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
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 http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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
Copyright (c) 2026 Halit Orhan http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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
Copyright (c) 2026 Engin Kaynar http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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
Copyright (c) 2026 Hassen Arfaoui http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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.
Copyright (c) 2026 Mohammad W. Alomari http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
Zinbiel superalgebras and superbialgebras
UDC 512.5 In the present paper, we propose and study a new structure of the Zinbiel superalgebras. Moreover, we define dual bimodules and matched pairs. In addition, we establish a relationship between the dual bimodules of Zinbiel superalgebras and the matched pairs of Lie speralgebras. Furthermore, we examine the Manin triples of Zinbiel superalgebras. Finally, we present the theorem that establishes the links between the matched pairs of Zinbiel superalgebras, the Zinbiel superbialgebras, and the matched pairs of subadjacent Lie superalgebras.
Zinbiel superalgebras; Lie supercoalgebras; bialgebras; Bimodule;Matched pair.
Copyright (c) 2026 Boujemaa Agrebaoui, Imed Basdouri, Jebli Mohamed Eljoudi http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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 http://creativecommons.org/licenses/by/4.0
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
Copyright (c) 2026 S. Golubev http://creativecommons.org/licenses/by/4.0
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 $\mathbb{Q}\mathcal{Y}_n.$
Copyright (c) 2026 L. Bedratyuk http://creativecommons.org/licenses/by/4.0
Spectral analysis of self-adjoint momentum operators with nonlocal potentials
UDC 517.98 Dedicated with great pleasure to Prof. Leonid P. Nizhnik on the occasion of his 90th birthday. We construct and present a detailed study of self-adjoint operators of momentum and their perturbations by nonlocal potentials on the entire axis with one limit point and on a finite interval. The domain of definition of these perturbed self-adjoint operators is described by extended homogeneous boundary conditions, which contain not only boundary values of the function but also linear functionals of functions in the entire domain of their definition. The spectral properties of exactly solvable models of this kind are investigated.
condition for self--adjointness, resolvent structure, eigenvalues
momentum operator, differential operator,
momentum operator, differential operator
Copyright (c) 2025 Каміла Дембовська, Ірина Нижник http://creativecommons.org/licenses/by/4.0
On some classes of non-commutative dimonoids
UDC 512.53 We study dimonoids, i.e., algebraic structures with two associative binary operations satisfying a given system of axioms. We analyze the properties of dual dimonoids. In the class of non-commutative dimonoids, we construct numerous Abelian, non-Abelian, and rectangular dimonoids. The structure of these objects is analyzed, their automorphism groups and halos are computed.
напівгрупа, дімоноїд, група автоморфізмів
Copyright (c) 2025 Володимир Гаврилків http://creativecommons.org/licenses/by/4.0
Fusion and specialization for type ADE shuffle algebras
UDC 512.5 Root vectors in quantum groups (of finite type) are generalized to fused currents in quantum loop groups [J. Ding, S. Khoroshkin, Transform. Groups, 5, №1, 35–59 (2000)]. We construct fused currents as duals to specialization maps of the corresponding shuffle algebras [B. Enriquez, Transform. Groups, 5, №2, 111–120 (2000), B. Enriquez, J. Lie Theory, 13, №1, 21–64 (2003), and B. Feigin, A. Odesskii, NATO Sci., Ser. II, Math. Phys. Chem., 35 (2001)] for the ADE types; an approach, which has a potential for generalization to arbitrary Kac–Moody types. Both root vectors and fused currents depend on a convex order of the positive roots and, in the present paper, we choose the Auslander–Reiten order [C. Ringel, J. reine und angew. Math., 470, 51–88 (1996)] corresponding to the orientation of the ADE-type Dynkin diagram.
Copyright (c) 2025 Andrei Neguţ, Alexander Tsymbaliuk http://creativecommons.org/licenses/by/4.0
Bohr's inequalities associated with the set of all sequences of nonnegative continuous functions
UDC 517.5 We establish several sharp versions of Bohr's inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\mathbb{D} := \{z \in \mathbb{C}\colon |z| < 1\}$ by using a sequence $\{\Psi_n(r)\}_{n=0}^\infty$ of nonnegative continuous functions defined on $[0,1)$ and such that the series $\sum_{n=0}^\infty \Psi_n(r)$ converges locally uniformly in $[0,1).$ As an application, we deduce several well-known results, as well as numerous improved and refined Bohr's inequalities for harmonic mappings in the unit disk $\mathbb{D}.$
Copyright (c) 2025 Raju Biswas, Rajib Mandal http://creativecommons.org/licenses/by/4.0
Sharp bound for the second Hankel determinant for a $q$-starlike function associated with the $q$-exponential function
UDC 517.5 We determine the sharp bound of the second-order Hankel determinant $H_{2,2}(f)$ and the coefficient functional for functions from the class of $q$-starlike functions $ \mathcal{S}^*(\mathcal{L},q)$ by refining the results obtained in [Srivastava et al., Upper bound of the third Hankel determinant for a subclass of $q$-starlike functions associated with the $q$-exponential function, Bull. Sci. Math., 167 (2021); https://doi.org/10.1016/j.bulsci.2020.102942]. Moreover, we obtain sharp estimates for the Fekete–Szegö functional and the coefficient differences for the analyzed class of functions.
Copyright (c) 2025 Karri Sanjay Kumar http://creativecommons.org/licenses/by/4.0
VITALII PAVLOVYCH MOTORNYI (28.07.1940–21.07.2025)
O. M. Tymokha, V. L. Makarov, I. O. Shevchuk, V. Ya. Gutlyanskii, V. F. Babenko, S. B. Vakarchuk, O. V. Davydov, V. A. Kofanov, N. V. Parfinovych, A. S. Romanyuk, A. S. Serdyuk, I. V. Sokolenko
Copyright (c) 2026 O. M. Tymokha, V. L. Makarov, I. O. Shevchuk, V. Ya. Gutlyanskii, V. F. Babenko, S. B. Vakarchuk, O. V. Davydov, V. A. Kofanov, N. V. Parfinovych, A. S. Romanyuk, A. S. Serdyuk, I. V. Sokolenko http://creativecommons.org/licenses/by/4.0
UDC 512.552.4 The $G$-construction allows us to obtain a system of subspaces using an abstract Gram operator. Any system of subspaces is unitarily equivalent to a system constructed by using this method. We propose another construction of a system of subspaces with the same property, provide criteria for the unitary equivalence of a pair of systems of subspaces in terms of the corresponding Gram operators, and study the problem of irreducibility of a system of subspaces in terms of the corresponding Gram operator. In addition, we provide an example illustrating how the obtained results apply to the study of a certain class of the systems of subspaces.
Copyright (c) 2025 O. Strilets http://creativecommons.org/licenses/by/4.0
Supplement to the article by Douak and Maroni (2020) on a new class of 2-orthogonal polynomials
UDC 510 We consider some issues related to the 2-orthogonal polynomials (2-OP). The answers to these issues can be regarded a supplement to the article [K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties, Integral Transform and Special Functions (2020)]. The conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the ``"classical" nature of the 2-OP in Hahn's sense are clarified. It is constructively proved that these recurrence relations do not cover all possible "classical" 2-OPs. An example of ``"classical" 2-OP generated by the generating function constructed by using a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the classical OPs. In particular, this concerns the fact that their zeros are real and their location. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the ``"classical" 2-OPs with similar properties.
2-orthogonal polynomials, 2-orthogonal polynomials, generating function, recurrence relation, ``classical'' 2-orthogonal polynomials, differential equation, zeros of orthogonal polynomials.
Copyright (c) 2025 V. Makarov http://creativecommons.org/licenses/by/4.0
Solvability of linear boundary-value problems for ordinary differential systems in the space $C^{n}$
UDC 517.927 We study linear boundary-value problems for systems of first-order ordinary differential equations with the most general boundary conditions in the normed spaces of continuously differentiable functions on a finite closed interval. The boundary conditions are allowed to be overdetermined or underdetermined with respect to the differential system and may contain arbitrary derivatives of the unknown functions. We prove that the problem operator is Fredholm on appropriate pairs of normed spaces, find its index and $d$-characteristics, and prove the limit theorems for sequences of characteristic matrices of the investigated boundary-value problems and $d$-characteristics of the corresponding Fredholm operators.
Copyright (c) 2025 V. Soldatov http://creativecommons.org/licenses/by/4.0
A quadratic completely integrated Burgers-type nonlinear model: conservation laws and Poisson structures
UDC 517.9 We study the complete integrability of a Burgers-type nonlinear quadratic dynamical system on a functional manifold. For this system, we prove the existence of an infinite hierarchy of functionally independent and involutive conservation laws and determine a compatible pair of Poisson operators, which allows us to rewrite the analyzed system in the bi-Hamiltonian form. We also construct a recursion operator generating an infinite hierarchy of commuting vector fields on a functional manifold.
нелiнiйна динамiчна система, закони збереження, асимптотичний аналiз, пуассоновi структури, бiгамiльтоновiсть, iнтегровнiсть, рекурсiйний оператор.
Copyright (c) 2025 Ya. Kokovska, M. Prytula http://creativecommons.org/licenses/by/4.0
Characteristics of equilibrium states in the models of struggle between alternative opponents in the presence of external assistance only to individual players
UDC 517.9 We determine the characteristics (type, number, position in the phase portrait, and stability) of the equilibrium states of a dynamical system simulating the competition between individuals in an abstract society a part of which receives external assistance. By analyzing a system with three individuals as an example, we determine the detailed description of the dependence of phase portrait of the system on the values of the parameter of assistance and its bifurcation thresholds for two cases where assistance is provided only to one or to two players. For possible applications, we propose an interpretation of the equilibrium states as compromises in the conflict competition of social clusters.
dynamical system, equilibrium states, difference equations, stability
Copyright (c) 2025 T. Karataieva, V. Koshmanenko http://creativecommons.org/licenses/by/4.0
UDC 517.9 We propose a new family of transcendental meromorphic functions $\lambda + \tan z^2$ for $ \lambda \in \mathbb C$ and study the dynamics of the family of functions. We explore both the dynamical plane ($z$-plane) and the parameter plane ($\lambda$-plane). We show that, in the dynamical plane, there are no Herman rings, and the Julia set forms a Cantor set when the parameter lies within the unbounded hyperbolic components. In addition, it is proved that these unbounded hyperbolic components are the only available components distributed over the four quadrants of the parameter space in the complex plane. Conversely, it is shown that the Julia set is connected for the maps whose parameter lies within the remaining hyperbolic components of the parameter space. We also perform the comprehensive analysis of the combinatorial structure of both the parameter space and the dynamical plane for this family of transcendental meromorphic maps.
Copyright (c) 2025 Santanu Nandi http://creativecommons.org/licenses/by/4.0
Moment inequality, central limit theorem, and the invariance principle for linearly positive quadrant dependent random fields
UDC 519.21 We obtain the moment inequality by considering the central limit theorem for the sums of linearly positive quadrant dependent random fields when the covariance satisfies certain conditions. Applying this moment inequality, we prove the invariance principle.
moment inequality, central limit theorem, linearly positive quadrant dependent random fields, invariance principle.
Copyright (c) 2025 Abdurakhman Mukhamedov http://creativecommons.org/licenses/by/4.0