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
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
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
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
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
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
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
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.
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
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
Point spectrum in the equilibrium states of dynamic conflict systems and its role in the models of beliefs formation
The structure of the point spectrum in the limit (in time) states of dynamic conflict systems is studied in terms of probability measures. It is shown that a necessary and sufficient condition for the appearance of measures with point spectra is a single priority strategy. In this case, we establish the exponential rate of concentration of the distributions with point spectrum and its density in the phase space. The possibility of using information about the structure of the point spectrum in a new mathematical model of the formation of the beliefs of individuals in an abstract society is proposed.
dynamic system, stochastic matrix, infinite product, point spectrum, belief formation
Copyright (c) 2025 Володимир Кошманенко, Оксана Сатур
Finite difference calculus in the continuum
UDC 517.9 We describe known and new results on the finite-difference calculus on configuration spaces. We also describe the finite-difference geometry on configuration spaces, relate finite-difference operators to the canonical commutation relations, find explicit form of certain finite-difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.
falling factorials; spatial combinatorics; configuration spaces; Newton series; Poisson measure; Wick ordering; canonical commutation relations
Copyright (c) 2025 Dmitri Finkelshtein, Yuri Kondratiev, Eugene Lytvynov, Maria João Oliveira
Editorial Board of the "Ukrainian Mathematical Journal"
Copyright (c) 2025 Редакційна колегія „Українського математичного журналу''
Kirkwood–Salzburg equation for connected correlation functions
An infinite system of nonlinear Kirkwood-Salzburg equations is deduced for the connected correlation functions of a continuous system of classical point particles interacting via a pair potential. We consider the equation for density that takes into account only 2-particle correlations between the particles. The qualitative analysis of the behavior of this equation is carried out from the viewpoint of finding possible critical points indicating the existence of phase transitions for different values of temperature and chemical potential. As a possible next step, we can mention the application of numerical methods and higher correlations.
Класична статистична механіка, зв'язні кореляційні функції.
Copyright (c) 2025 Олексій Ребенко, Юрій Погорєлов
Functional limit theorems for a time-changed multidimensional Wiener process
UDC 519.21 We study the asymptotic behavior of a properly normalized time-changed multidimensional Wiener process. The time change is described by an additive (in time) functional of the Wiener process itself. On the level of generators, the time change means that we consider the Laplace operator, which generates a multidimensional Wiener process, and multiply it by a (possibly degenerate) state-space dependent intensity. It is assumed that the intensity admits limits at infinity in each octant of the state space but the values of these limits may be different. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time-changed multidimensional Wiener process. Among possible limits, there is a multidimensional analog of the skew Brownian motion.
Multidimensional Wiener process; time change; functional limit theorem; multidimensional skew Brownian motion
Copyright (c) 2025 Yuliya Mishura, René L. Schilling
Extended objects as ultradistribution-marked configuration spaces
UDC 517.9 Configuration spaces of point particles with and without internal quantum numbers (marks) were studies earlier as a framework for the description of the dynamics of multiagent systems. Here, we propose to extend this framework to configuration spaces of extended objects of any shape. This is attained by identifying the shape with the support of an ultradistribution of compact support. Then, by using the multipole expansion property of these ultradistributions, we conclude that the configuration spaces of extended objects are a simple extension of the marked configuration spaces.
Configuration spaces; Extended objects; Ultradistributions
Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group
UDC 517.98 Suppose that we have a canonical Gibbs measure $\mu$ defined on a marked configuration space $\Omega$ that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on $\Omega.$ Then $\mu$ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to $\mu$-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version $\beta$ of this cocycle. Further, we characterize all the measures on $\Omega$ that possess $\beta$ as their cocycle. If $\mu$ is obtained (e.g.) from a particular two-body potential $\hat{V}$ (satisfying some mild regularity assumptions), then $\beta$ takes a certain explicit form, and the class of canonical Gibbs measures characterized by $\beta$ contains exactly the measures associated with the potential $\hat{V}.$ Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general $G$-spaces for local infinite-dimensional groups.
Canonical Gibbs measures, marked Poisson measures, diffeomorphism groups, quasi-invariant measure, Radon-Nikodym derivatives
Copyright (c) 2025 Tobias Kuna, Gerald A. Goldin, Yuri G. Kondratiev, José L. Silva
Gibbs measure over the cone of vector-valued discrete measures
UDC 517.9 We consider a gas each particle of which is characterized by a pair $(x,v_x),$ where $x\in \mathbb R^d$ is the position and $v_x\in \mathbb R^d_0=\mathbb R^d\setminus \{0\}$ is the velocity. Gibbs measures are defined on the cone of vector-valued measures. Our aim is to prove their existence. We introduce a family of probability measures $\mu_\lambda$ on the cone $\mathbb K(\mathbb R^d)$ and define local Hamiltonian and partition functions for a positive, symmetric, bounded, and measurable pair potential. By using the definitions mentioned above, we define Gibbs measure as a solution to the Dobrushin–,Lanford–Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is nonempty and relatively compact.
Local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential
Gibbs’s measure as a solution to the Dobrushin-Lanford-Ruelle equation.
Copyright (c) 2025 Luca Di Persio, Yuri Kondratiev, Viktorya Vardanyan
On a continuous analog of the parabolic Anderson model
UDC 519.21; 517.9 We consider a stochastic equation in $\mathbb{R}^{d},$ whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in $\mathbb{R}^{d}.$ We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as $t\rightarrow \infty $ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results.
Copyright (c) 2025 Yu. Kondratiev, L. Pastur
Stochastic dynamics on product manifolds: twenty five years after
UDC 519.21; 517.9 We consider an infinite system of stochastic differential equations in a compact manifold $\mathcal{M}.$ The equations are labeled by vertices of a geometric graph with unbounded vertex degrees and coupled via the nearest neighbor interaction. We prove the global existence and uniqueness of strong solutions and construct in this way the stochastic dynamics associated with Gibbs measures that describes equilibrium states of a (quenched) system of particles with positions, which form a typical realization of a Poisson or Gibbs point process in $\mathbb{R}^{d}.$
On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$
UDC 517.9 We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $\beta\colon \mathbb{R}\to\mathbb{R},$ $-D\colon \mathbb{R}^d\to\mathbb{R}^d,$ and $b\colon \mathbb{R}\to\mathbb{R},$ is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu, M. Röckner, The invariance principle for nonlinear Fokker–Planck equations, J. Different. Equat., 315, 200–221 (2022)] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb{R}^d)$ does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions.
Copyright (c) 2025 Viorel Barbu, Michael Röckner
Some personal reminiscences on Yuri Kondratiev and the impact of his research
UDC 517.9 We sketch a strong impact of the works by Yuri Kondratiev in various fields of science. In particular, after a more personal literature review of his studies, we briefly give two examples in which the theoretical results of Yuri Kondratiev changed the situation in the corresponding research theories.
Infinite Dimensional Analysis, Mathematical Physics
Copyright (c) 2025 Sergio Albeverio, Wolfgang Bock, Ludwig Streit
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.