Serdica Mathematical Journal

Volume 41, Numbers 2−3, 2015

C O N T E N T S

• Argyros, I. K., S. George. Expanding the applicability of Steffensen's method for finding fixed point of operators in Banach space (pp. 159−184)
• Makhijani, N., R. K. Sharma, J. B. Srivastava. The unit group of F_q[D₃₀] (pp. 185−198)
• Zagorodnyuk, S. M. On the truncated matrix power moment problem with a gap (pp. 199−228)
• Martino, I. Vertex collapsing and cut ideals (pp. 229−242)
• Atçeken, M., S. Dirik. Pseudo-slant submanifolds of a nearly Kenmotsu manifold (pp. 243−262)
• Garçon, M., J. Garnier, A. Omrane. Fixed points in multiple equilibria probability-migration models (pp. 263−276)
• da Silva e Silva, D. D. P. On the central polynomials with involution of M_1,1(E) (pp. 277−292)
• Kurdachenko, L. A., A. A. Pypka, I. Ya. Subbotin. On some relations between the factors of the upper and lower central series in Lie algebras (pp. 293−306)
• Slaoui, Y. Large and moderate deviation principles for averaged stochastic approximation method for the estimation of a regression function (pp. 307−328)
• Al Fares, A., E. Golvin, M. Krebs. A class of 2-groups of derived length three (pp. 329−332)

A B S T R A C T S

EXPANDING THE APPLICABILITY OF STEFFENSEN'S METHOD FOR FINDING FIXED POINT OF OPERATORS IN BANACH SPACE
Ioannis K. Argyros ioannisa@cameron.edu,
Santhosh George sgeorge@nitk.ac.in

2010 Mathematics Subject Classification: 65H10, 65G99, 47H17, 49M15.
Key words: Steffensen's method, Banach space, fixed point, local, semilocal convergence, divided difference.

We present a new local as well as a semilocal convergence analysis for Steffensen's method in order to locate fixed points of operators on a Banach space setting. Using more precise majorizing sequences we show under the same or less computational cost that our convergence criteria are weaker than in earlier studies such as [1]−[13], [19, 20, 26]. Numerical examples are provided to illustrate the theoretical results.

THE UNIT GROUP OF F_q[D₃₀]
Neha Makhijani nehamakhijani@gmail.com,
R. K. Sharma rksharma@maths.iitd.ac.in,
J. B. Srivastava jbsrivas@gmail.com

2010 Mathematics Subject Classification: 16S34, 20C05.
Key words: Group algebras, Wedderburn decomposition, Jacobson radical.

Let F_q be a finite field with q=pⁿ elements and D₃₀ be the dihedral group of order 30. A complete characterization of the unit group of the group algebra F_q[D₃₀] has been obtained.

ON THE TRUNCATED MATRIX POWER MOMENT PROBLEM WITH A GAP
S. M. Zagorodnyuk Sergey.M.Zagorodnyuk@univer.kharkov.ua

2010 Mathematics Subject Classification: 44A60, 47A57, 30E05.
Key words: moment problem, matrix measure, generalized resolvent.

In this paper we study the truncated matrix power moment problem with an odd number of prescribed moments. A Nevanlinna-type formula is derived for this moment problem in the case when the moment problem has more than one solution (the indeterminate case). The coefficients of the corresponding linear fractional transformation are expressed explicitly in terms of the given moments. In the determinate case the solution to the moment problem is constructed, as well. Those solutions of the moment problem, which satisfy the following condition: M(Δ)=0, where Δ is a prescribed open subset of R, and M is the matrix measure generated by a solution of the moment problem, are described.

VERTEX COLLAPSING AND CUT IDEALS
Ivan Martino ivan.martino@unifr.ch

2010 Mathematics Subject Classification: 05C25, 14M25.
Key words: Cut ideals, algebraic statistics, collapsing, non-classical cut ideals.

In this work we study how some elementary graph operations (like the disjoint union) and the collapse of two vertices modify the cut ideal of a graph. They pave the way for reducing the cut ideal of every graph to the cut ideal of smaller ones.
To deal with the collapse operation we generalize the definition of cut ideal given in literature, introducing the concepts of edge labeling and edge multiplicity: in fact we state the non-classical behavior of the cut ideal. Moreover we show the transformation of the toric map hidden behind these operations.

PSEUDO-SLANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD
M. Atçeken mehmet.atceken@gop.edu.tr,
S. Dirik suleyman.dirik@amasya.edu.tr

2010 Mathematics Subject Classification: 53C15, 53C25, 53C17, 53D15, 53D10.
Key words: nearly Kenmotsu manifold, slant submanifold, proper-slant submanifold, pseudo-slant submanifold.

In this paper, we study pseudo-slant submanifolds of nearly Kenmotsu manifolds. We characterize a totally umbilical proper-slant submanifolds and find a necessary and sufficient condition for such submanifold totally geodesic. Also we study integrability conditions of distributions of a pseudo-slant submanifold.

FIXED POINTS IN MULTIPLE EQUILIBRIA PROBABILITY-MIGRATION MODELS
M. Garçon manuel.garcon@gmail.com,
J. Garnier garnier@math.univ-paris-diderot.fr,
A. Omrane aomrane@guyane.univ-ag.fr

2010 Mathematics Subject Classification: 37C25, 40A99, 91B14, 91B50, 91B74.
Key words: Human capital, education, migration, indeterminacy, economic growth, fixed point.

We analyze a probability-migration model in which the probability of migration depends on human capital (education essentially). In this model, the human capital can converge to two possible values (fixed points), a low or high value. The aim of this paper is to analyze how belief mechanisms can lead to the selection of a particular fixed point.
In particular, we prove that for any belief mechanism, there exists a critical value H such that the result can always be expressed by an assertion of the form: if initially, the human capital is smaller than the critical value H, then the human capital will converge to the low fixed point, while if the initial human capital is larger than H, then the human capital will converge to the high fixed point value. The fixed points do not depend on the belief mechanism, while the critical value strongly depends on it.

ON THE CENTRAL POLYNOMIALS WITH INVOLUTION OF M_1,1(E)
Diogo Diniz Pereira da Silva e Silva diogo@dme.ufcg.edu.br

2010 Mathematics Subject Classification: 16R50, 16R10.
Key words: algebras with involution, central polynomials with involution, polynomial identities with involution.

Let K be an infinite field of characteristic ≠ 2. In this paper we study the *-central polynomials for the algebra M_1,1(E) with the involution induced by the transposition superinvolution on M_1,1(K). More precisely, we determine a finite set of polynomials that together with the\break *-identities generate the *-central polynomials. In the case when the field K has characteristic zero a finite set of generators for the *-central polynomials is determined.

ON SOME RELATIONS BETWEEN THE FACTORS OF THE UPPER AND LOWER CENTRAL SERIES IN LIE ALGEBRAS
L. A. Kurdachenko lkurdachenko@gmail.com,
A. A. Pypka pypka@ua.fm,
I. Ya. Subbotin isubboti@nu.edu

2010 Mathematics Subject Classification: 17B30, 17B65, 17B99.
Key words: Baer's Theorem, upper central series, lower central series, Z-decomposition, upper hypercenter, nilpotent residual, hypercentral residual.

A generalization for Lie algebras of the group theoretical Baer's theorem regarding relations between the factors of the upper and lower central series is proved.

LARGE AND MODERATE DEVIATION PRINCIPLES FOR AVERAGED STOCHASTIC APPROXIMATION METHOD FOR THE ESTIMATION OF A REGRESSION FUNCTION
Yousri Slaoui Yousri.Slaoui@math.univ-poitiers.fr

2010 Mathematics Subject Classification: 62G08, 62L20, 60F10.
Key words: Nonparametric regression, stochastic approximation algorithm, large and moderate deviations principles.

In this paper we prove large deviations principles for the averaged stochastic approximation method for the estimation of a regression function introduced by Mokkadem et al. (2009). We show that the averaged stochastic approximation algorithm constructed using the weight sequence which minimize the asymptotic variance gives the same pointwise LDP as the Nadaraya-Watson kernel estimator. Moreover, we give a moderate deviations principle for these estimators. It turns out that the rate function obtained in the moderate deviations principle for the averaged stochastic approximation algorithm constructed using the weight sequence which minimize the asymptotic variance is larger than the one obtained for the Nadaraya-Watson estimator and the one obtained for the semi-recursive estimator.

A CLASS OF $2$-GROUPS OF DERIVED LENGTH THREE
Ahmed Al Fares aalfare6@calstatela.edu,
Ezekiel Golvin zeke.golvin@gmail.com,
Mike Krebs mkrebs@calstatela.edu

2010 Mathematics Subject Classification: 20D15.
Key words: finite group, p-group, two-grouper, derived series.

We say that a finite group is a 2-grouper if it can be constructed by successive semidirect products with the cyclic group Z₂ of order 2, beginning with Z₂. In this paper, we investigate the finite 2-groupers of derived length 3. For such groups, we determine the possible orders of G/G′, G′/G′′, and G′′/G′′′, where H′ denotes the derived subgroup of a group H. Given positive integers a, b, c, we find two sufficient conditions on a, b, c to guarantee that there exists a 2-grouper G with |G/G′|=2^a and |G′/G′′|=2^b and |G′′/G′′′|=2^c: when a≥3 and b=3, or when a≥3 and b≥3 and c≤3.