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
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
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
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
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
Application of the even-type delayed mean in the approximation of functions from the generalized Zygmund class with weight
UDC 517.5 We introduce a new class of functions, namely, the generalized Zygmund class with weight. Moreover, we determine the degree of approximation for functions belonging to this class. The degree of approximation is expressed in terms of two moduli of continuity of the second order $w_1$ and $w_2$ such that $w_1/w_2$ is positive and nondecreasing. To realize this task, for approximation we use the even-type delayed mean of their Fourier series.
Degree of approximation, Fourier series, Zygmund class with weight, modulus of continuity, even-delayed mean.
Copyright (c) 2025 Xhevat Z. Krasniqi http://creativecommons.org/licenses/by/4.0
Bicomplex Green's theorem and its application
UDC 517.9 We consider a bicomplex extension of Green's theorem, which is a key result in the integral calculus. By its application, we illustrate the bicomplex Cauchy–Goursat theorem and employ the bicomplex version of Green's theorem in order to efficiently determine numerous integrals involving various forms of bicomplex functions.
Copyright (c) 2025 Mahesh Puri Goswami, Raj Kumar http://creativecommons.org/licenses/by/4.0
Timelike Bonnet surfaces in the Lorentzian 3-space
UDC 519.11 We generalize the criteria proposed by Z. Soyu\c{c}ok in the 3-dimensional Euclidean space to the Lorentz 3-space for the timelike surfaces. With an aid of this criterion we investigate nonisothermic timelike Bonnet surfaces and their associate surfaces. Finally, we consider the classification of helicoids proposed by Beneki, et al. We present nonisothermic timelike helicoids as examples illustrating our results. Thus, they satisfy the conditions as timelike Bonnet surfaces. In addition, we present the isothermic timelike minimal helicoids.
Bonnet surface, timelike surface, timelike helicoid, associate surface, A-net
Copyright (c) 2025 Filiz Kanbay, Burcu Yüksekdağ http://creativecommons.org/licenses/by/4.0
Bergman theory induced by a quaternionic perturbed fractional $\psi$-Fueter operator
UDC 517.5 A useful tool proved in the present paper is the Borel–Pompeiu formula for the resulting function class of null solutions of the quaternionic perturbed fractional $\psi$-Fueter operator. Motivated by this formula, we developed major steps of the associated Bergman theory.
perturbed fractional $\psi$-Fueter operator
Copyright (c) 2025 José O. González Cervantes, Marco A. Pérez de la Rosa, Juan Bory Reyes http://creativecommons.org/licenses/by/4.0
Topological structure of simple pro-Hamiltonian flows on the Möbius strip
UDC 515.1 We investigate the topological properties of flows on the Möbius strip, whose lift to a double cover, which is a cylinder, consists of Hamiltonian flows with a Hamiltonian that is a Morse function, constant on the boundary components. We construct a topological classification of such simple flows using distinguishing graphs made up of rooted trees, which are Reeb graphs. The resulting recursive formula calculates the number of topologically non-equivalent flows with a given number of saddles.
Copyright (c) 2025 Олександр Пришляк, Сергій Стась http://creativecommons.org/licenses/by/4.0
On the domain of analytic continuation of Lauricella–Saran’s hypergeometric functions $F_M$ and their ratios
UDC 517.5 We consider the problem of extending the Lauricella-Saran's hypergeometric functions $F_M$ by branched continued fractions. In three-dimensional complex space, the domain of analytical continuation of the Lauricella-Saran's hypergeometric functions $F_M$ and their ratios is established.
Copyright (c) 2025 Роман Дмитришин, Іван Нижник http://creativecommons.org/licenses/by/4.0
Lax–Sato integrable dispersionless systems on supermanifolds related to a centrally extended generalization of the loop superconformal Lie algebra
UDC 517.9 We propose a new Lie-algebraic approach to the construction of Lax–Sato integrable dispersionless systems on functional supermanifolds by means of the centrally extended semidirect sum of the loop Lie algebra of superconformal vector fields on a supertorus and its regular dual space, which is based on the general Adler–Kostant–Symes Lie-algebraic scheme. By using this approach, we obtain the Lax–Sato integrable superanalogs for some systems of Mikhalev–Pavlov-type dispersionless equations given on functional supermanifolds of four commuting and numerous anticommuting independent variables and find the left gradients of the Casimir invariant reduced to the orbits of the coadjoint action of the central extension, related to these systems, as well as the associated pairs of compatible Poisson operators.
Copyright (c) 2025 Оксана Гентош http://creativecommons.org/licenses/by/4.0
A version of Cartan–Nochka's theorem for non-Archimedean holomorphic curves with integrated reduced counting functions
UDC 517.5 Let $\mathbb{K}$ be an algebraically closed field of characteristic $0,$ completed with respect to a non-Archimedean absolute value and let $\mathbb{P}^n(\mathbb{K})$ be an $n$-dimensional projective space over $\mathbb{K}.$ A collection $\mathcal H = \{H_1,\ldots,H_q\} \in \mathbb{P}^n(\mathbb{K}),$ $q \geq N+1,$ is said to be in $N$-subgeneral position if, for any $1\leq i_1<\ldots<i_{N+1}\leq q,$ we have $\bigcap_{j=1}^{N+1} H_{i_j} = \varnothing.$ We prove a version of the second main theorem for non-Archimedean holomorphic curves intersecting hyperplanes in $N$-subgeneral position with integrated reduced counting functions.
Holomorphic curves, $p$-adic value distribution, $p$-adic Nevanlinna-Cartan theory
Copyright (c) 2025 Ha Tran Phuong, Bui The Hung, Padaphet Inthavichit http://creativecommons.org/licenses/by/4.0
Embedded trapped modes for water waves in an infinite channel
UDC 517.9 We present a simple perturbation scheme for calculating the frequencies of water waves trapped by a small obstruction of the water bed. The corresponding solutions describe embedded trapped modes that are localized at the obstruction and decay at infinity. A simple formula for their frequencies is obtained.
Copyright (c) 2025 M. I. Romero Rodríguez http://creativecommons.org/licenses/by/4.0
$(p,q)$-Analytic functions and a complex $(p,q)$-binomial formula
UDC 517.5 We introduce the notions of $(p,q)$-differential for complex discrete functions and $(p,q)$-complex differential operators in a sense of Pashaev. The $(p,q)$-analog of the $q$-binomial formula is presented and the $(p,q)$-Taylor expansion is constructed for $(p,q)$-analytic functions. These results provide a generalized framework for $(p,q)$-calculus, contributing to its applications in mathematical analysis and related fields.
Copyright (c) 2025 İlker Gençtürk http://creativecommons.org/licenses/by/4.0
Casorati curvatures of pointwise slant submanifolds in para-complex space forms
UDC 515.1 We define and study pointwise slant and pointwise semi-slant submanifolds in para-Kaehler manifolds. Some theorems, characterizations, and examples are obtained for pointwise slant and pointwise semi-slant submanifolds. We also obtain some results concerning the Casorati curvature for a pointwise slant submanifold in the para-Kaehler space form.
2020 Mathematic Subject Classification. 53C43,53C15,53C40.
Copyright (c) 2025 Sedat Ayaz http://creativecommons.org/licenses/by/4.0
On $\mathit{G}$-derivations of Lie–Yamaguti superalgebras
UDC 512.5 Let $\mathit{G}$ be an automorphism group. We study $\mathit{G}$-derivations associated with the Lie–Yamaguti superalgebras. The concept of $\mathit{G}$-derivation, which is a derivation under both bilinear and trilinear operations is defined for the Lie–Yamaguti superalgebras. We also study some important properties of $\mathit{G}$-derivations, as well as with their relationship with other derivations of the Lie–Yamaguti superalgebras.
Lie-Yamaguti superalgebra; Derivation; Quasi-derivation
Copyright (c) 2025 Meher Abdaoui, Jamel Boujelben http://creativecommons.org/licenses/by/4.0
Evaluation of the growth of solutions of systems of linear differential equations of the $n$th order
UDC 517.925.7 For a system of linear differential equations of the $m$th order with coefficients holomorphic in a neighborhood of $\infty,$ we obtain an a priori estimate of growth for the $n$-valued ($n \in \mathbb{N}$) analytic vector solutions of the system via the estimate for the order of growth of its coefficients.
systems of ordinary differential equations
theory of distribution of values ​​of meromorphic functions
Copyright (c) 2025 Анатолій Мохонько, Олексій Мохонько http://creativecommons.org/licenses/by/4.0
Classical solutions of time-fractional quasilinear reaction-diffusion systems
UDC 517.9 We analyze a quasilinear reaction-diffusion system with the time-fractional Caputo derivative. We prove the existence and uniqueness result to initial-boundary problems with Dirichlet and Robin (Neumann) boundary conditions under suitable assumptions on the given data. The existence of the solution to our problem is proved by the Leray–Schauder fixed-point theorem. Positivity property allows us to apply the method of upper-lower solutions. We provide an example of upper and lower solutions to some specific time-fractional reaction-diffusion system.
Caputo derivative, Holder spaces, classical solution
Copyright (c) 2025 Микола Краснощок http://creativecommons.org/licenses/by/4.0
Riemann operator function. Cayley’s transformation
UDC 510 We study a new class of special functions called Laguerre–Bessel polynomials (PLBs), which play the same role as Laguerre polynomials in the case of application of the Cayley transformation to the evaluation of the operator exponent. The generating function for PLBs is a Bessel operator function of the first kind of zero order obtained by replacing its operator argument by its Cayley transformation. It is shown that the PLBs coincide with a new class of 2-orthogonal polynomials with an accuracy of up to a linear transformation. It is shown that the PLBs are classical in the Hahn–Maroni sense because their normalized derivatives also form a class of 2-orthogonal polynomials with an accuracy of up to a linear transformation. We determine the following characteristics traditional for the polynomial classes: explicit representation, recurrence relations, and the differential equations of the minimal third order.
2-orthogonal polynomials, Laguerre--Ceyley polynomials, recurrent relation, Hahn property, classical $d$-orthogonal polynomials, monic polynomials, differential equation
Copyright (c) 2025 Володимир Макаров http://creativecommons.org/licenses/by/4.0
UDC 517.956.223+512.64 We present known forms of the Lopatynsky condition of ellipticity of a problem for a linear regularly elliptic system of equations and determine new forms of this condition. Several examples of its application to various elliptic systems of equations are considered.
еліптичні системи, еліптичні задачі, умова Лопатинського, розклад матричного многочлена
Copyright (c) 2025 Володимир Ільків http://creativecommons.org/licenses/by/4.0
Averaged characteristics of smoothness in $L_2$ and estimations for the widths values of function classes
UDC 517.5 In the space of $2\pi$-periodic functions $L_2,$ we investigate the characteristic of smoothness $\omega^{*}_{\mathcal{M}}(f,t):=$ $(1/t) \displaystyle\int\nolimits_0^t \omega_{\mathcal{M}}(f,\tau) d \tau $ obtained as a result of averaging of the generalized modulus of continuity $\omega_{\mathcal{M}}(f)$ formed by using a generalized finite-difference operator $\Delta^{\mathcal{M}}_h \colon L_2 \to L_2.$ We also study some properties of the functions $\omega_{\mathcal{M}}(f)$ and $\omega^{*}_{\mathcal{M}}(f).$ For the classes of functions $W(\omega^{*}_{\mathcal{M}}, \Phi),$ where $\Phi$ is a majorant, we determine the lower and upper estimates for the values of a series of $n$-widths and the establish the condition for $\Phi$ under which we obtain the exact values of these estimates. Several exact results are illustrated by specific examples.
generalized finite difference operator, generalized smoothness characteristic of function, Fourier series, Fourier coefficients, majorant, widths of function classes.
Copyright (c) 2025 Сергій Вакарчук, Валентина Забутна, Михайло Вакарчук http://creativecommons.org/licenses/by/4.0
Some inequalities for a certain subclass of Bazilevič functions involving multiplicative calculus
UDC 517.5 We define a new class of functions expressed as a convex combination of differential characterizations affected by the multiplicative derivative. Multiplicative derivative is defined in a domain, which excludes zero. Hence, in this case, the defined subclass does not involve replacing the ordinary derivative with a multiplicative derivative. However, we have used the motivation behind the purpose of this restrictive calculus given by the circumstances that we have a more versatile calculus of Newton and Euler. Estimates involving the initial coefficients of the functions, which have Maclaurin series and belong to the given function class are our main results. Some examples, along with the graphs, are used to establish the inclusion and the closure properties. Moreover, we obtain the logarithmic and inverse coefficients for the analyzed function class.
Copyright (c) 2025 Kadhavoor R. Karthikeyan, Sakkarai Lakshmi, Seetharam Varadharajan http://creativecommons.org/licenses/by/4.0