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
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.
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
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
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
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
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
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.
Copyright (c) 2026 Eureka Pattnayak, Trailokya Panigrahi, Rabha M. El-Ashwah
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.
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
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
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
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
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
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.