Serdica Mathematical Journal

Volume 34, Number 2, 2008

C O N T E N T S

• Sarkar, A. K. A unified group-theoretic method on improper partial semi-bilateral generating functions (pp. 373-382)
• Bensalem, N. Régularité Lipschitzienne des géodésiques minimisantes pour quelques distributions affines (pp. 383-394)
• Elabbasy, E. M., W. W. Elhaddad. Oscillation criteria for nonlinear differential equations of second order with damping term (pp. 395-414)
• Dzhumadil'daev, A. S. q-Leibniz algebras (pp. 415-440)
• Marinov, R. Ts. An iterative procedure for solving nonsmooth generalized equation (pp. 441-454)
• Argyros, I. K., S. Hilout. Steffensen methods for solving generalized equations (pp. 455-466)
• Kovacheva, R. K. Zeros of sequences of partial sums and overconvergence (pp. 467-482)
• Kurbanov, S. Limit theorems for non-critical branching processes with continuous state space (pp. 483-488)
• Gourdin, D., H. Kamoun, O. Ben Khalifa. Solutions globales régulières pour quelques équations lineaires d'évolution du type pseudo-différentiel singulier (pp. 489-508)
• Veleva, E. Test for independence of the variables with missing elements in one and the same column of the empirical correlation matrix (pp. 509-530)

A B S T R A C T S

A UNIFIED GROUP-THEORETIC METHOD ON IMPROPER PARTIAL SEMI-BILATERAL GENERATING FUNCTIONS
Asit Kumar Sarkar asit_kumar_sarkar@yahoo.com

2000 Mathematics Subject Classification: 33A65, 33C20.
Key words: Generating relation, Improper partial semi-bilateral generating function.

A unifed group-theoretic method of obtaining more general class of generating functions from a given class of improper partial semi-bilateral generating functions involving Laguerre and Gegenbauer polynomials are discussed.

RÉGULARITÉ LIPSCHITZIENNE DES GÉODÉSIQUES MINIMISANTES POUR QUELQUES DISTRIBUTIONS AFFINES
Naceurdine Bensalem naceurdine_bensalem@yahoo.fr

2000 Mathematics Subject Classification: 49J15, 49J30, 53B50.
Key words: Contrôle optimal, Régularité Lipschitzienne, Distribution affine, Géodésique.

In the context of sub-Riemannian geometry and the Lipschitzian regularity of minimizers in control theory, we investigate some properties of minimizing geodesics for certain affine distributions. In particular, we consider the case of a generalized H2-strong affine distribution and the case of an affine Plaff system of maximal class.

OSCILLATION CRITERIA FOR NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH DAMPING TERM
E. M. Elabbasy emelabbasy@mans.edu.eg
W. W. Elhaddad

2000 Mathematics Subject Classification: 34C10, 34C15.
Key words: Oscillation, second order nonlinear differential equation.

Some new criteria for the oscillation of all solutions of second order differential equations of the form

(d/dt)(r(t)ψ(x)|dx/dt|^α−2(dx/dt))+ p(t)φ(|x|^α−2x,r(t) ψ(x)|dx/dt|^α−2(dx/dt))+q(t)|x|^α−2 x=0,

and the more general equation

(d/dt)(r(t)ψ(x)|dx/dt|^α−2(dx/dt))+p(t)φ(g(x),r(t) ψ(x)|dx/dt|^α−2 (dx/dt))+q(t)g(x)=0,

are established. our results generalize and extend some known oscillation criterain in the literature.

q-LEIBNIZ ALGEBRAS
A. S. Dzhumadil'daev askar@math.kz, askar56@hotmail.com

2000 Mathematics Subject Classification: Primary 17A32, Secondary 17D25.
Key words: Leibniz algebras, Zinbiel algebras, Omni-Lie algebras, polynomial identities, q-commutators

An algebra (A,ο) is called Leibniz if aο(bοc) = (a ο b)ο c-(a ο c) ο b for all a,b,c ∈ A. We study identities for the algebras A^(q) = (A,ο_q), where a ο_q b = a ο b+q b ο a is the q-commutator. Let Char K ≠ 2,3. We show that the class of q-Leibniz algebras is defined by one identity of degree 3 if q² ≠ 1, q ≠−2, by two identities of degree 3 if q = −2, and by the commutativity identity and one identity of degree 4 if q = 1. In the case of q = −1 we construct two identities of degree 5 that form a base of identities of degree 5 for −1-Leibniz algebras. Any identity of degree < 5 for −1-Leibniz algebras follows from the anti-commutativity identity.

AN ITERATIVE PROCEDURE FOR SOLVING NONSMOOTH GENERALIZED EQUATION
Rumen Tsanev Marinov marinov_r@yahoo.com

2000 Mathematics Subject Classification: 47H04, 65K10.
Key words: Set-valued maps, generalized equation, linear convergence, Aubin continuity.

In this article, we study a general iterative procedure of the following form 0 ∈ f(x_k)+F(x_k+1), where f is a function and F is a set valued map acting from a Banach space X to a linear normed space Y, for solving generalized equations in the nonsmooth framework. We prove that this method is locally Q-linearly convergent to x^* a solution of the generalized equation 0 ∈ f(x)+F(x) if the set-valued map [f(x^*)+g(·)−g(x^*)+F(·)]⁻¹ is Aubin continuous at (0,x^*), where g:X→ Y is a function, whose Fréchet derivative is L-Lipschitz.

STEFFENSEN METHODS FOR SOLVING GENERALIZED EQUATIONS
Ioannis K. Argyros ioannisa@cameron.edu,
Saïd Hilout said_hilout@yahoo.fr

2000 Mathematics Subject Classification: 65G99, 65K10, 47H04.
Key words: Steffensen's method, Banach space, set--valued mapping, generalized equations, Aubin continuity, divided difference, Newton's method.

We provide a local convergence analysis for Steffensen's method in order to solve a generalized equation in a Banach space setting. Using well known fixed point theorems for set-valued maps [13] and Hölder type conditions introduced by us in [2] for nonlinear equations, we obtain the superlinear local convergence of Steffensen's method. Our results compare favorably with related ones obtained in [11].

ZEROS OF SEQUENCES OF PARTIAL SUMS AND OVERCONVERGENCE
Ralitza K. Kovacheva rkovach@math.bas.bg

2000 Mathematics Subject Classification: 30B40, 30B10, 30C15, 31A15.
Key words: Overconvergence, Hadamard-Ostrowski gaps, equilibrium measure.

We are concerned with overconvergent power series. The main idea is to relate the distribution of the zeros of subsequences of partial sums and the phenomenon of overconvergence. Sufficient conditions for a power series to be overconvergent in terms of the distribution of the zeros of a subsequence are provided, and results of Jentzsch-Szegö type about the asymptotic distribution of the zeros of overconvergent subsequences are stated.

LIMIT THEOREMS FOR NON-CRITICAL BRANCHING PROCESSES WITH CONTINUOUS STATE SPACE
S. Kurbanov kurbanov@kma.zcu.cz

2000 Mathematics Subject Classification: Primary 60J80, Secondary 60G99.
Key words: Random variable, branching process, decreasing immigration, independent increment, factorial moment.

In the paper a modification of the branching stochastic process with immigration and with continuous states introduced by Adke S. R. and Gadag V. G. (1995) is considered. Limit theorems for the non-critical processes with or without non-stationary immigration and finite variance are proved. The subcritical case is illustrated with examples.

SOLUTIONS GLOBALES RÉGULIÈRES POUR QUELQUES ÉQUATIONS LINEAIRES D'ÉVOLUTION DU TYPE PSEUDO-DIFFÉRENTIEL SINGULIER
D. Gourdin gourdin@math.jussieu.fr,
H. Kamoun,
O. Ben Khalifa

2000 Mathematics Subject Classification: 35C15, 35D05, 35D10, 35S10, 35S99.
Key words: Linear Cauchy problem, global solution, Sobolev spaces, Integral Kirchhof solutions.

We give here examples of equations of type (1) ∂_tt² y -p(t, D_x) y = 0, where p is a singular pseudo-differential operator with regular global solutions when the Cauchy data are regular, t ∈ R,  x ∈ R⁵ .

TEST FOR INDEPENDENCE OF THE VARIABLES WITH MISSING ELEMENTS IN ONE AND THE SAME COLUMN OF THE EMPIRICAL CORRELATION MATRIX
Evelina Veleva eveleva@abv.bg

2000 Mathematics Subject Classification: 62H15, 62H12.
Key words: Multivariate normal distribution, Wishart distribution, correlation matrix completion, maximum likelihood ratio test.

We consider variables with joint multivariate normal distribution and suppose that the sample correlation matrix has missing elements, located in one and the same column. Under these assumptions we derive the maximum likelihood ratio test for independence of the variables. We obtain also the maximum likelihood estimations for the missing values.