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
A method for the construction of exact solutions to the nonlinear heat equation $u_t = \left(F(u)u_x \right)_x +G(u)u_x +H(u)$
UDC 517.9 We propose a method for the construction of exact solutions to the nonlinear heat equation based on the classical method of separation of variables and its generalization. We consider substitutions used to reduce the nonlinear heat equation to a system of two ordinary differential equations and construct the classes of exact solutions by the method of generalized separation of variables.
Copyright (c) 2019 Barannyk T. A.; Barannyk A. F.; Yuryk I. I.
A problem of extreme decomposition of the complex plane with free poles
UDC 517.54 We study a problem of nonoverlapping domains with free poles on radial systems. Our main results strengthen and generalize several known results obtained in the investigation of this problem.
On the stability of the equilibrium state of a rigid body with a multilayer ideal fluid separated by elastic plates
UDC 531.36:531.38:533.6.013.42 L. N. Stretensky’s problem and the problem of physical pendulum oscillations are generalized to the case of a multilayer ideal fluid separated by elastic plates. Assuming the positive definiteness of the potential energy (L. N. Stretensky’s problem) and changed potential energy (the physical pendulum), we obtain the conditions of stability for the equilibrium state in these problems. A more detailed study is performed in the case of a cylindrical cavity of arbitrary cross-section. We show that for the stability of the equilibrium state in L. N. Stretensky’s problem it is necessary and sufficient that there exists a stable equilibrium state of the elastic plates and liquid in the stationary rigid body and it is sufficient that a heavier liquid is located below a lighter one. For the stability of the equilibrium state in the problem of physical pendulum oscillations, it is also necessary that there is a stable equilibrium state of the elastic plates and liquid in the stationary rigid body. It is also shown that we can use pre-tensioning of plates to stabilize an unstable equilibrium position of the physical pendulum.
багатошарова ідеальна рідина, пружні пластини, стійкість положення рівноваги
Copyright (c) 2021 Юрій Микитович Кононов
A note on variational formalism for sloshing with rotational flows in a rigid tank with an unprescribed motion
UDC 517.9 The Bateman – Luke-type variational formulation of the free-boundary ‘sloshing’ problem is generalized to irrotational flows and unprescribed tank motions, i.e., to the case where both the tank and liquid motions should be found simultaneously for a given set of external forces applied to fixed points of the rigid tank body. We prove that the variational equation, which corresponds to the formulated problem, implies both the dynamic (force and moment) equations of the rigid body and the free-boundary problem, which describes sloshing in terms of the Clebsch potentials.
Copyright (c) 2021 Олександр Миколайович Тимоха, Іван Олександрович Луковський
Mathematical problems of thermomechanics for deformable bodies under thermal irradiation
UDC 535.51, 535.55, 539.3 It is a review of the results of researches in the field of mathematical problems of the thermomechanics of magnetizable and polarizable electroconductive deformable bodies under electromagnetic irradiation, which are carried out at Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine. In this paper, we formulate the problems of mathematical physics that describe the thermal and thermal stress state in such bodies taking into account the peculiarities of electromagnetic action in different frequency ranges and analyze the methods of study of the thermomechanical behavior of bodies (in particular, of different transparency, as semitransparent as well as opaque) in these ranges under thermal irradiation.
Copyright (c) 2021 О. Р. Гачкевич , Р. М. Кушнір , Р. Ф. Терлецький
Integral equation of elastic medium containing a deformable thread-like inclusion
UDC 517.968.23: 539.3 In this study, we propose a method of mathematical modeling of deformable thread-like (wire) inclusions based on replacing their influence on the elastic medium with the tension/compression forces distributed along their axis. We derive a regularized integral equation of the problem, which is external with respect to the inhomogeneity, and also a mathematical model of inhomogeneity, which, taking into account the contact conditions, makes it possible to determine the desired distribution of forces along the axis of inhomogeneity. An approach to solving these equations is proposed. For the case of a rectilinear elastic thread-like inclusion, an approximate solution of the problem is obtained in a close analytical form.
Copyright (c) 2021 Ярослав Пастернак, Георгій Сулим
Reduction of the two-dimensional thermoelasticity problems for solids with corner points to key integrodifferential equations
UDK 539.3 A generalization of the direct integration method is given with regard to solving the original equations of two-dimensional thermoelasticity problems for solids with corner points (i.e., plane problems for a rectangular domain and an annular sector and the axisymmetric problem for a solid cylinder of finite length). The problems are thereby reduced to a governing integrodifferential equation for a key function, which is unique for each problem. By making use of equilibrium equations, the expressions for stress-tensor components are derived in terms of the key function, while the complete sets of boundary conditions are identically reduced to the corresponding sets of integral conditions for the key function. The algorithms for separating variables in the derived governing equations are suggested by utilizing the complete sets of the eigen- and associated functions in order to construct the solutions to the formulated problems in the form of explicit dependencies on thermal loading in view of the boundary conditions.
Copyright (c) 2021 Роман Кушнір, Yuriy Tokovyy, Микола Юзв'як, Анатолій Ясінський
On a particular case of motion in the three-body problem
UDC 531.36; 531.011 We study the three-body problem in a particular case where two bodies have equal masses, which implies the existence of a manifold of symmetric motions. We are looking for the conditions of existence of bounded (unbounded) symmetric motions. Our analysis of boundedness (unboundedness) of motions shows that both the structure of the manifold of symmetrical motions and the integrals of energy and angular momentum are essential.
Symmetrical motion, a Hill stable pair, distal motion, bounded motion
Asymptotic stabilization of a flexible beam with an attached mass
UDC 517.977 A mathematical model of a simply supported Euler – Bernoulli beam with attached spring-mass system is considered. The model is controlled by distributed piezo actuators and a lumped force. We address the issue of asymptotic behavior of solutions of this system driven by a linear feedback law. The precompactness of trajectories is established for the operator formulation of the closed-loop dynamics. Sufficient conditions for strong asymptotic stability of the trivial equilibrium are obtained.
Euler--Bernoulli beam; stabilization; asymptotic stability; Lyapunov functional
Copyright (c) 2021 Юлія Калоша, Олександр Леонідович Зуєв
Review of researches on mathematical problems of mechanics and control theory at Institute of Mathematics NAS of Ukraine
We present a review of researches on mathematical problems of mechanics and control theory conducted at Institute of Mathematics NAS of Ukraine from the beginning of its foundation.
Copyright (c) 2021 О. М. Тимоха, І. О. Луковський, О. Г. Мазко
Structure of integrals of equations of oscillations of a conical shell closed at a vertex
We consider a system of differential equations, which describes the free oscillations of a thin-walled conical shell of rotation with a vertex. Based on the analytical theory of systems of differential equations with a small parameter at the highest derivative and equations with a regular singular point, we establish the formal structure of regular integrals of the original equations.
варіаційні метроди, сингулярно збурені крайові задачі, тонкостінні оболонки обертання
Weighted damping of external and initial disturbances in descriptor control systems
UDC 517.925.51; 681.5.03 We investigate the problem of $H_{∞}$ -control of a generalized type for a class of linear descriptor systems and suggest a criterion and sufficient conditions for the existence of control laws ensuring that the closed-loop system is regular, stable and impulse-free with a desired estimate for the weighted damping level of external and initial disturbances. The main computational procedures for the synthesis of regulators are reduced to solving linear and quadratic matrix inequalities that have no rank constraints. An example of robust stabilization of a hydraulic system with three tanks is given.
Copyright (c) 2021 Олексій Григорович Мазко
UDC 517.5 For a familiy of holomorphic functions on an arbitrary domain, we introduce Fatou and Julia like sets, and establish some of their interesting properties.
On $\Pi$-permutable subgroups in finite groups
UDC 512.542 Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $\Pi$ be a nonempty subset of the set $\sigma.$ A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{complete Hall $\Pi$-set} of $G$ if every member of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $\sigma _{i}\in \Pi$ and ${\cal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma _{i}\in \Pi$ such that $\sigma_i\cap\pi(G)\neq\varnothing.$ A subgroup $A$ of $G$ is called (i) {${\cal H}^{G}$-permutable} if $AH^{x}=H^{x}A$ for $H\in {\cal H}$ and $x\in G$; (ii) {$\Pi$-permutable in $G$} if $A$ is ${\cal H}^{G}$-permutable for some complete Hall $\Pi$-set $\cal H$ of $G.$ We study the influence of $\Pi$-permutable subgroups on the structure of $G.$ In particular, we prove that if $\pi= \displaystyle\bigcup\nolimits_{\sigma_{i}\in \Pi} \sigma_{i}$ and $G =AB,$ where $A$ and $B$ are ${\cal H}^{G}$-permutable $\pi$-separable (respectively, $\pi$-closed) subgroups of $G,$ then $G$ is also $\pi$-separable (respectively, $\pi$-closed). Some known results are generalized.
Copyright (c) 2021 Цзяньхун Хуан, Бин Ху, Никита М. Адарченко
Automorphisms and endomorphisms of partitions of topological spaces
UDC 515.1 Let $X$ be a topological space, let $\Delta$ be a partition of $X$, and let $Y =X/\Delta$ be a quotient space with the corresponding quotient topology. Then the automorphism group $\mathcal{H}(\Delta)$ of $\Delta$ (i.e., the homeomorphisms of $X,$ which permute the elements of partition) acts in a natural way upon $Y$ by homeomorphisms. We determine the cases in which the corresponding homomorphism of the action $\psi\colon\mathcal{H}(\Delta) \to \mathcal{H}(Y)$ into the group of homeomorphisms of $Y$ is continuous with respect to the compact-open topologies. The obtained results have applications to the foliation theory.
Copyright (c) 2026 S. Maksymenko, E. Polulyakh
Groups, rings, and algebras: common strategic approaches and mutual influences
UDC 512.5 We focus on recent promising trends in the application of the key concepts and approaches from the classical infinite-group theory to various branches of algebra, such as modules over group rings, infinite-dimensional linear groups, Leibniz algebras, other generalizations of the Lie algebras, and braces. The efficacy of these trends has been well-documented in a series of recent books from reputable publishers. In our article, we present a concise overview of these emerging trends. The analysis of the mutual influence of algebraic systems promotes deeper understanding of their individual and collective significance and illustrates the unity and diversity typical of contemporary mathematics. We believe that the subsequent development of investigations in this field would promote the appearance of new discoveries and innovations clarifying the fundamental role played by the groups, rings, algebras and other algebraic structures in modern mathematics.
Copyright (c) 2026 M. Dixon, M. Semko, I. Subbotin
Asymptotic behaviour of rapidly varying solutions of a differential equation with rapidly varying nonlinearity
UDC 517.925 We study the asymptotic behavior of one class of solutions for a two-term nonautonomous ordinary differential equation of the fourth order with rapidly varying nonlinearity.
неавтономні диференціальні рівняння, експоненціальна нелінійність, $P_\omega(Y_0,\lambda_0)$- розв'язки, асимптотична поведінка $P_\omega(Y_0,\lambda_0)$- розв'язків
Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams
UDC 512.81 We propose two realizations of representations of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $\Lambda_n$ by differential operators. For each of these realizations, the actions on Schur polynomials are determined and the decomposition of $\Lambda_n$ into irreducible representations is obtained. By using the $\mathfrak{sl}_2$-isomorphism between $\Lambda_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with at most $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n.$
UDC 519.17. 519.951 Let $H=(V, E)$ be a connected finite hypergraph, which is an extension of the graph theory in which the edges may connect more than two vertices and form hyperedges. We study the Kazdan-Warner equation\begin{gather*}\Delta \phi=c-he^{\phi}\end{gather*} on $H,$ where $c$ is a constant and $h$ is a known function defined on $H$. Based on the work by Grigor'yan, Lin, and Yang [A. Grigor'yan, Y. Lin, Y. Yang, Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations, 55, № 4, Article 92 (2016)], we employ the variational calculus to extend the main results concerning the solutions to the Kazdan-Warner equation from finite graphs to hypergraphs. We obtain similar results for the cases where $c>0$ and $c<0$ provided that $h$ satisfies certain conditions on hypergraphs. However, for the case where $c=0,$ we cannot get the same results.
Copyright (c) 2026 Haigang Zhang, Juan Zhao
On the forgotten topological index of signed graphs
UDC 519.17 The forgotten topological index denoted by $F(G)$ of a graph $G=(V,E)$ is defined as follows: $F(G)=\displaystyle\sum\nolimits_{i=1}^{n}\!d_v^3,$ where $d_v$ denotes the degree of the vertex $v.$ We extend the notion of forgotten topological index to signed graphs and introduce the $MS$-index of a signed graph. Moreover, we determine the forgotten topological index for the tensor product, Cartesian product, lexicographic product, strong product, symmetric difference, and the joint of the graphs $G_1$ and $G_2$ in terms of the forgotten topological index, the first Zagreb index, and the $M\kern-1ptS$-index of signed graphs $\Sigma_1=(G_1,\sigma_1)$ and $\Sigma_2=(G_2,\sigma_2),$ along with their sizes and orders.
Copyright (c) 2026 Shariefuddin Pirzada, Mir Riyaz ul Rashid
Coefficient estimates in new subclasses of analytic and bi-univalent functions defined by the combinations of the $q$-derivative and the Sǎlǎgean derivative operator
UDC 517.53 We introduce and study two new subclasses of the function class $\Sigma $ of bi-univalent functions defined in the open unit disk and associated with the Sǎlǎgean and $q$-derivative operators. Furthermore, we find estimates for the Taylor–Maclaurin coefficients $|a_{2}|$ and $|a_{3}|$ of the functions from these new subclasses. The accumulated results are quite interesting and new.
Coefficient bounds and coefficient estimates
On certain geometric properties and Hardy spaces of generalized Bessel–Wright function
UDC 517.53, 517.58 The main objective of the present research is to examine a specific sufficiency criterion for the pre-starlikeness, strong starlikeness, strong convexity, starlikeness, and convexity of the generalized Bessel–Wright function. Furthermore, we determine the conditions for the generalized Bessel–Wright functions to be included in the Hardy spaces, as well as bounded analytic functions. These applications are given in the form of examples and corollaries.
Copyright (c) 2026 M. U. Nawaz, Mohsan Raza, K. Mehrez
Carleson measures on Hardy spaces with bicomplex scalars
UDC 517.53 We explore Carleson measures given on the Hardy spaces defined over the bicomplex space and investigate the properties of the bicomplex Toeplitz operator. In the first section, we provide a comprehensive introduction and formulate the key definitions necessary for the subsequent analysis. In the second section, we present the results for $\mathbb{D}$-Carleson measures and establish their relationship with bicomplex Hardy spaces. Finally, in the last section, we extend the concept of the Toeplitz operator to the bicomplex space, thus offering broader prospects.
Bicomplex numbers, Carleson measures, Hardy spaces, Holomorphic functions
Copyright (c) 2026 Stanzin Dolkar, Sanjay Kumar
UDC 512.55 Let $R$ be a ring and let $M$ be a left $R$-module. We say that $M$ is {\it ps-lifting} if every submodule $N$ of $M$ contains a direct summand $X$ of $M$ such that $\dfrac{N}{X}$ is projective semisimple. We present some properties of these modules. It is shown that: (1) if a projective module is ps-lifting, then it is hereditary; (2) for a ring $R,$ every left $R$-module is ps-lifting if and only if every $R$-module is a direct sum of an injective module and a projective semisimple module; (3) $_{R}R$ is ps-lifting if and only if $\dfrac{R}{\rm Soc(R)}$ is semisimple and $R$ is hereditary.
projective semisimple module, lifting module, projective module
UDC 512.552 We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p \nmid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations.
Group rings; Unit group; Prime fields; Polynomials.
Copyright (c) 2026 Abhinay Kumar Gupta, R. K. Sharma
Characterization and classification of sigmoid function and generalized discrete probability distributions series defined by Awolere–Oladipo derivative operator in the analytic univalent function space
UDC 517.53 The sigmoid function is a special function emulating biological neurons in terms of signal processing or sending alerts to various departments of the brain for responses. It is also good for machine learning, while the generalized discrete probability distribution series is useful for modeling infectious diseases, genetic modeling, clinical trials, risk analysis, optimal pricing, ecological and climatic modeling, and conservation biology. We explore the convex combination and several other types of behaviors of the sigmoid function associated with the generalized discrete probability distribution series in the space of univalent function theory. This is done by introducing and studying a new derivative operator. Several coefficient inequalities and consequences of various choices of the parameters involved are discussed. Graphically, by using the Python software, various convex combinations of $K_{\phi}$ and $g_{\phi}$ are presented in terms of disease and intervention, policy and policy intervention, investment return, disease spread, infection and intervention, free market and intervention, and cost and efficiency relations.
Copyright (c) 2026 Ibrahim Tunji Awolere, Adeniyi Musibau Gbolagade, Adeniyi Ayeni, Abiodun Tinuoye Oladipo, Şahsene Altınkaya
Stability of Caputo fractional hyperbolic systems
UDC 517.95 We study the problems of well-posedness and stability for fractional linear hyperbolic systems (FLHS) with the Caputo time-fractional derivative. First, the existence and uniqueness are established for the global solution of the FLHS. Then we obtain some polynomial stability results for the FLHS in different Hilbert spaces. Finally, some numerical examples are presented to confirm the theoretical results.
Hyperbolic system, Caputo fractional derivative, Polynomial stability, Finite difference scheme
Solutions of a time-asymmetric boundary-value problem for a third-order equation with variable coefficients
UDC 517.951 We study a boundary-value problem with asymmetric time conditions for a third-order inhomogeneous equation with multiple characteristics of lowest-order terms. The unique solvability of the problem is proved by the energy-integral method. It is shown that if the uniqueness condition is violated, then the homogeneous problem has a nontrivial solution. The existence is proved by the Fourier method. The solution of the stated problem is obtained in the explicit form by using the constructed Green's function. The uniform convergence of the solutions and their derivatives contained in the equation is proved.
Differential equation, third order, multiple characteristics, asymmetric boundary value problem, regular solution, uniqueness, existence, Green’s function.
Copyright (c) 2026 Yusupjon Apakov, Raxmatilla Umarov
On the Grüss type inequalities within $p$-disks with applications
UDC 517.5 We extend the classical Gr\"{u}ss inequality from Euclidean disks in $\mathbb{R}^2$ to generalized $p$-disks and present rigorous formulations for functions satisfying H\"{o}lder-type conditions. Several new Gr\"{u}ss-type inequalities are established and applied to establish bounds for the covariance of random variables defined either by probability density functions or by joint distributions.
On the behavior of the modulus of $m$-th derivatives of the algebraic polynomials in the whole complex plane without recurrence formula in the weighted Bergman space
UDC 517.5 We study the problem of growth of the $m$th derivatives of an arbitrary algebraic polynomial in weighted Bergman spaces in unbounded domains of the complex plane, without using the recurrence relation. We consider $k$-quasidisks and quasidisks with some additional functional conditions. These conditions enable us to combine several known classes of curves into a single class and obtain estimates for the growth of $m$th derivatives in this common class. By using similar estimates obtained for bounded quasidisks, we also provide estimates valid in the whole complex plane.
Bernstein-Markoff inequality, Walsh inequality, Algebraic polynomial, Quasiconformal mapping, Quasicircle,
Copyright (c) 2026 F. G. Abdullayev, M. Imashkyzy