Serdica Mathematical Journal

Volume 42, Numbers 3−4, 2016

C O N T E N T S

• Shchukin, M. On n-homogeneous C^*-algebras over two-dimensional non-oriented compact (pp. 203−210)
• Maheshwari, S., R. K. Sharma. Lie regular generators of general linear group GL(4, Z_n) (pp. 211−220)
• Fonseca, L. F. G. On Z₂-graded polynomial identities of sl₂(F) over a finite field (pp. 221−234)
• Johnson, S., A. Shaheen, G. Subuyuj. The isoperimetric number of a generalized Paley graph (pp. 235−250)
• Vladeva, D. I. Derivations in finite endomorphism semirings (pp. 251−260)
• Baghdad, A., M. Chraïbi Kaadoud, M. T. Loumi. Centre de masse et nouvelles inégalités pour la norme et le rayon numerique d'un opérateur linéaire borné (pp. 261−276)
• Kolev, E. Deletion-correcting codes and dominant vectors (pp. 277−286)
• Argyros, I. K., S. George. Extending the applicability of Newton-secant methods for functions with values in a cone (pp. 287−300)
• Ganchev, G., K. Kanchev, O. Kassabov. Transition to canonical principal parameters on maximal spacelike surfaces in Minkowski space (pp. 301−310)
• Hayssam, E. Overdetermined strata for degree 6 hyperbolic polynomials (pp. 311−324)
• Kurdachenko, L. A., I. Ya. Subbotin. A note on residuals in groups with quasicentral Abelian normal subgroups (pp. 325−334)
• Iordanov, I., N. Chervenov. Copulas on Sobolev spaces (pp. 335−360)

A B S T R A C T S

ON n-HOMOGENEOUS C^*-ALGEBRAS OVER TWO-DIMENSIONAL NON-ORIENTED COMPACT MANIFOLDS
Mikhail Shchukin mvs777777@gmail.com

2010 Mathematics Subject Classification: Primary 46L05, Secondary 19K99.
Key words: n-homogeneous C^*-algebras, algebraic bundle, two-dimensional manifold, classification of C^*-algebras, operator algebras.

We consider algebraic bundles over two-dimensional compact non-oriented connected manifold. Every non-oriented compact manifold can be realized as sphere S² with k projective planes on it. Let P^k be the sphere S² with k projective planes. Let ζ be algebraic bundle over P^k with fiber Mat(n). If n = 2m + 1 then the bundle ζ is trivial. If n = 2m then there are two non-isomorphic algebraic bundles over P^k with fiber Mat(n). J. Fell, J. Tomiyama, M. Takesaki showed in 1961 the correspondence between the classes of algebraic bundles and n-homogeneous C^*-algebras. Hence we can classify non-isomorphic n-homogeneous C^*-algebras over P^k.

LIE REGULAR GENERATORS OF GENERAL LINEAR GROUP GL(4, Z_n)
Swati Maheshwari swatimahesh88@gmail.com,
R. K. Sharma rksharmaiitd@gmail.com

2010 Mathematics Subject Classification: Primary 20H25, 16U60, 20F05.
Key words: linear group, Lie regular elements.

In this article, we discuss the existence of Lie regular matrices in M(4, Z_m). It is shown that the general linear group GL(4, Z_m) is generated by Lie regular matrices for all m > 1.

ON Z₂-GRADED POLYNOMIAL IDENTITIES OF sl₂(F) OVER A FINITE FIELD
Luís Felipe Gonçalves Fonseca luisfelipe@ufv.br

2010 Mathematics Subject Classification: 16R10, 17B01, 15A72, 17B70.
Key words: Graded identities, Lie algebras, finite basis of identities.

Let F be a finite field of char F > 3 and sl₂(F) be the Lie algebra of traceless 2 × 2 matrices over F. In this paper, we find a basis for the Z₂-graded identities of sl₂(F).

THE ISOPERIMETRIC NUMBER OF A GENERALIZED PALEY GRAPH
Spencer Johnson,
Anthony Shaheen ashahee@calstatela.edu,
Gustavo Subuyuj gsubuyu@calstatela.edu

2010 Mathematics Subject Classification: Primary 05C99; Secondary 11A07.
Key words: isoperimetric number, expansion number, Paley graph, Generalized Paley graph.

Let p be an odd prime, m ≥ 2 be an integer, and d = gcd(m, p-1). Suppose that d divides (p − 1)/2. We define the generalized Paley graph on p and m to be the Cayley graph whose vertex set is Z_p and whose generating set is the set of non-zero m-th powers modulo p. We derive basic properties of these graphs. We give bounds on the isoperimetric number of a generalized Paley graph.

DERIVATIONS IN FINITE ENDOMORPHISM SEMIRINGS
Dimitrinka I. Vladeva d_vladeva@abv.bg

2010 Mathematics Subject Classification: 06A05, 12H05, 16Y60, 16W20.
Key words: Semiring, endomorphism semiring of a finite chain, differential algebra, simplicial complex.

The aim of this article is to construct examples of derivations in finite semirings.

CENTRE DE MASSE ET NOUVELLES INEGALITES POUR LA NORME ET LE RAYON NUMERIQUE D'UN OPERATEUR LINEAIRE BORNE
Abderrahim Baghdad bagabd66@gmail.com,
Mohamed Chraïbi Kaadoud chraibik@uca.ac.ma,
Moulay Taieb Loumi loumi@uca.ac.ma

2010 Mathematics Subject Classification: 47A05, 47A12, 47A30, 30B10.
Key words: Domaine numérique, domaine numérique maximal, rayon numérique, centre de masse, et distance aux scalaires.

In this paper, we give a necessary and sufficient condition for having ||B - c_B(A^*B)A + λ A ||² = inf_{μ ∈ C} ||B − μ A||² + inf_{||x|| = 1} ||Ax||² |λ|², where λ is a complex number, A, B ∈ B(H), the algebra of bounded linear operators on a complex Hilbert space H and c_B(A^*B) is the center of mass relatively to B. We generalize and improve some inequalities established by Chraïbi [Géométrie du spectre dans une algèbre de Banach et domaine numérique. Stud. Math. 162, 1 (2004), 1−14], Dragomir [Norm and numerical radius inequalities for a product of two linear operators in Hilbert spaces. J. Math. Inequal. 2, 4 (2008), 499−510], [Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Demonstratio Math. 40, 2 (2007), 411−417], [Inequalities for the norm and the numerical radius of composite operators in Hilbert spaces. Inequalities and Applications (Eds C. Bandle, L. Losonczi, A. Gilґanyi et al.) International Series of Numerical Mathematics, vol. 157, 2008, 135−146], [Power inequalities for the numerical radius of a product of two operators in Hilbert spaces. Sarajevo J. Math. 5(18), 2 (2009), 269−278] and Albadawi-Shebrawi [Numerical Radius and Operator Norm Inequalities. J. Inequal. Appl. (2009), Article ID 492154, 11 pp.] linking the norm and the numerical radius of a bounded linear operator.

DELETION-CORRECTING CODES AND DOMINANT VECTORS
Emil Kolev emil@math.bas.bg

2010 Mathematics Subject Classification: 94B05.
Key words: insertion/deletion codes, Varshamov-Tenengolts codes, multiple insertion/deletion codes.

In this paper we describe all pairs of binary vectors (u, v) such that the set of vectors obtained by t deletions in v is a subset of the set of vectors obtained by t deletions in u for t = 1, 2. Such pairs play an important role for finding the value of L₂(n, t), the maximum cardinality of binary t-deletion-correcting codes of length n.

EXTENDING THE APPLICABILITY OF NEWTON-SECANT METHODS FOR FUNCTIONS WITH VALUES IN A CONE
Ioannis K. Argyros iargyros@cameron.edu,
Santhosh George sgeorge@nitk.ac.in

2010 Mathematics Subject Classification: 65G99, 90C30, 49J53.
Key words: Newton-secant method, semilocal convergence, variational inclusion, generalized continuity conditions.

In this study, we consider Newton-secant method for solving the nonlinear variational inclusion problems in Banach space. Using generalized continuity conditions, we prove the convergence of the method with the following advantages: tighter error estimates on the distances involved and the information on the location of the solution is at least as precise. These advantages were obtained under the same computational cost.

TRANSITION TO CANONICAL PRINCIPAL PARAMETERS ON MAXIMAL SPACELIKE SURFACES IN MINKOWSKI SPACE
Georgi Ganchev ganchev@math.bas.bg,
Krassimir Kanchev kbkanchev@yahoo.com,
Ognian Kassabov okassabov@abv.bg,

2010 Mathematics Subject Classification: Primary 53A10, Secondary 53A05.
Key words: Maximal surfaces, canonical principal parameters.

The first author has recently proposed to use special geometric parameters in the study of maximal spacelike surfaces in Minkowski 3-space. In canonical principal parameters any maximal spacelike surface is determined up to its position in the space by the normal curvature of the surface. Here we prove a theorem that permits a transition from general isothermal parameters to canonical principal parameters and we make some applications on parametric polynomial maximal spacelike surfaces. Thus we show that this approach implies an effective method to prove the coincidence of two maximal spacelike surfaces given in isothermal coordinates by different parametric equations.

OVERDETERMINED STRATA FOR DEGREE 6 HYPERBOLIC POLYNOMIALS
Ezzaldine Hayssam haysam82@hotmail.com

2010 Mathematics Subject Classification: 12D10, 14P05, 26C10, 13P15, 13P10.
Key words: hyperbolic polynomials, Gegenbauer polynomials, overdetermined strata, resultants.

A resultant-based method to calculate the overdetermined strata for degree 6 hyperbolic polynomials in one variable is revealed. This method is a new method to calculate overdetermined strata. The overdetermined strata in degree 6 have not been calculated before as the geometric method used until now can not be generalized to degree n ≥ 6.

A NOTE ON RESIDUALS IN GROUPS WITH~QUASICENTRAL ABELIAN NORMAL SUBGROUPS
L. A. Kurdachenko lkurdachenko@i.ua,
I. Ya. Subbotin isubboti@nu.edu

2010 Mathematics Subject Classification: 20E15, 20F19.
Key words: hypocentral series, hypercentral series, transitivity of normality, nilpotent residual.

In 1973, I. N. Abramovskii initiated the study of groups in which the transitivity condition is imposed on abelian normal subgroups only. He studied these locally finite groups under the additional restriction of commutativity of their Sylow p-subgroups. Much later the groups in which all abelian subnormal subgroups are normal were considered by M. Chaboksavar and F. de Giovanni. In the article we consider a broader class of groups, namely, groups in which every normal abelian subgroup is quasicentral.

COPULAS ON SOBOLEV SPACES
Iordan Iordanov haysam82@hotmail.com,
Nikolay Chervenov nikolay.tchervenov@gmail.com, nikolay.chervenov@bnpparibas.com

2010 Mathematics Subject Classification: 35L05, 35L20, 62F10, 62F12, 62F99.
Key words: copula, boundary value problem, 2-increasing function.

Defining a copula function and investigating its properties are both non-trivial tasks, as there is no general method for constructing them. We present a method which allows us to obtain a class of copulas as a solution of a boundary value problem in an appropriate Sobolev space. Furthermore, our method allows us to reduce the otherwise complex task of checking the 2-increasing of C-volume of the copula to simple differentiation. We demonstrate the applicability of our method to a number of examples.