Serdica Mathematical Journal

Volume 45, Number 2, 2019

This issue contains contributions to the
Second International Conference Mathematics Days in Sofia
held from 10th to 14th of July, 2017.

C O N T E N T S

A B S T R A C T S

LOG-CONVEXITY OF WEIGHTED AREA INTEGRAL MEANS OF H^p FUNCTIONS ON THE UPPER HALF-PLANE
Martin At. Stanev martin_stanev@yahoo.com

2010 Mathematics Subject Classification: 30H10, 30H20.
Key words: log-convexity,weighted area integral means,holomorphic function,upper half-plane.

In the present work weighted area integral means M_p,φ(f; Im z) are studied and it is proved that the function y → log M_p,φ(f; y) is convex in the case when f belongs to a Hardy space on the upper half-plane.

SOME MATHEMATICAL PROBLEMS IN CANCER RESEARCH
Mariia Beliaeva maria.beljaeva29@gmail.com

2010 Mathematics Subject Classification: 92C50, 97M60, 62P10.
Key words: cancer mathematical models, mistakes, multistage model, carcinogenesis, Bad Luck theory.

Some mathematical models used in cancer research are considered. Some mathematical mistakes made in those models are analyzed. Also a new version of the well-known multistage model of carcinogenesis is presented.

COUNTABLY COMPACTNESS AND BAIRE SPACE PROPERTY
Mitrofan M. Choban mmchoban@gmail.com

2010 Mathematics Subject Classification: 54D20, 54D45, 54G15, 54A25.
Key words: τ-bounded sets, τ-closure, P-space, Stone-Čech compactification, Baire space, meager space.

In the present article τ-bounded spaces are investigated. It is shown that for every infinite cardinal τ there exists a meager Hausdorff τ-bounded space.

COMPARISON PRINCIPLE FOR WEAKLY-COUPLED NON-COOPERATIVE ELLIPTIC AND PARABOLIC SYSTEMS
Georgi Boyadzhiev gpb@math.bas.bg,
Nikolay Kutev kutev@math.bas.bg

2010 Mathematics Subject Classification: 35J47, 35K40.
Key words: Comparison principle, elliptic systems, parabolic systems, cooperative and non-cooperative systems.

In this review article is considered the comparison principle for linear and quasi-linear weakly coupled systems of elliptic and of parabolic PDE. It is demonstrated that a cooperativeness is a kind of a watershed quality for the comparison principle. Roughly speaking comparison principle holds for cooperative systems, while it does not hold for every non-cooperative one.
Considering a cooperative system one can apply the theory of a positive operator in a positive cone and prove the validity of the comparison principle. One particularly important result for cooperative systems is the existence of positive first eigenvalue and positive first eigenvector.
Investigation of the validity of the comparison principle for non-cooperative system is more complicated. In this paper is mentioned the idea of division of the non-cooperative system in a cooperative and competitive part. Then the spectral properties of the cooperative part are employed in order to derive conditions for validity of comparison principle for the non-cooperative system.
Some applications of comparison principle are given as well.

VARIETIES OF BICOMMUTATIVE ALGEBRAS
Vesselin Drensky drensky@math.bas.bg

2010 Mathematics Subject Classification: 17A30, 17A50, 20C30.
Key words: Free bicommutative algebras, varieties of bicommutative algebras, codimension sequence, codimension growth, two-dimensional algebras.

Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (x₁x₂)x₃ = (x₁x₃)x₂ and x₁(x₂x₃) = x₂(x₁x₃). Let 𝔅 be the variety of all bicommutative algebras over a field K of characteristic 0 and let F(𝔅) be the free algebra of countable rank in 𝔅. We prove that if 𝔙 is a subvariety of 𝔅 satisfying a polynomial identity f = 0 of degree k, where 0≠ f ∈ F(𝔅), then the codimension sequence c_n(𝔙), n = 1, 2, ..., is bounded by a polynomial in n of degree k − 1. Since c_n(𝔅) = 2ⁿ − 2 for n ≥ 2, and exp(𝔅) = 2, this gives that exp(𝔙) ≤ 1, i.e., 𝔅 is minimal with respect to the codimension growth. When the field K is algebraically closed there are only three pairwise nonisomorphic two-dimensional bicommutative algebras A which are nonassociative. They are one-generated and with the property dim A² = 1. We present bases of their polynomial identities and show that one of these algebras generates the whole variety 𝔅.