Serdica Mathematical Journal

Volume 46, Number 2, 2020

C O N T E N T S

A B S T R A C T S

A NOTE ON DOUBLE FOURIER COEFFICIENTS
Kiran N. Darji darjikiranmsu@gmail.com,
Rajendra G. Vyas drrgvyas@yahoo.com

2020 Mathematics Subject Classification: 42B05, 26B30, 26D15.
Key words: double Fourier series, order of magnitude, functions of (ϕ, ψ)-(Λ, Γ)-bounded variation.

We estimate the order of magnitude of double Fourier coefficients of functions of (ϕ, ψ)-(Λ, Γ)-bounded variation in the sense of Vitali and Hardy.

THE CENTER OF THE TOTAL RING OF FRACTIONS
Oswaldo Lezama jolezamas@unal.edu.co,
Helbert Venegas hjvenegasr@unal.edu.co

2020 Mathematics Subject Classification: Primary: 16S85, 16U70. Secondary: 16P90, 16S36.
Key words: Ore domains, total ring of fractions, center of a ring, Gelfand−Kirillov dimension, skew PBW extensions.

Let A be a right Ore domain, Z(A) be the center of A and Q_r(A) be the right total ring of fractions of A. If K is a field and A is a K-algebra, in this short paper we prove that if A is finitely generated and GKdim(A) < GKdim(Z(A)) + 1, then Z(Q_r(A)) ≅ Q(Z(A)). Many examples that illustrate the theorem are included, most of them within the skew PBW extensions.

2020 Mathematics Subject Classification: Primary 11A05; Secondary 11Y16.
Key words: binary gcd algorithm, extended gcd algorithm.

In this paper we are looking for fast gcd algorithms in certain quadratic number fields. These algorithms do not belong to the Euclidean algorithm family rather the proposed algorithms can be viewed as generalization of the binary gcd algorithm.

HYPERBOLIC POLYNOMIALS AND CANONICAL SIGN PATTERNS
Vladimir Petrov Kostov vladimir.kostov@unice.fr

2020 Mathematics Subject Classification: 26C10, 30C15.
Key words: real polynomial in one variable, hyperbolic polynomial, sign pattern, Descartes' rule of signs.

A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly c positive and exactly p negative roots counted with multiplicity, where c and p are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all c + p roots are distinct and ordered on the positive half-axis, then at which positions can the p moduli of negative roots be depending on the positions of the positive and negative signs of the coefficients of the polynomial? We are especially interested in the choices of these signs for which exactly one order of the moduli of the roots is possible.

QUASI-MINIMAL LORENTZ SURFACES IN PSEUDO-EUCLIDEAN 4-SPACE WITH NEUTRAL METRIC
Yana Aleksieva yana_a_n@fmi.uni-sofia.bg,
Velichka Milousheva vmil@math.bas.bg

2020 Mathematics Subject Classification: Primary 53B30, Secondary 53A35, 53B25.
Key words: Quasi-minimal surface, marginally trapped surface, pseudo-Euclidean 4-space, Fundamental theorem.

A Lorentz surface in the pseudo-Euclidean 4-space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. We prove that any quasi-minimal Lorentz surface whose Gauss curvature K and normal curvature ϰ satisfy the condition K² − ϰ² ≠ 0 at every point is determined (up to a rigid motion) by five geometric functions satisfying a system of four partial differential equations.

ON 3-DISTANCE SPHERICAL 5-DESIGNS
Peter Boyvalenkov peter@math.bas.bg,
Navid Safaei navid_safaei@gsme.sharif.edu

2020 Mathematics Subject Classification: 05B30, 52C17.
Key words: spherical designs, maximal codes, few distance codes.

Inspired by a recently formulated conjecture by Bannai et al. we investigate spherical codes which admit exactly three different distances and are spherical 5-designs. Computing and analyzing distance distributions we provide new proof of the fact (due to Levenshtein) that such codes are maximal and rule out certain cases towards a proof of the conjecture.

LEIBNIZ ALGEBRAS WHOSE SUBALGEBRAS ARE LEFT IDEALS
L. A. Kurdachenko lkurdachenko@i.ua,
I. Ya. Subbotin isubboti@nu.edu,
V. S. Yashchuk viktoriia.s.yashchuk@gmail.com

2020 Mathematics Subject Classification: 17A32, 17A60, 17A99.
Key words: Leibniz algebra, Lie algebra, ideal, left ideal, nilpotent Leibniz algebra, extraspecial algebra.

In this paper we have described the Leibniz algebras, whose subalgebras are left ideals.

METRIC DIMENSION OF ULTRAMETRIC SPACES
Bogdana Oliynyk oliynyk@ukma.edu.ua,
Bogdan Ponomarchuk ponomarchuk.bogdan@gmail.com

2020 Mathematics Subject Classification: 05B25, 05C12, 68R12.
Key words: metric dimension, ultrametric space, rooted tree, polynomial-time algorithm.

For an arbitrary finite metric space (X, d) a subset A, A ⊂ X, is called a resolving set if for any two points x and y from the space X there is an element a from subset A, such that distances d(a, x) and d(a, y) are different. The metric dimension md(X) of the space X is the minimum cardinality of a resolving set.
It is well known that the problem of finding the metric dimension of a metric space is NP-complete [7] (M. R. Garey, D. S. Johnson. Computers and intractability. A guide to the theory of NP-completeness. San Francisco, W. H. Freeman and Company, 1979.). In this paper, the metric dimension for finite ultrametric spaces is completely characterized. It is proved that for any finite ultrametric space there exists a polynomial-time algorithm for determining the metric dimension of this spaces.