahdi, S. Conditional confidence interval for the scale parameter of a Weibull distribution (pp. 55-70)
• Elabbasy, E. M., T. S. Hassan. Oscillation criteria for first order delay differential equations (pp. 71-86)
• Karim, D. Zero-dimensionality and Serre rings (pp. 87-94)
• Khadzhiivanov, N., N. Nenov. Sequences of maximal degree vertices in graphs (pp. 95-102)

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.shu-bg.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)dx-f(x,y)dy], where d(h) is the closed connected component of the level curve {[(x²+y²)/2]-[(x³)/3]+axy² = 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, Lagrange-Fritz-John 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 BERNSTEIN-TYPE OPERATORS
Zoltán Finta fzoltan@math.ubbcluj.ro

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36. Key words: Goodman-Sharma operator, direct and converse approximation theorems, $K$-functional.

We establish direct and converse theorems for generalized parameter dependent Bernstein-type operators. The direct estimate is given using a K-functional and the inverse result is a strong converse inequality of type A, in the terminology of [2].

WEIERSTRASS POINTS WITH FIRST NON-GAP FOUR ON A DOUBLE COVERING OF A HYPERELLIPTIC CURVE
Jiryo Komeda komeda@gen.kanagawa-it.ac.jp
Akira Ohbuchi ohbuchi@ias.tokushima-u.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, 4-semigroup.

Let H be a 4-semigroup, 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® P¹ 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 two-sided 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 = q₀ where q₀ 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 q₀. However, when q is in the vicinity of q₀, 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 first-order delay differential equation of the form

. x   ( t) +p( t) x( t(t) ) = 0,

where p,  t Î C[ [ t₀,¥) ,R⁺], R⁺ = [ 0,¥), t( t) is nondecreasing, t( t) < t for t ³ t₀ 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 > e-1 e-2 æ ç è k+ 1 l1 ö ÷ ø - 1 e-2 ,

holds, where l₁ is the smaller root of the equation l = e^kl.

ZERO-DIMENSIONALITY AND SERRE RINGS
D. Karim ikarim@ucam.ac.ma

2000 Mathematics Subject Classification: Primary 13A99; Secondary 13A15, 13B02, 13E05. Key words: Zero-dimensional ring, semi-quasilocal ring, Nagata ring, von Neumann regular ring, directed union of Artinian subrings, Serre ring.

This paper deals with zero-dimensionality. 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 zero-dimensional rings and each X_a is an indeterminate over R_a.

SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS
Nickolay Khadzhiivanov hadji@fmi.uni-sofia.bg
Nedyalko Nenov nenov@fmi.uni-sofia.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 v₁,...,v_r is a vertex sequence in G such that G(v₁,...,v_r) = Æ and v_i is a maximal degree vertex in G(v₁,...,v_i-1), we prove that e(G) £ e(K(p₁,...,p_r)) where K(p₁,...,p_r) is the complete r-partite graph with p_i = |G(v₁,...,v_i-1)\G(v_i)|.