Serdica Mathematical Journal

Volume 45, Number 4, 2019

C O N T E N T S

A B S T R A C T S

ON THE CHOW RING OF FANO VARIETIES OF TYPE S6
Robert Laterveer robert.laterveer@math.unistra.fr

2010 Mathematics Subject Classification: Primary 14C15, 14C25, 14C30.
Key words: Algebraic cycles, Chow groups, motives, Beauville's splitting property, multiplicative Chow−Künneth decomposition, Fano varieties of K3 type.

Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian Gr(2,10). These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.

UNIT GROUPS OF SEMISIMPLE GROUP ALGEBRAS OF~CERTAIN DIHEDRAL GROUPS
Meena Sahai meena_sahai@hotmail.com,
Sheere Farhat Ansari sheere_farhat@rediffmail.com

2010 Mathematics Subject Classification: Primary 16S34; Secondary 20C05.
Key words: group algebra, unit group, dihedral group, semisimple.

Let F be a finite field of characteristic p > 0 having q = pⁿ elements. Let D_n denote the dihedral group of order 2n. In this article, we have obtained the structure of the unit groups of the semisimple group algebras FD₁₁, FD₁₃, FD₁₇, FD₁₉ and FD₂₃.

2010 Mathematics Subject Classification: 16W80, 16N20, 16N40.
Key words: Largest regular ideal, upper nilradical, Brown-McCoy radical, Hartman-Mycielski functor.

The well-known construction of Hartman-Mycielski delivers a method of embedding of a topological ring (R, 𝒯) into an arcwise connected, locally arcwise connected topological ring (R̃, 𝒯 ̃). An important question in the theory of topological rings is: When is the radical ρ(R) of a topological ring closed? We study in this paper the following question: Let ρ be a radical in the sense of Kurosh in the class of associative rings and (R, 𝒯) be a topological ring. Under which conditions is closed the radical ρ(R̃) of the image of R under the Hartman-Mycielski functor? We give in this paper a complete answer for a few classical radicals. This question has been solved for the Jacobson radical in the master thesis of the third author.

THE ALGEBRAIC SOLUTION OF UNILATERAL MATRIX POLYNOMIAL EQUATIONS
Ajai Choudhry ajaic203@yahoo.com

2010 Mathematics Subject Classification: 15A24.
Key words: unilateral matrix equations, matrix polynomial equations, quadratic matrix equations, solvents of matrix polynomials.

In this paper we describe an algebraic method of obtaining all solutions of the unilateral matrix polynomial equation A₀X^m + A₁X^m-1+ A₂X^m-2 + ⋯ + A_m-1X + A_m = 0 where the coefficients A_i and X are square matrices of order 2 and m is a positive integer ≥ 2. We also show how an extension of the method can be used to solve the quadratic matrix equation AX² + BX + C = 0 where A, B, C and X are 3 × 3 matrices. We give several illustrative examples which show that the algebraic method described in the paper is quite practical and effective in solving such matrix equations.

CANONICAL COORDINATES ON MINIMAL TIME-LIKE SURFACES IN THE N-DIMENSIONAL LORENTZIAN SPACE
Georgi Ganchev ganchev@math.bas.bg,
Krasimir Kanchev kbkanchev@yahoo.com

2010 Mathematics Subject Classification: Primary 53A10, Secondary 53B30.
Key words: Minimal time-like surfaces in the $n$-dimensional Lorentzian space, canonical coordinates.

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Lorentzian space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface.
On any time-like surface we introduce a special complex function over the algebra of the double numbers and apply the analysis over the algebra of the double numbers as a convenient tool to study these surfaces. Then the canonical coordinates on minimal time-like surfaces are characterized by a natural condition for this complex function.
We consider the hyperbola of the normal curvature of any minimal time-like surface and give a geometric interpretation of the canonical coordinates in terms of the elements of this hyperbola.

SOME RESULTS ON UNIQUENESS OF DIFFERENCE POLYNOMIAL
Harina P. Waghamore harinapw@gmail.com, harinapw@bub.ernet.in
Ramya Maligi ramyamalgi@gmail.com

2010 Mathematics Subject Classification: Primary 30D35.
Key words: Meromorphic function, entire function, difference polynomials, uniqueness.

In this paper, we consider the uniqueness of the difference polynomial fⁿ P(f) ∏^d_j=1 f(z+c_j)^s_j. The results of the paper improve and generalize some recent results due to Y. Liu, J. P. Wang and F. H. Liu [10] (Some results on value distribution of the difference operator. Bull. Iraninan Math. Soc. 41, No 3 (2015), 603−611).