Ukrains’kyi Matematychnyi Zhurnal

On the asymptotic values of meromorphic functions in the $n$-fold punctured plane

2024-12-06T10:16:34+02:00

UDC 517.5

We present some results on the asymptotic paths and asymptotic values for meromorphic functions in the $n$-fold extended punctured plane $\widehat{\mathbb{C}}\backslash\{p_{1},\ldots,p_{n}\} .$ These results extend some classical results obtained for analytic and meromorphic functions in the complex plane $\mathbb{C}.$ In particular, in this more general setting, we give a version of F. Iversen's result on the existence of asymptotic values for entire functions in the plane. We also obtain a bound for the number of isolated directly critical singularities of a meromorphic function of finite order $k$ and a finite number of poles.

Potentials for solenoidal fields using the three-dimensional $\varphi$-harmonic cyclic algebra

2024-12-06T10:16:39+02:00

UDC 517.5

Given a PDE, in [E. López-González, E. A. Martínez-García, R.~Torres-Córdoba, Chaos, Solitons and Fractals, 73, Article 113757 (2023)], the authors proposed a method for the construction of solutions by considering an associative real algebra $\mathbb A$ and a suitable affine vector field $\varphi$ with respect to which the components of all functions $\mathcal L\circ\varphi$ are solutions, where $\mathcal L$ is differentiable in a sense of Lorch with respect to $\mathbb A.$ If we consider the 3D cyclic algebra and a suitable 3D affine map $\varphi,$ then we get families of solutions for the Laplace equation with three independent variables.

The metric dimension of the total graph of a semiring

2024-12-06T10:16:36+02:00

UDC 512.5

We calculate the metric dimension of the total graph of a direct product of finite commutative antinegative semirings with their sets of zero-divisors closed under addition.

Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field

2024-12-06T10:16:40+02:00

UDC 512.5+514

We determine the complete degeneration picture inside the variety of nilpotent associative algebras of dimension three over an algebraically closed field. As compared with the discussion in [N. M. Ivanova, C. A. Pallikaros, Adv. Group Theory and Appl., 18, 41-79 (2024)], for some arguments in the present article, it is necessary to develop alternative techniques, which are now valid over an arbitrary algebraically closed field. There is a dichotomy of cases concerning the obtained results corresponding to the cases where the characteristic of the field is $2$ or not.

Integer divisor connectivity graph

2024-12-06T10:16:33+02:00

UDC 512.5

Let $n$ be a nonprime integer. We introduce a new simple undirected graph and denote it by $MD(n),$ where the vertices are the proper divisors of $n$ and two vertices $x$ and $y$ are adjacent if $xy$ divides $n.$ We explore the connectedness of $MD(n)$ and provide detailed calculations for the degree of each vertex. In addition, we focus on the special case where $n = p^{\alpha},$ where $p$ is a prime positive integer and $\alpha\geq 3$ is a positive integer. For these instances, we explicitly determine the chromatic number $\chi$ and the clique number $\omega$ of $MD(n).$ Finally, we conclude that $\chi(MD(n)) = \omega(MD(n)).$

Local nearrings, their structure, and the GAP system

2024-12-06T10:16:45+02:00

UDC 512.6

We present an overview of the current state of research of finite local nearrings. In particular, it contains all recently obtained results on the classification of local nearrings.

Line graph of extensions of the zero-divisor graph in commutative rings

2024-12-06T10:16:32+02:00

UDC 512.6

We consider a finite commutative ring with unity denoted by $\mathscr{P}.$ Within this framework, the essential graph of $\mathscr{P}$ is represented as $E{G}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy)$ is an essential ideal of $\mathscr{P}.$ At the same time, the weakly zero-divisor graph of $\mathscr{P}$ is denoted by $\text{W}{\Gamma}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set and an edge is defined between two distinct vertices $u$ and $v$ if and only if there exist $r \in ann(u)^*$ and $s\in ann(v)^*$ such that $rs=0$, where $ann(u) = \{v \in \mathscr{P}\colon uv = 0\}$ for $u \in \mathscr{P}.$ In our research, we deal with the conditions under which both $E{G}(\mathscr{P})$ and $\text{W}{\Gamma}(\mathscr{P})$ can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs.

Actual problems of the theory of approximations in metrics of discrete spaces on the sets of summable periodic and almost periodic functions

2024-12-05T10:16:24+02:00

UDC 517.5

We present a review highlighting the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p$ of periodic and almost periodic summable functions, respectively, where the $l_p$-norms of the sequences of Fourier coefficients are finite. In particular, the present review contains the available results concerning the best and best $n$-term approximations, as well as the widths of the classes of functions of one and many variables defined by means of the $\psi$-derivatives and generalized moduli of smoothness in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p.$ Special attention is given to the development of investigations related to the derivation of direct and inverse approximation theorems in these spaces.

New results on Bullen-type inequalities for coordinated convex functions obtained by using conformable fractional integrals

2024-12-05T10:16:23+02:00

UDC 517.9

Our aim is to investigate novel Bullen-type inequalities for coordinated convex mappings by employing conformable fractional integrals. Initially, an identity incorporating the conformable fractional integrals was established to serve for this purpose. By using this identity, new inequalities аre derived expanding the scope of previously established results obtained with the help of Riemann–Liouville integrals by making specific choices of variable and applying the Hölder inequality and the power-mean inequality.