Ukrains’kyi Matematychnyi Zhurnal

Crossingless sheaves and their classes in the equivariant $K$-theory

2025-01-03T13:17:58+02:00

UDC 517.9

We introduce crossingless sheaves in certain equivariant derived categories, which are analogous to the Bezrukavnikov–Mirkovic exotic sheaves for two-block nilpotents. The classes of crossingless sheaves are computed in the equivariant $K$-theory of Cautis–Kamnitzer varieties.

Semigroup of weak endomorphisms of a partial equivalence relationship

2025-01-03T13:19:51+02:00

UDC 512.53

We study the weak endomorphism semigroup of a partial equivalence relation. We describe necessary and sufficient conditions for the existence of these endomorphisms and, in general, endotopisms. We establish the conditions under which the set of all idempotents of the weak endomorphism semigroup of a strict partial equivalence is its subsemigroup, as well as the conditions of regularity and coregularity of the weak endomorphism semigroup. In terms of the wreath product of a symmetric transformation semigroup and a small category, we describe a faithful representation of the weak endomorphism semigroup of an arbitrary relation of a strict partial equivalence.

Domination number on an octagonal chain and an octagonal grid

2025-01-02T14:44:56+02:00

UDC 519.1

Domination of the graph and topological indices are essential topics in the graph theory. We analyze the problem of $k$-domination, $k\in\{1,2,3\}$, on octagonal chains and an octagonal grid. We determine the minimal $k$-dominating sets and $k$-domination $k$ numbers for a chain of octagons with two common vertices. By using the obtained results, we determine the $k$-domination numbers for the grid of octagons $O_{mxn}$ with $m,n\in N$.

Elements of Lévy analysis on the spaces of nonregular test and generalized functions

2025-01-03T13:22:04+02:00

UDC 517.98

We present a survey of some author's results related to the development of the L\'evy analysis on the spaces of nonregular test and generalized functions. The indicated spaces are constructed by using Lytvynov's generalization of the chaotic representation property, which is a direct analog of decomposition of square-integrable random variables in the Hermite orthogonal polynomials in the Gaussian analysis. In this approach, numerous definitions and statements are quite similar to their prototypes from the Gaussian analysis, which is very convenient for applications. The survey covers a fairly broad range of issues, namely, an extended stochastic integral, a Hida stochastic derivative and their generalizations, the operators of stochastic differentiations and their analogs (generalizations), elements of the Wick calculus, the relationship between the Wick calculus and integration, etc.

On a subclass of starlike functions associated with a strip domain

2025-01-03T13:25:42+02:00

UDC 517.5

We introduce a new subclass of starlike functions defined as $\mathcal{S}^{*}_{\tau}:=\{f\in \mathcal{A}:zf'(z)/f(z) \prec 1+\arctan z=:\tau(z)\},$ where $\tau(z)$ maps the unit disk $\mathbb {D}:= \{z\in \mathbb{C}:|z|<1\}$ onto a strip domain. We deduce structural formulas, as well as the growth and distortion theorems for $\mathcal{S}^{*}_{\tau}.$ In addition, inclusion relations with some well-known subclasses of $\mathcal{S}$ are established and sharp radius estimates are obtained, as well as the sharp coefficient bounds for the initial five coefficients and the second and third order Hankel determinants of $\mathcal{S}^{*}_{\tau}.$

Stabilization of homogeneous conformable fractional-order systems

2024-12-31T10:18:56+02:00

UDC 517.9

We propose an explicit homogeneous feedback control under the assumption that a control Lyapunov function exists for an affine control conformable fractional-order system and satisfies a homogeneity condition. Furthermore, we demonstrate that the existence of a homogeneous control Lyapunov function for a homogeneous affine conformable fractional-order system results in a homogeneous closed-loop system when applying the previous feedback control.

On the application of one-dimensional dynamics in the study of infinite- dimensional dynamical systems and modeling of the distributed chaos

2025-01-02T17:15:28+02:00

UDC 517.9

We propose a brief survey of the applications of one-dimensional dynamics to the investifation of infinite-dimensional dynamical systems on the spaces of continuous and smooth functions, which were developed in the Department of Dynamical Systems Theory of the Institute of Mathematics of the National Academy of Sciences of Ukraine under the leadership of O. M. Sharkovsky.

Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices

2024-12-31T10:18:56+02:00

UDC 512.5

Let $n$ be a positive integer, let $R$ be a (unitary associative) ring, and let $M_n(R)$ be the ring of all $n$ by $n$ matrices over $R.$ For a permutation $\sigma$ in the symmetry group $\Sigma_n$ and a ring automorphism $\varphi$ of $R,$ we introduce the definition of $\sigma$-$\varphi$ permutation matrices. The set $B_n(\sigma, \varphi, R)$ of all $\sigma$-$\varphi$ permutation matrices is proved to be a subring of $M_n(R).$ We show that the extension $B_n(\sigma, \varphi, R) \subseteq M_n(R)$ is a separable Frobenius extension. Moreover, if $R$ is a commutative cellular algebra over the invariant subring $R^\varphi$ of $R,$ then $B_n(\sigma, \varphi, R)$ is also a cellular algebra over $R^\varphi.$

Index of volume 76 of „Ukrainian Mathematical Journal”

2025-01-02T17:16:19+02:00