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, 517.98, 519.8 Let $G$ be a simple finite graph and let $\mathcal U_G$ be the corresponding quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of the quantum automorphisms of $\mathcal U_G.$ It is shown that, for the complete graph $K_n,$ the algebra $C(\mathrm{Qut}(\mathcal U_{K_n}))$ is not commutative even for all $n\geq 3,$ unlike $C(\mathrm{Qut}(K_n))=C(S_n^+).$ Moreover, we prove that, for any graph $G$ with $|V(G)|\geq 3,$ the quantum graph $\mathcal U_G$ admits nonlocal symmetry, which means that there exists a perfect quantum no-signaling correlation for the quantum automorphism game for $\mathcal U_G,$ which is not local.
Copyright (c) 2026 Ольга Островська, Василь Островський, Людмила Туровська
Even positive-definite hyperbolically convex functions in a Hilbert space. Representations of Hilbert spaces by self-adjoint operators
UDC 517.98 The paper consists of two parts. In the first part, we establish sufficient and necessary conditions for the integral representation of even positive-definite hyperbolically convex (h.c.) functions $k(x),$ $x\in H$. These functions are continuous in the $j$-topology. The positive definiteness of a function is understood in the pointwise sense. The analyzed theorem is a modified version of the Berezansky theorem presented in [Yu. M. Berezansky and I. M. Gali, Ukr. Mat. Zh., 24, № 4, 351–372 (1972)]. The integral representation for some other positive-definite kernels were considered in [O. V. Lopotko and I. I. Rudynskyi, Ukr. Mat. Zh., 34, № 3, 310–312 (1982)] and [O. V. Lopotko, Dop. Akad. Nauk Ukr., Ser. A, 8, 11–13 (1991)]. In the second part, we prove the integral representation for a family of commutative self-adjoint operators $\mathcal{A}_x$ connected by algebraic relations. For this purpose, we construct a rigging (chain) $H_{\kappa}=H_0\supset H_+=L_2\supset \mathcal{D}$ for $x\in H.$ Our proof is based on the integral representation of even positive-definite h.c. functions of infinitely many variables (see [Yu. M. Berezansky and Yu. G. Kondratyev, Spectral methods in infinite-dimensional analysis, Naukova Dumka, Kyiv (1988)] and [O. V. Lopotko, Bukov. Mat. Zh., 11, № 1, 26–38 (2023)]). Some other forms of generalizations of this kind were considered in [Yu. M. Berezansky and A. A. Kalyuzhny, Ukr. Mat. Zh., 36, № 4, 417–421 (1984); A. A. Kurepa, Canad. Math. J., 12, 45–50 (1960)]; and [Yu. S. Samoilenko, Spectral Theory of the Sets of Self-Adjoint Operators, Naukova Dumka, Kyiv (1984)].
integral representation, positively definite kernel, topology, rigging, operators
Inverse scattering problem for a hyperbolic system of equations on the semiaxis
UDC 517.98, 517.95 We study а scattering problem for the hyperbolic systems of first-order differential equations on the semiaxis under the assumption that all solutions exhibit asymptotic behaviors at infinity corresponding either to incident or to scattered waves. The problem is reduced to finding the solution of the system in the case where all incident waves are known together with a certain homogeneous boundary condition. Under these assumptions, the solution exists and is unique. Therefore, we observe the appearance of a scattering operator on the semiaxis, which transforms all incident waves into scattered waves. For a special case in which the numbers of incident and scattered waves are equal, we establish the detailed properties of the scattering operator, including its factorization into Volterra factors of the opposite polarities. This factorization makes it possible to solve the inverse scattering problem on the semiaxis, i.e., to unambiguously reconstruct all potentials in the system according to the two known scattering operators corresponding to two different boundary conditions.
scattering problem, inverse problem, integral representations of solutions.
Copyright (c) 2026 Леонід Нижник, Нізамеддін Ш. Іскендеров
Model of a dynamical conflict system with regime of chaotic behavior
UDC 517.938, 519.8 We construct a one-dimensional model of conflict dynamics with regime of chaotic behavior. It is shown that this type of behavior appears as a consequence of the nonlinear law of interaction between a pair of abstract opponents. The main factor in the emergence of chaos is the influence of the parameter of external support provided to the minimal element of the system. The properties of the chaotic behavior are graphically illustrated.
різницеве рівняння, динамічна система конфлікту, режим хаотичної поведінки
difference equation, conflict dynamical system, regime of chaotic behavior
хаотична поведінка траєкторій динамічної системи
Copyright (c) 2026 Тетяна Каратаєва, Володимир Кошманенко
First boundary-value problem on the semiaxis for an isotropic superdiffusion equation of order greater than one
UDC 517.95, 517.983, 519.21 The Dirichlet problem is studied for a one-dimensional isotropic superdiffusion equation of order $\alpha\in(1;2)$ on the semiaxis. The analytic solutions of this problem are obtained by the mapping method in the classes of continuous Hўоlder functions, which may have an integrable singularity with respect to time. The probability interpretation of the solutions is clarified and the uniqueness of the regular solution is proved on an infinite time interval.
Fractional Laplacian, Riesz fractional derivative operator, random Levy processes, Dirichlet boundary value problem, isotropic superdiffusion equation
Strengthened Berezansky's theorem on the tensor product of operators
UDC 517.98 We prove a strengthened version of Berezansky's theorem on the tensor product of linear continuous operators in separable Hilbert spaces, as well as a corollary of this theorem on the embedding of tensor products of Hilbert spaces.
tensor product of operators; tensor product of Hilbert spaces
тензорний добуток операторів; тензорний добуток гільбертових просторів
Solving the Dirichlet problem for the Lamé system by the hypercomplex method
UDC 517.54, 517.95 A hypercomplex method for solving the Dirichlet problem for the Lamé system in a bounded simply connected domain is developed, which is based on representing solutions via components of a monogenic function in a commutative biharmonic algebra. The Dirichlet problem is reduced to a system of integral equations by using a hypercomplex Cauchy type integral. Sufficient conditions for the Fredholm property of the specified system are established for bounded domains with a smooth boundary belonging to a class essentially wider than the class of Lyapunov curves, in the case where the given boundary functions belong to wider classes than the H\"older classes. The solution of the Dirichlet problem for a disk is obtained in explicit form.
Copyright (c) 2026 Сергій Грищук, Сергій Плакса
Nonstationary temperature field of a multilayer plate under the periodic action of a heat flux in the presence of self-radiation
UDC 539.3, 517.95 We construct the solution of a one-dimensional nonlinear nonstationary heat-conduction problem for a multilayer plate in the case where one of its bounding surfaces is heated by a heat flux whose intensity varies as a sinusoidal function of time, whereas the opposite surface is cooled down in the process of convective heat exchange with the medium whose temperature varies according to the exponential law. Moreover, we take into account the effect of thermal emission from the heated surface. The solution is constructed by using generalized functions, the appropriate Green's function for the multilayer plate, and linear splines. The exact sums of the series required to improve the rate of convergence of slowly convergent series are found. The results of numerical analyses of the temperature fields in a five-layer plate are also presented.
Copyright (c) 2026 Роман Кушнір, Борис Процюк, Юрій Процюк
Mathematical modeling of rotation of a free elastic solid with cavities containing fluids in a resistive medium
UDC 531.36, 531.38, 517.93 Based on the well-known P. V. Kharlamov equations of motion of a system of coupled gyrostats and the S. L. Sobolev function of state, we develop a mathematical model of rotation in a resistive medium of a free elastic solid with two cavities completely filled with ideal incompressible fluid. The mathematical model of an elastic solid with an ideal fluid is presented in the form of a system of two elastically coupled solids filled with a fluid. The solids are connected with an elastic restoring Hooke hinge and subjected to the action of dissipative moments and moments supporting their rotation. In the case of two Lagrangian gyroscopes with arbitrary axisymmetric cavities containing fluids, we derive a transcendental characteristic equation and analyze it by taking into account the fundamental tone of the oscillations of fluid. It is proved that necessary conditions for the asymptotic stability can always be satisfied by increasing the elasticity coefficient of the hinge provided that the fundamental tone of the oscillations of fluids is greater than 1. The absence of internal resonance is demonstrated in the case where the first tones of the oscillations of fluids coincide. The analysis of necessary conditions for the asymptotic stability with respect to the angular velocity of uniform rotation turned out to be more complicated. These conditions impose restrictions not only on the fundamental tone of the oscillations of fluids but also on the attached equatorial moment of inertia of solids and the parameter of inertial coupling of the fluid. The investigation of the proposed quite simple mathematical model makes it possible to estimate the influence of elasticity and fluids on the stability of rotation of an elastic solid with fluid with an accuracy sufficient for the practical purposes.
Upper bounds for singular subspaces under unitarily invariant norms with applications to high-dimensional covariance matrix estimation
UDC 519.21, 517.98, 512.64 In [J. Cape, et al., Ann. Statist., 47, Article 2405 (2019)], the authors established perturbation bounds for the $\ell_{2 \to \infty}$-norm of singular subspaces and applied this framework to high-dimensional covariance matrix estimation under multivariate normal distributions. We establish perturbation bounds for unitarily invariant norms and apply these bounds to multivariate normal and sub-Gaussian random vectors. Combined with our upper bounds, these results enable us to get the improved estimation of high-dimensional covariance matrices. Finally, we perform simulation experiments demonstrating the improved scaling behavior as compared to Cape, et al.'s bounds in the presented experiments.
Upper bounds, Singular subspace, Unitarily invariant norm, Covariance matrix
Copyright (c) 2026 Chunguang Ren, Pei Zhang, Shiguo Huang
Mannheim curves with modified orthogonal frame in Galilean space $G_{3}$
UDC 514.75 We introduce admissible Mannheim partner curves according to a modified orthogonal frame in the 3-dimensional Galilean space $G_{3}.$ Moreover, we use a modified orthogonal frame according to torsion. Further, we analyze these partner curves with respect to the modified orthogonal frame by curvature and torsion and obtain some characterizations of these curves. These findings have led us to the discovery of some curious remarks concerning Mannheim curves.
Galilean space, Mannheim partner curves, modified orthogonal frame.
Copyright (c) 2026 Murat Turan, Hülya Gün Bozok
Hilbert basis theorem for generalized Noetherian rings
UDC 512.552 We explore the Hilbert Basis Theorem for uniformly $S$-Noetherian rings, $S$-$w$-Noetherian rings, and uniformly $S$-$w$-Noetherian rings. Moreover, we also show that uniformly $S$-($w$-)Noetherian rings may be not ($w$-)Noetherian even in the case where $S$ consists of nonzero divisors.
On the monophonic global domination number of a graph
UDC 519.17 We introduce the monophonic global domination number $\overline{\gamma}_m(G)$ by combining monophonic convexity (via chordless paths) with the global domination in a graph and its complement. A monophonic global dominating set is defined, and $\overline{\gamma}_m(G)$ is regarded as the minimum size of sets of this kind. We also establish the bounds, relate $\overline{\gamma}_m(G)$ to the classical domination number, and characterize the graphs that attain extreme values. The realization theorem is proved for prescribed parameter values. The behavior of $\overline{\gamma}_m(G)$ under graph operations, in particular, for the corona product, is analyzed. The applications to the network monitoring are discussed and several open problems are proposed for further research.
monophonic global domination number, global domination number, monophonic number, domination number.
Copyright (c) 2026 V. Selvi, J. John, V. Sujin Flower
Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications
UDC 517.51, 517.16 The theory of multiplicative calculus has recently gained loads of attention. In this framework, the operations of addition and subtraction are systematically replaced with multiplication and division, respectively. The interest in this system is to obtain a multiplicative analog of the results obtained in the classical sense. Our aim is to contribute to the body of knowledge in this direction. Specifically, we establish some novel generalizations of the Hermite–Hadamard–Mercer-type inequalities involving new multiplicative tempered fractional integrals. A new lemma is also established. By using this lemma, Hölder's inequality, and the power-mean inequality, we obtain more inequalities of the Trapezoid–Mercer type. Our results generalize many other results available from the literature. We present some numerical examples, utilizing the Wolfram Mathematica software for computations and graphing, to prove the validity of our results by comparing the outcomes for different values of the operating parameter $\lambda\in[0,1]$ with 0.1 as the step size. Finally, we obtain additional estimates by applying our results to special means, such as arithmetic, harmonic, logarithmic, and $p$-logarithmic means.
Hermite--Hadamard type inequalities; Mercer type inequalities; Multiplicative convex functions; Trapezoid inequalities; Special means.
Copyright (c) 2026 Eze R. Nwaeze, Bosede O. Fagbemigun
Majorization for Levinson-type inequalities and generalization for Ky-Fan-type inequalities
UDC 517.51, 517.16 Two Levinson-type inequalities are extended to the majorization type; as a special case, two Ky-Fan-type inequalities are also extended to the majorization type.
Convexity and monotonicity of analytic functions related to Le Roy-type hypergeometric functions
UDC 517.588, 517.54 We investigate the convexity and monotonicity properties for two classes of analytic functions related to the Le~Roy-type hypergeometric function. Numerical examples and graphical illustrations support our theoretical findings and motivate a proposed conjecture.
Analytic functions, Convexity, Monotonicity, Le Roy-type hypergeometric function, Special functions, Geometric function theory.
Le Roy-type hypergeometric function; univalent functions; analytic functions; convex functions, monotonicity
Copyright (c) 2026 Khaled Mehrez, Abdelaziz Alenazi
Embeddings of Hausdorff semitopological semilattices in hyperspaces
UDC 515.12, 512.56 Beer and Ok showed that a locally compact and order-connected Hausdorff topological semilattice $X$ can be embedded in the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion. First, we show that this result can be generalized to the case of Hausdorff semitopological semilattices. Second, we also prove that a locally compact Hausdorff semitopological semilattice is a topological poset. Third, we conclude that a mapping defined by $x\rightarrow{\downarrow}x$ from a locally compact lower semiclosed space $(X, \tau, \leq)$ to the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion is continuous if and only if $X$ is upper open and $\leq $ is closed in $X\times X.$ Finally, we introduce the concept of $H$-closedness for a $T_{0}$ topological space.
Hausdorff semitopological semilattice, Embedding, Hyperspace, $H$-closed;
Copyright (c) 2026 Xiangping Chu, Qingguo Li
Оn Hom-pre-Poisson conformal algebras and related structures
UDC 512.554, 514.7 We introduce and study Hom-Poisson conformal algebras. They are generalizations of Poisson conformal algebras in the Hom-setting. First, we provide several construction results for Hom-Poisson conformal algebras and characterize a special class called quadratic Hom-Poisson conformal algebras. We consider the notion of $\mathcal{O}$-operators on Hom-Poisson conformal algebras and explore them to define and construct Hom-pre-Poisson conformal algebras. Finally, we introduce the notion of Hom-dendriform conformal formal deformations of Hom-Zinbiel conformal algebras and show that the Hom-pre-Poisson conformal algebras are the corresponding semi-classical limits.
Hom-Poisson conformal algebra, Hom-pre-Poisson conformal algebra, Hom-Gel'fand-Dorfman Poisson algebra, deformation quantization.
Copyright (c) 2026 Taoufik Chtioui, Atef Hajjaji, Sami Mabrouk
Distribution of the intertwining number on perfect matchings
UDC 519.11, 519.17, 512.54 Ehrenborg and Readdy defined the intertwining number on set partitions. Considering this statistic on perfect matchings, we provide an additional combinatorial description and present an explicit generating function. In particular, we show that the intertwining number on perfect matchings is essentially equidistributed with the rank function of the strong Bruhat order on fixed-point-free involutions of the symmetric group.
Copyright (c) 2026 Yonah Cherniavsky, Yuval Khachatryan-Raziel
$q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach
UDC 517.588, 517.52 By using the multiplicative form of the extended Carlitz inverse-series relations, we establish two general ``dual'' theorems on Jackson's summation formula for the well-poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n = \Big\lfloor\dfrac{n}2 \Big\rfloor + \Big\lfloor\dfrac{n + 1}2 \Big\rfloor $ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in the classical $\pi$-related Ramanujan-like series with convergence rate ``"$dfrac1{16}$", including the series for $1/\pi^2$ discovered by Guillera (2003). The triplicate dual formulas under the partition pattern $n = \Big\lfloor{\dfrac{n}3}\Big\rfloor + \Big\lfloor{\dfrac{n + 1}3} \Big\rfloor + \Big\lfloor{\dfrac{n + 2}3} \Big\rfloor $ are examined via the "reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts with the convergence rate "$\dfrac{-1}{27}$" as $q\to1.$
Bisection series; Jackson’s formula for well–poised 8phi7-series
Copyright (c) 2026 Xiaojing Chen, Wenchang Chu
Bertrand and hyperelastic curves on Riemannian manifolds
UDC 514.76, 514.75 We introduce the notion of Bertrand curves on Riemannian manifolds. The necessary conditions for a curve on a Riemannian manifold to be a Bertrand curve are obtained in the form of differential equations. Then, necessary and sufficient conditions are found for the Bertrand curve to remain a Bertrand curve in the ambient space. Moreover, we examine the influence of Bertrand and hyperelastic curves on the geometric properties of isometric immersions. We study the relationship between these types of curves and umbilical submanifolds. Finally, we present the criteria for the Bertrand curve to be a hyperelastic curve in the ambient space along an immersion.
Hybrid two-point methods using only one inverse for solving nonlinear equations
UDC 519.6, 517.98 The task of solving of nonlinear equations often involves the use of iterative methods that require multiple inversions of linear operators, which can be computationally costly, especially for large matrices or complex operators. We propose hybrid two-point methods that overcome the indicated limitation by requiring only a single inversion of a fixed linear operator throughout the entire process. This approach preserves the rapid convergence and avoids the repeated inversions of $Q'(y_n)$ in each step. We provide both semilocal and local convergence analyses, utilizing a majorant function to control the Fréchet derivative of the operator. Numerical experiments confirm that the performance of our methods is comparable with the classical approaches but is characterized by much lower computational costs. These findings highlight the potentials of this technique as a practical alternative to Newton's method and other iterative methods requiring operator inversions.
Two Point Methods; Banach space; Divided difference; Hybrid two Point Method; Convergence
Copyright (c) 2026 I. K. Argyros, S. Shakhno, M. Havdiak, M. I. Argyros
Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function
UDC 517.53, 517.54 Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ be a class of normalized analytic functions $f$ in an open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ The aim of the present article is to find upper bounds for the module of some coefficients, of the fourth Hankel determinant $H_{4,1}(f)$ for the function class $\mathcal{\mathcal{S}}_{\cos}^{\ast},$ and of some functionals defined by using the expansion coefficients in Taylor series. Moreover, we also find the upper bounds for the fifth and sixth logarithmic coefficients obtained for the functions from the same class. Note that some of our results are the best possible. The results of our paper improve numerous versions of the results recently presented in [K. Marimuthu, J. Uma, and T. Bulboacă, Hacet. J. Math. Stat., 52, No. 3, 596 (2023)]. The tools used in the proofs have been recently obtained in [N. E. Cho, B. Kowalczyk, A. Lecko, and B. ´Śmiarowska, Filomat, 34, No. 6, 2061 (2020)]. They are combined with the results from [F. Carlson, Ark. Mat. Astr. Fys., 27A, No. 1, 8 (1939)], and with the method aimed at finding the extrema of real functions of many variables.
Copyright (c) 2026 Rashid Ali, Mohsan Raza, Teodor Bulboacă
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