Ukrains’kyi Matematychnyi Zhurnal (Ukrainian Mathematical Journal) Editor-in-Chief: A. N. Timokha ISSN: 0041-6053, 1027-3190 Ukrainian Mathematical Journal is published by Springer Nature and Institute of Mathematics of the National Academy of Sciences of Ukraine (est. 1934). The journal is a comprehensive international platform whose aim consists of publishing original research and review articles across various branches of pure and applied mathematics. Submitted manuscripts can be written in either English or Ukrainian. Their single-blind peer review is handled by the Editorial Board. After being accepted, Ukrains’kyi Matematychnyi Zhurnal (est. 1949) publishes the full-scale Ukrainian-language articles and abstracts of English submissions, whereas Ukrainian Mathematical Journal collects and issues, in full, the accepted English-language articles together with the Ukrainian-language manuscript translations. The journal scopes cover but not restricted to the eight main heading: Algebra; ODE & Dynamical Systems; Geometry, Topology & Fractals; Functional Analysis, Operator Theory & PDEs; Theory of Functions; Mathematical Physics; Probability & Statistics; Numerical & Applied Mathematics The authors are obligated to mention one of these heading and the corresponding editor when submitting the manuscript. UMZh indexed in: MathSciNet, zbMATH, Scopus, Web of Science, Google Scholar.
Institute of Mathematics, NAS of Ukraine
Optimal control of Goursat–Darboux-type polyhedral differential inclusions
UDC 517.96, 517.97 We study the problem of optimal control of the Goursat–Darboux-type differential inclusions (DFIs) given by polyhedral set-valued mappings and, for this purpose, pose an auxiliary problem with discrete Goursat–Darboux inclusion. By using the Farkas theorem, we compute locally adjoint mappings (LAMs) and establish necessary and sufficient optimality conditions for the polyhedral discrete Goursat–Darboux inclusions. Further, by the discretization method, for the Goursat–Darboux-type polyhedral DFI, we formulate (by using solely the specific features of its polyhedral property), first for a discrete-approximate problem and then, for a continuous problem, necessary and sufficient optimality conditions in the form of Euler–Lagrange-type polyhedral inclusions.
Ritz method in the problem of free vibrations of thin elastic shells
UDC 539.385:534.13, 517.97 Based on the variational formulation of the problem of free vibrations of an arbitrary elastic thin-walled shell, we construct differential operators without restrictions imposed on their shape, which are valid both for the technical and general linear theories of shells. As a special case, we demonstrate the principles of construction of solutions to the problem posed for the shells of revolution by using the obtained differential operators.
On the generalization of the Newton–Kantorovich theorem for nonlinear operator equations
UDC 517.98, 519.61 We construct a modification of the classical Newton-Kantorovich method for nonlinear operator equations whose linearization gives equations that are, as a rule, solvable. For finding the solution of a nonlinear operator equation, we propose to use an iterative scheme with quadratic convergence. In particular, we establish solvability conditions and construct an iterative procedure aimed at finding the solution of the generalized Riccati equation.
Copyright (c) 2026 Сергій Чуйко, Ольга Нєсмєлова
Delay factors and genesis of the limit sets for a nonideal “tank with liquid–electric motor” system
UDC 517.929, 517.938, 534.14 We study a "tank with liquid–electric motor" nonideal deterministic system. Two delay-approximation models are considered. The impacts of the delays on the emergence, evolution, and decay of both regular and chaotic limit sets (attractors) of the system are investigated. The main dynamic characteristics of the steady-state modes of the system are computed and analyzed. Various scenarios of transition to chaos are investigated. The realization of the generalized intermittency scenario driven by delay factors is established.
Copyright (c) 2026 Ільмі Сеїт-Джеліль, Олександр Швець
UDC 539.37, 517.9 We propose an approach to the construction of an analytic solution to the three-dimensional thermoelasticity problem for a bar with rectangular cross section infinitely extended in the axial direction under the conditions of stationary distribution of the temperature field. The problem is solved by the method of direct integration with the help of the Vihak functions, for which the governing integrodifferential equations are deduced on the basis of the continuity equations. A procedure for constructing solutions to boundary-value problems for the obtained governing equations is developed with the help of successive approximations. The application of the proposed approach makes it possible to construct the solution to the original problem, which exactly satisfies the boundary conditions and also satisfies the original equations with prescribed accuracy. The numerical investigation of the solution is performed for a rectangular bar and compared with the limiting cases of a rectangular parallelepiped and a layer with infinite (in plan) cross section.
Copyright (c) 2026 Юрій Токовий, Микола Юзв'як
Differential equation of minimal order for a system of polynomials
UDC 517.92, 517.58 Classical orthogonal polynomials, such as Laguerre, Legendre, Hermite, Hegenbauer, Jacobi, and Bessel polynomials, served as a fundamental tool for solving applied problems. They satisfy second-order differential equations. It is of interest to analyze the questions concerning the minimal order of equations of this kind: Is it possible that these polynomials are solutions of the first-order equations and what equations of the minimal order are satisfied by other polynomials, in particular, by the reciprocal classical polynomials. The present paper is devoted to the analysis of these problems.
Polynomial differential equation, minimal order differential equation, classical orthogonal polynomials, Bessel polynomials, recurrence relation
Ключові слова: поліноміальне диференціальне рівняння, диференціальне рівняння мінімального порядку, класичні ортогональні поліноми, поліноми Бесселя, рекурентне співвідношення
Copyright (c) 2026 Володимир Макаров, Олена Кашпур
Attainment of weighted performance measures and spectrum localization in descriptor control systems
UDC 517.925.51, 681.5.03 We study the problems of estimation of the weighted damping level of bounded disturbances and spectrum localization for linear descriptor systems with indefinite initial vector. In terms of linear matrix inequalities, we establish criteria and sufficient conditions for the validity of the desired estimation of the generalized performance measure and localization of the system spectrum in a given region of the complex plane. New methods are proposed for the synthesis of static state and output controllers guaranteeing the validity of the indicated properties of the closed-loop descriptor system. We also present a numerical example of descriptor control system.
Cumulant expansions of operator groups of quantum many-particle systems
UDC 530.145:517.98 We present a method of cluster expansions for groups of operators associated with the von-Neumann equations for states and the Heisenberg equations for observables aimed at construction of generating operators for the solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.
cluster expansion, cumulant expansion, semigroup of operators, hierarchy of evolution equations, quantum many-particle system
Copyright (c) 2026 Ігор Гап'як, Віктор Герасименко
Differential-symbol method for solving the Nicoletti problem for a homogeneous equation with second-order time partial derivatives
UDC 517.95 The Nicoletti problem is investigated for a homogeneous differential equation with partial time derivatives of the second order and, in general, of the infinite order in the spatial variable. We select classes of quasipolynomial functions in which the solution to the problem exists and is unique. The case where the solution to the Nicoletti problem exists in the spaces of continuously differentiable functions is considered separately. A method for constructing solutions to the problem based on the differential-symbol method is described. We also present examples of solving the problems.
диференціально-символьний метод, задача Ніколетті, двоточкові умови, квазіполіномні розв'язки
задача Ніколетті, двоточкові умови, квазіполіномні розв'язки
Block matrices corresponding to the strong two-dimensional real moment problem
UDC 517.98 We generalize the relationship between the strong one-dimensional real moment problem and the spectral theory of Jacobi-type block matrices for the two-dimensional case. An analog of Jacobi-type matrices corresponding to the problem in this formulation is proposed, and the corresponding system of polynomials orthogonal with respect to a measure with compact support on the real plane is described.
Copyright (c) 2026 Валентина Козак, Ольга Дюженкова
Boyanov–Naydenov problem and the inequalities of various metrics
UDC 517.51 We solve the Boyanov–Naidenov problem $\|x\|_{q, \delta} \to \sup$ on classes of differentiable functions $ W^r_{p, \omega}(A_0, A_r) := \{x\in L^r_{\infty}\colon \|x\|_{p, \omega} \le A_0,$ $ \|x^{(r)}\|_{\infty} \le A_r \}, $ where $\|x\|_{p, \, \delta} := \sup \{\|x\|_{L_p[a, b]}\colon a, b \in {\rm \bf R},$ $ 0< b-a \le \delta \}; \; p, \delta > 0; $ $q \ge p;$ $\omega := \pi / \lambda,$ the number $\lambda$ satisfies the condition $ A_0 = A_r \|\varphi_{\lambda, r}\|_{p, \, \pi / \lambda}, $ $\varphi_{\lambda, r}(t) := \lambda^{-r}\varphi_{r}(\lambda t),$ and $\varphi_{r}$ is the ideal Euler spline of order $r.$ In addition, we prove that the Boyanov–Naidenov problem is equivalent to the problem of finding the sharp constant $C = C(\lambda)$ in the inequality of different metrics\begin{align}\|x\|_{q, \delta} \leq C \|x\|_{p, \omega}^{\alpha} \big\|x^{(r)}\big\|_\infty^{1\alpha},\quad x\in L^{r,\lambda}_{p,\varepsilon},\tag{1}\end{align}where $\alpha = \dfrac{r + 1/q}{r + 1/p},$ $L^{r, \lambda}_{p, \varepsilon} := \{x\in L^r_{\infty}\colon \|x\|_{p, \varepsilon} = \|\varphi_{\lambda, r}\|_{p, \varepsilon}\|x^{(r)}\|_{\infty} \},$ and $\lambda > 0.$ In particular, we obtain inequalities of the form (1) sharp in the classes $L^{r, \lambda}_{p, \varepsilon}.$ We also solve the Boyanov–Naidenov problem in the spaces of trigonometric polynomials and splines and establish theorems on the relationship between this problem and sharp inequalities of different metrics. As a consequence, sharp inequalities of this kind are proved for polynomials and splines.
Boyanov-Naydenov problem, inequalities of different metrics, classes of functions with a given comparison function, Sobolev classes, polynomials, splines
Задача Боянова-Найдьонова та нерівності різних метрик
Copyright (c) 2026 Дорошенко Данііл, Володимир Кофанов
Group classification of a class of systems of nonlinear diffusion equations. I
UDC 517.95:512.81 We study the Lie symmetries of a class of systems of $(1+1)$-dimensional nonlinear diffusion equations. The equivalence algebra and the kernel of maximal invariance algebras of this class are found. We prove that there are nine inequivalent subclasses of this class for which the corresponding diffusion systems admit Lie symmetry extensions. We also perform the exhaustive group classification of the systems of nonlinear diffusion equations that belong to the first identified subclass.
(1 1)-dimensional nonlinear evolution equations
Copyright (c) 2026 Олександр Волошин, Микола Сєров, Олена Ванєєва
Construction of exact solutions of nonlinear heat conduction equations
UDC 517.958 We consider substitutions reducing nonlinear heat-conduction equations with a source to a system of two ordinary differential equations. Classes of exact solutions of these equations with generalized separation of variables are constructed.
Copyright (c) 2026 Анатолій Баранник, Тетяна Баранник, Іван Юрик
The dual affine log-Minkowski inequality
UDC 517.98, 514.17 We deduce an affine dual log-Minkowski inequality by introducing two new concepts of dual affine measure and Orlicz dual affine measure and using the newly established Orlicz dual affine Minkowski inequality. In a special case, the Orlicz affine dual log-Minkowski inequality yields the $L_{p}$-affine dual log-Minkowski inequality. The affine dual log-Minkowski inequality is also obtained.
dual affine quermassintegral, Orlicz dual affine quermassintegral, log-Minkowski inequality,
dual affine quermassintegral, Orlicz dual affine quermassintegral, log-Minkowski inequality, log-dual Minkowski inequality, Orlicz affine dual Minkowski inequality.
Critical point equation on $K$-contact generalized Sasakian space form
UDC 514.7 We investigate the critical-point equation within the framework of $K$-contact generalized Sasakian space forms. It is demonstrated that a complete $K$-contact generalized Sasakian space form satisfying the Miao–Tam equation is necessarily Einstein and isometric to the unit sphere $S^{2n+1}$. In addition, we show that if this space form satisfies the Euler–Lagrange equation corresponding to the total scalar curvature functional, then it is Einstein and also satisfies the Fischer–Marsden equation. An illustrative example is provided to support and validate our theoretical findings.
$K$-contact generalized Sasakian space form,
Copyright (c) 2026 R. C. Pavithra, H. G. Nagaraja
Some properties of subclasses of spiral-like functions involving beta-negative binomial distribution
UDC 517.53 By using beta-negative binomial distribution, we introduce two novel subclasses of spiral-like functions; namely, spiral-starlike functions and spiral-convex functions denoted by $S^{\eta}_{\lambda, \gamma, \mu}(\alpha, \beta)$ and $E^{\eta}_{\lambda, \gamma, \mu}(\alpha, \beta),$ respectively, and defined in the domain of open unit disk $\mathbb{D}=\{z \in \mathbb{C}\colon |z|<1\}.$ We establish sufficient conditions for functions to be members of families mentioned above. Further, the bounds of some initial coefficients and Fekete–Szegö functionals for the classes described above are obtained.
Analytic function, Subordination, Fekete-Szeg¨o functional, Spiral-like functions. 1
Copyright (c) 2026 Eureka Pattnayak, Trailokya Panigrahi, Rabha M. El-Ashwah
On the dissipative Sturm–Liouville problem with transmission conditions depending on the eigenparameter
UDC 517.927, 517.98 A class of dissipative Sturm–Liouville boundary-value problems with eigenparameter-dependent transmission conditions is investigated. By transforming the Sturm–Liouville problem with eigenparameter-dependent transmission conditions to the corresponding linear operator associated with the problem, we prove that this operator is dissipative. Further, certain eigenvalue properties are presented and the theorems on completeness of this operator are established by applying Krein's theorem.
eigenparameter-dependent transmission conditions
Copyright (c) 2026 Fei-fan Li, Ji-jun Ao
Convolution theorem and uncertainty principles for the windowed linear canonical Lions transform
UDC 517.44 We introduce the windowed linear canonical Lions transform, which generalizes the classical Lions transform introduced in [K. Trimeche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harmon. Anal., 4, № 1, 97–112 (1997)]. Some basic properties, such as Plancherel, inversion, and convolution theorems involving this integral operator are formulated and proved. In addition, the Donoho–Stark uncertainty principle, the Lieb uncertainty principle, and the Heisenberg-type inequality via the $k$-entropy are discussed and proved for the proposed transform.
Convolution theorem, Uncertainty principle
Linear canonical Lions transform, Generalized Fourier transform
Copyright (c) 2026 Abdelaali Dades, Othman Tyr
Semi-central Bott–Duffin $(e,f)$-inverses
UDC 512.552 We study Bott–Duffin $(e,f)$-inverses in the context of left and right semi-central idempotents. A new class of generalized inverses named semi-central Bott–Duffin $(e,f)$-inverses is introduced and studied. An example is given to show that the Bott–Duffin $(e,f)$-inverses are not necessarily semi-central Bott–Duffin $(e,f)$-inverses. It is shown that the semi-central Bott–Duffin $(e,f)$-inverses exhibit additional properties beyond the properties of general Bott–Duffin $(e,f)$-inverses. As applications, we examine semi-central Bott–Duffin $(e,f)$-inverses for several classes of matrices.
semi-central Bott-Duffin (e,f)-inverse, Bott-Duffin (e,f)-inverse, matrix
Geometric properties of a generalized linear operator acting on univalent functions
UDC 517.53 Let \(f\) be an analytic and univalent function in an open unit disk \(\mathbb{D}\) that belongs to certain subclasses, such as starlike, convex, or close-to-convex functions. For the parameters \(\alpha, \beta \in [0,1]\) such that \(\alpha + \beta \le 1,\) we define a function\[g_{\alpha,\beta}(z) = (1-\alpha-\beta)f(z) + (\alpha+\beta) z f'(z),\] which represents a convex-type combination of the identity operator and the classical differential operator. We investigate the conditions under which the function \(g_{\alpha,\beta}(z)\) generated by a generalized linear operator preserves the geometric properties of the original function \(f\) with particular emphasis on radius problems related to univalence and distortion behavior. Explicit radius bounds are deduced by using classical analytic techniques. In addition, AI-assisted numerical experiments are used to verify the sharpness of the theoretical results and to illustrate the dependence of the radius functions on the parameters \(\alpha\) and \(\beta.\) Representative numerical values and graphical visualizations are provided.
Analytic function, convex function, close-to-convex function, starlike function, univalent functions, radius of univalence.
Copyright (c) 2026 Alaattin Akyar, Oya Mert Coşkun
New developments of dynamic inequalities on time scales
UDC 517.98 We establish new results for $\diamond_\alpha$-inequalities on time scales and formulate some dynamic Hilbert-type inequalities on the $\diamond_\alpha$-calculus of time scales for functions $\diamond_\alpha$-differentiable with respect to one and two variables. We obtain discrete and continuous inequalities as exceptional cases of our results ($\mathbb{T}=\mathbb{Z},$ $\mathbb{T}=\mathbb{R},$ and $\mathbb{T}=k\mathbb{Z},$ where $k>0$). In addition, we can derive some other inequalities on different time scales, such as $\mathbb{T}=q^{\mathbb{Z}},$ where $q>1.$ These inequalities are proved by using H\"older's inequality and the mean inequality.
Copyright (c) 2026 Lütfi Akin, Hilal Orhan
Optimal recovery of mappings, optimal information operators, and extremal subspaces
UDC 517.5 We consider the problems of optimal recovery of an operator $A$ (generally speaking, nonlinear) defined on a unit ball $B_H$ of the Hilbert space $H$ based on the information about elements of this unit ball $B_H$ given by a linear bounded operator $T\colon H\to Y,$ where $Y$ is a Banach space. For a fixed information operator $T,$ it is shown that the optimal method of recovery is offered by the so-called $T$-interpolating splines. For a fixed $Y$ we also solve the problem of finding the optimal information operator. Moreover, for a bounded linear self-adjoint operator $A,$ it is shown that if $T$ is the optimal information operator for the recovery of $A$ on $B_H,$ then any other operator $TA^n,$ $n\in\mathbb N,$ is also an optimal information operator.
Abstract splines, optimal recovery, optimal information operators, widths, extremal subspaces
Copyright (c) 2025 Владислав Бабенко, Юлія Бабенко, Наталія Парфінович
A survey of the theory of operator polynomial interpolation
UDC 517.988 The present survey is devoted to the 85th birthday of Volodymyr Volodymyrovych Khlobystov, prominent Ukrainian mathematician, renowned for his pioneering contributions to computational mathematics. He was a unique scholar who, despite severe visual impairments (almost total blindness) managed to get leading results in the theory of operator polynomial interpolation. Together with his colleagues and numerous students, he laid the foundations of the contemporary theory of operator polynomial interpolation. The present survey, as compared with the most well-known works of other researchers, highlights a much greater depth and power of Khlobystov’s results. As the most important of his results, we can mention the creation of a new direction in interpolation theory based on continuous nodes. For the first time, this approach balanced the continuous amount of information about the interpolated object with the continuous set of interpolation nodes in the proposed interpolants. The operator interpolation problems of Lagrange, Hermite, and Hermite-Birkhoff types were solved, the conditions for the existence and uniqueness of solutions were established, the constructive description of the entire set of corresponding interpolants was provided, a subset of interpolation polynomials preserving the polynomials of a given degree was selected, the accuracy of the interpolation formulas was analyzed, and the convergence of the interpolation processes was investigated.
Lagrange, Hermite, and Hermite-Birkhoff interpolation operator problems
Copyright (c) 2025 Володимир Макаров, Олена Кашпур
On convergence of a sequence of mappings with inverse modulus inequality to a discrete mapping
UDC 517.5 We consider mappings satisfying the Poletsky-type inverse inequality in a domain of the Euclidean space. It is shown that the uniform limit of the family of these mappings is a discrete mapping. We separately consider domains that are locally connected on their boundaries and regular domains in the quasiconformal sense.
Copyright (c) 2025 E. Sevost’yanov, V. Targonskii
Mather sets of a cascade generated by the FTRL algorithm of a two-agent zero-sum game with mixed strategies
UDC 517.9 We establish conditions under which a two-dimensional cascade generated by the FTRL algorithm of a two-agent zero-sum game with mixed strategies, is determined by a twist mapping of the plane. For the indicated cascade, we prove the existence of Mather sets corresponding to the rotation numbers from a certain interval. In this way, we explain why, in the computer experiments, the orbits of the analyzed cascade are located on closed curves and, in a certain sense, have the property of local convexity.
Copyright (c) 2025 V. Kolner, I. Parasyuk
On the asymptotic behavior of the diameter of the image of a ball at infinity
UDC 517.5 We study the asymptotic behavior of the diameter of the image of a ball under ring $Q$-homeomorphisms with respect to the $p$-modulus for $p>n$ in the space $\mathbb{R}^{n}$, $n\geq 2$. We also establish the lower bound for the distortion of the diameter of ball image and solve extremal problems of minimization of the functionals of distortion of the diameter of ball image on some classes of ring $Q$-homeomorphisms with respect to the $p$-modulus.
ring $Q$-homeomorphisms with respect to $p$-modulus
Copyright (c) 2025 B. Klishchuk, R. Salimov, M. Stefanchuk
Best approximation by trigonometric polynomials of convolution classes generated by some linear combinations of periodic kernels
UDC 517.5 For any nontrivial linear combinations of finitely many Poisson kernels $P_{q_i,\beta}(t)=\displaystyle\sum\nolimits^\infty_{k=0}{q^k_i{\cos \left(kt-\frac{\beta\pi}{2}\right)}},$ $\beta\in {\mathbb R},$ $q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N},$ we establish the Nagy condition $N^*_n$ for all numbers $n$ starting from a certain number $n_0.$ In addition, for any $n\in {\mathbb N},$ we prove the existence of linear combinations $m\ (m\in\mathbb{N}\setminus\{1\})$ of Bernoulli kernels $D_{r_i}(t)=\displaystyle\sum\nolimits_{k=1}^\infty{(-1)}^{\frac{r_i-1}{2}} \dfrac{{\sin k }t}{k^{r_i}},$ $r_i=2l_i-1,\ l_i\in {\mathbb N},$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ where $r_i\ne r_j$ for $i\ne j,$ as well as linear combinations $m$ of conjugate Poisson kernels $P_{q_i,1}(t)=\displaystyle\sum\nolimits^{\infty }_{k=1}q^k_i{\sin k}t,$ $ q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ where $q_i\ne q_j$ for $i\ne j,$ which satisfy the Nikolsky condition $A^*_n$ but do not satisfy the Nagy condition $N^*_n.$ As a result, in each analyzed case, we determine the exact values of the best approximations, on average, of these linear combinations by the trigonometric polynomials of orders not higher than $n-1$ and compute the exact values of the best approximations for the classes of convolutions generated by the indicated linear combinations in metrics of the spaces $C$ and $L.$
the best approximation by trigonometric polynomials, Bernoulli kernel, Poisson kernel, convolution of functions
Copyright (c) 2025 Анатолій Сердюк, Віктор Сорич, Ніна Сорич
Generalized Vandermonde operator and bounded solutions of differential equations
UDC 517.929.2 We study the problem of existence of a unique bounded solution to a linear differential equation of any order with bounded operator coefficients. We consider the case where the corresponding ``algebraic'' operator equation has multiple separated pairwise commuting roots. By using the generalized Vandermonde operator constructed with the help of these roots, we obtain a representation of the unique bounded solution.
generalized Vandermonde operator, differential equation, operator coefficient, bounded solutions
A nondegenerate interpolation continued fraction
UDC 517.518:519.652 We prove that the Thiele's interpolation continued fraction has either \(2k-1\) approximants when the function is a polynomial of the \(k\)th degree or \(2k\) approximants for the function \(g(z) =a/(z-\alpha)^k.\) We specify the conditions under which the coefficients of the continued fraction are finite and different from zero. For a given set of values of the functions at the nodes, we propose an algorithm that either constructs a nondegenerate interpolation continued fraction or establishes the impossibility of this construction. We also present some examples.
ланцюговий дріб, невироджений інтерполяційний ланцюговий дріб, алгоритм побудови
Copyright (c) 2025 Юлія Мисло, Михайло Пагіря
On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set
UDC 512.536 We study the topologization of the semigroup $\mathscr{O\!\!I}\!_n(L)$ of finite partial order isomorphisms of bounded rank of an infinite linear ordered set $(L,\leqslant).$ In particular, we show that every $T_1$ left-topological (right-topological) semigroup $\mathscr{O\!\!I}\!_n(L)$ is an Urysohn, functionally Hausdorff, totally disconnected, and scattered space. It is proved that, on the semigroup $\mathscr{O\!\!I}\!_n(L),$ there exists a unique Hausdorff countably compact (pseudocompact) shift-continuous topology, which is compact, and that the Bohr compactification of the Hausdorff topological semigroup $\mathscr{O\!\!I}\!_n(L)$ is the trivial semigroup.
Copyright (c) 2025 Олег Гутік, Максим Щипель