Transactions of Institute of Mathematics, the NAS of Ukraine
Transactions of Institute of Mathematics of NAS of Ukraine (TRIM) publishes original research and reivew papers (as an exception, essays in the history of mathematics), whose topics are associated with current research interests of the institute (see, Editorial Board) in the fields of pure and applied mathematics, as well as the history and methodology of teaching mathematics. The annular volume consists contains one continuous issue. The Editorial Board considers manuscripts written in Ukrainian and English.
Algebraic structure of the fundamental group of orbits of smooth functions on the 2-torus
This review paper is devoted to the description of the algebraic structure of the fundamental group of orbits of functions with isolated singularities on 2-torus with respect to the action of the group of diffeomorphisms of the 2-torus
Copyright (c) 2024 Богдан Фещенко http://creativecommons.org/licenses/by/4.0
Nonlinear properties of trajectories in models of dynamic conflict system
This article focuses on the study of trajectory properties of conflict dynamical systems. Both attractive and repulsive interactions between system components are considered. The results reveal a wide range of behavioral scenarios, including stable, cyclic, quasi-periodic, and chaotic regimes. Special attention is given to the role of interaction parameters and initial conditions in determining the long-term dynamics of the system. The findings provide a foundation for modeling conflict systems in various applied domains, such as economics, ecology, and social sciences.
Copyright (c) 2024 Оксана Сатур http://creativecommons.org/licenses/by/4.0
Fractal analysis of functions with complex local structure
We consider class of fractal functions whose argument and value are written in the same (or different) number representation systems with zero redundancy (Q2-representation, Q2*-representation, A-continued fraction representation of numbers). For the given examples of function constructions, their topological, metric, fractal and differential properties are described.
Copyright (c) 2024 Софія Ратушняк http://creativecommons.org/licenses/by/4.0
Spatial oscillations of a liquid in a vertical cylindrical container
Using the nonlinear Narimanov–Moiseev modal system with linear damping coefficients corresponding to the logarithmic decrement of the sloshing eigenmodes, steady-state damped resonant sloshing in a cylindrical tank has been investigated. The usage of Miles formula to estimate the damping coefficients shows that these effects can be of significant importance in laboratory conditions. Asymptotic steady-state solutions of the modal system have been obtained for a given motion of a cylindrical tank with four degrees of freedom when the forcing frequency approaches the lowest eigenfrequency of the fluid. This solution turns out to be asymptotically equivalent to a horizontal elliptical trajectory of the container. The response curves, which depend on the ratio of the semiaxes of the elliptical orbit and are related to the amplitudes of the two lowest eigenmodes of the sloshing, have been analyzed.
Copyright (c) 2024 Ігор Райновський http://creativecommons.org/licenses/by/4.0
Estimates of characteristics of nonlinear approximation of classes of periodic functions
In the paper we make an overview of results concerning order estimates of the best m-term (sparse) trigonometric approximation and the best orthogonal trigonometric approximation of the Stepanets classes of multi- and univariate functions with bounded generalized derivatives in the Lebesque space. In addition, we formulate exact-order estimates for these approximation characteristics for the isotropic Nikol'skii-Besov classes with bounded differences in a certain Lebesque subspace. In particular, the case of small smoothness is considered of functions from the respective classes.
Copyright (c) 2024 Катерина Пожарська http://creativecommons.org/licenses/by/4.0
Representation and cohomologies of groups and semigroups
This article surveys results related to the cohomology of lattices over group rings, particularly over the group ring of the Klein four--group. It is shown that in this setting, syzygies coincide with the Auslander-Reiten translation, enabling the full computation of Tate cohomology for all lattices and the explicit description of the action of group automorphisms. Special attention is paid to regular lattices and their applications to the classification of multidimensional crystallographic and Chernikov groups. The article also explores the connection between the cohomology and representations of crossed group rings and those of coefficient rings, especially for semidirect product group rings. The approach is illustrated by a new classification of integral representations of the alternating group A4, leading to explicit cohomological invariants. The results are of interdisciplinary interest for group theory, homological algebra, crystallography, and coding theory.
Copyright (c) 2024 Андріяна Плакош http://creativecommons.org/licenses/by/4.0
Representation of some classes of quaternionic hyperholomorphic functions
In the algebra of complex quaternions H(C) we consider the left- and right-ψ-hyperholomorphic functions, and left-Λ — ψ-hyperholomorphic functions. We justify the transition in left- and right-ψ-hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan's basis we find the solution of Cauchy-Fueter equation. By the same method we find representations of left- and right-ψ-hyperholomorphic functions, and representation of left-Λ — ψ-hyperholomorphic functions.
Copyright (c) 2024 Тетяна Кузьменко, Віталій Шпаківський http://creativecommons.org/licenses/by/4.0
On the asymptotic behavior at infinity of ring and lower Q-homeomorphisms with respect to the p-modulus
In the paper the asymptotic behavior at infinity of ring Q-homeomorphisms with respect to the p-modulus as p≥n and lower Q-homeomorphisms with respect to the p-modulus as n−1<p≤n in the space Rn, n≥2, has been investigated.
Copyright (c) 2024 Богдан Кліщук, Руслан Салімов, Марія Стефанчук http://creativecommons.org/licenses/by/4.0
Extremal decomposition of the complex plane with free poles on the unit circle
This paper is an attempt to follow the path taken by the open problem of V.N. Dubinin on extremal decomposition of the complex plane with free poles on the unit circle. Currently, the problem is not completely solved. In this work, its partial solution for $n\geqslant40$ is obtained for the case of simply connected domains.
Copyright (c) 2024 Ірина Денега, Ярослав Заболотний http://creativecommons.org/licenses/by/4.0
Dependence of effective properties on regular perturbations
In this review, we present some of our results on the behavior of the effective properties in the presence of perturbations of the geometric and physical parameters. We will first show that the average longitudinal flow of a Newtonian fluid flowing at low Reynolds numbers around a periodic array of cylinders depends analytically on the perturbations of the periodicity structure and the cross section of the cylinders. Next, we will show the analytical dependence of the effective conductivity of a periodic two-phase composite with ideal and non-ideal contacts at the interface on the perturbations of the shape of the inclusions, the periodicity structure, and the conductivity of each material.
Copyright (c) 2024 Маттео Далла Ріва, Паоло Луцціні, Паоло Мусоліно, Роман Пухтаєвич http://creativecommons.org/licenses/by/4.0
On group classification of differential equations: algebraic approach
Using the classical Lie theorem on realizations of Lie algebras by vector fields on the line, we substantially simplify the proof of the known results on the group classification of the classes of (1+1)-dimensional nonlinear evolution equations ut=H(uxx) and ut+uux=H(uxx).
Copyright (c) 2024 Олександра Локазюк, Софія Гурака http://creativecommons.org/licenses/by/4.0
Kinetic cluster expansions of generating operators of solutions of hierarchies of evolution equations
The article provides an overview of the method of kinetic cluster expansions for generating operators of solutions to the hierarchies of evolutionary equations. The developed method allows for a rigorous justification of the derivation of nonlinear kinetic equations from the microscopic dynamics of a system of many particles.
Copyright (c) 2024 Ігор Гап'як http://creativecommons.org/licenses/by/4.0
Boundary value problems for integrodifferential equations
This paper presents a detailed survey of research on boundary value problems for integro-differential and integro-dynamic systems. The existence of solutions to these problems is examined using the theoretical apparatus of pseudo-inverse matrices and operators, alongside the Vishik-Lyusternik's method for the case of weakly perturbed systems.
Copyright (c) 2024 Іванна Бондар http://creativecommons.org/licenses/by/4.0
Integral theorems in finite-dimensional commutative algebra
For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.
Copyright (c) 2023 Сергій Плакса, Віталій Шпаківський http://creativecommons.org/licenses/by/4.0
Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary
Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliation $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$.
Copyright (c) 2023 Sergiy Maksymenko http://creativecommons.org/licenses/by/4.0
Expanded capabilities of analytical mechanics of a solid medium
Advanced potentials of analytical mechanics of continuum medium and sources promoted the appearance, development and high achievements of this direction of research namely in Ukraine are under discussion. We traced connections of this direction with D. O. Grave and his follower M. O. Kilchevsky. We shortly state and show by examples new elements of this approach and its advantages. It is shown that similar research are developed now in Ukraine.
Copyright (c) 2023 Олег Лімарченко http://creativecommons.org/licenses/by/4.0
The theory of dynamical systems of conflict in the framework of functional analysis
In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hilbert spaces and probabilistic distributions on the territory of life resources. The phenomenon of conflict as a contradiction between opponents appears in mathematical terms as an intersection the domains of definition for operators and overlapping of corresponding measures. Conflict interaction between opponents in the physical sense is described by the specific transformation of states in a Hilbert space. In turn, this is a mapping that changes the spectral measurements. Thus, a complex dynamical system arises, which we call a dynamical system of conflict. Then the following main problems arise as fundamental questions. What reasonable law of engagement (game or war) should be adopted to resolve the initial intersections? What is a fair limiting distribution of the resource territory? In a more general formulation, solving conflict problems means the detailed describing of all possible outcomes on opponents states of the type: victories, defeats, states of equilibrium, compromises as fixed points together with their basins of attraction.
Copyright (c) 2023 Volodymyr Koshmanenko http://creativecommons.org/licenses/by/4.0
Clark-Ocone type formulas on the spaces of regular basic and generalized functions in the analysis of Levy white noise
In the classical Gaussian analysis the Clark-Ocone formula can be written in the form $$ F=\mathbf{E}{F}+\int\mathbf{E}\big(\partial_t F|_{\mathcal F_t}\big)\,dW_t, $$ where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf{E}$ denotes the expectation; $\mathbf{E}\big(\circ|_{\mathcal F_t}\big)$---the conditional expectation with respect to the full $\sigma$-algebra $\mathcal F_t$ that is generated by the Wiener process $W$ up to the point of time $t$; $\partial_{\cdot} F$ is the Hida derivative of $F$; $\int\circ (t)dW_t$ denotes the It\^o stochastic integral with respect to the Wiener process. This formula has many applications, in particular, in the stochastic analysis and in the financial mathematics. In this paper we generalize the Clark-Ocone formula to spaces of regular test and generalized functions of the Levy white noise analysis. More exactly, we obtain different Clark-Ocone type formulas on the above-mentioned spaces, study the properties of the integrands in these formulas, establish the conditions under which a Clark-Ocone type formula takes a classical form, etc. In particular, we show that the restrictive condition of differentiability by Hida for a random variable is not really significant.
Copyright (c) 2023 Микола Качановський http://creativecommons.org/licenses/by/4.0
Advances in theory of evolution equations of many colliding particles
The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations.
Copyright (c) 2023 Віктор Герасименко, Ігор Гап'як http://creativecommons.org/licenses/by/4.0
The origin and development of the ideas of the theory of stochastic differential equations in the Ukrainian school of mathematics.
It is universally recognized that the theory of stochastic differential equations was created in the Japanese mathematical school during the forties of the last century, and it is much less-known that the idea of a stochastic differential equation arose independently in the Ukrainian school of mathematics about the same time. This paper is devoted to a description of the initial stages in forming the school of the theory of stochastic differential equations and its subsequent development in Ukraine.
Copyright (c) 2023 Микола Портенко http://creativecommons.org/licenses/by/4.0
From Grave' seminar to derived categories
This article arose from my lecture at the First Grave Readings, in which I tried to trace the path that began with D. Grave's lectures and seminar at Kyiv University and led to research in the most modern branches of mathematics. Of course, I chose that one a branch of numerous directions developed by Grave's students and their scientific heirs, which is close to the Kyiv School of Theory of Representations and to my own research. The choice of material in the article is also completely subjective, and it does not pretend to be historical review. Rather, it is the memories of a participant in the events.
Copyright (c) 2023 Юрій Дрозд http://creativecommons.org/licenses/by/4.0
CLX anniversary of Academician D. O. Grave
On the occasion of the 160th anniversary of the birth of academician D. O. Grave, this foreword contains some biographical facts from the life and work of the outstanding scientist. In general, this volume of the Proceedings of the Institute of Mathematics, a periodical founded in 1938 by academician D. O. Grave, contains lectures of Grave's readings and works on topical directions of development of modern mathematics in Ukraine.
Copyright (c) 2023 Віктор Герасименко, Сергій Максименко http://creativecommons.org/licenses/by/4.0
Канторвали як множини неелементарних ланцюгових дробів з обмеженим алфавітом
Let $G_{\mathcal{A}}$ be a set of values of continued fractions whose elements belong to a bounded set $\mathcal{A}$ of positive real numbers. We prove that $G_{\mathcal{A}}$ is a continuum bounded and perfect set. For $\mathcal{A}_3=\{0{,}5; 1; 8\}$, the set $G_{\mathcal{A}}$ is a Cantorval, namely, it is homeomorphic to the set $$E= \{x: x=\sum\limits_{k=1}^{\infty}(\frac{3\alpha_{2k-1}}{4^k}+\frac{2\alpha_{2k}}{4^k}),\alpha_k\in\{0,1\}\},$$ where $E$ contains a finite set of intervals whose complements are continuum nowhere dense sets.
Copyright (c) 2019 2019 http://creativecommons.org/licenses/by/4.0
Про одну сингулярну функцію канторівського типу, пов'язану з рядами Енгеля
In the paper, we construct and study the class of continuous on $[0, 1]$ functions with continuum set of peculiarities. Any function $f$ belonging to this class is defined by the Engel series representation of its argument and by special convergent series. Conditions for $f$ to be a~Cantor-type singular function are found.
Copyright (c) 2019 О. М. Барановський, М. В. Працьовитий http://creativecommons.org/licenses/by/4.0
Метрична та ймовірнісна теорії $G_2$-зображення чисел
For numbers of interval $[0,g_0]$, $g_0<1$, we consider a system of encoding with two bases having different signs $g_0$ and $g_1\equiv g_0-1$ by the means of alphabet $A=\{0,1\}$: \begin{equation}\label{ex:ab:1} x=\alpha_1 g_{1-\alpha_1}+\sum\limits^\infty_{k=2}(\alpha_k g_{1-\alpha_k}\prod\limits^{k-1}_{j=1}g_{\alpha_j})\equiv \Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, \end{equation} where $\alpha_n\in A$. We develop a probabilistic theory for this representation, i.e., we consider distributions of digits of random variable $X$ with a given distribution as well as distribution of random variable $\xi$ determined by distributions of digits of its $G_2$-representation (\ref{ex:ab:1}) if the digits are independent. Lebesgue structure of the probability distribution is described. Differential properties of the probability distribution function as well as properties of its spectrum and support are also studied.
Copyright (c) 2023 М. В. Працьовитий, І. М. Лисенко, Ю. П. Маслова http://creativecommons.org/licenses/by/4.0
Про розподіли значень функцій типу Серпінського
We construct an infinite-parameter family of continuous nowhere monotonic and, in general, non-differentiable functions which are generalizations of classical nowhere differentiable Sierpi\'nski function. We study topological and metric properties of the distribution of values of functions belonging to constructed family.
Copyright (c) 2023 Н. А. Василенко http://creativecommons.org/licenses/by/4.0
Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property
Investigated in this paper is a class $\mathfrak{M}$ of matrices (over an arbitrary field) in which all minors of some fixed order $k$ are equal and nonzero. It is established that the rank of such matrices equals to $k$. The possible values for the dimension of a matrix in $\mathfrak{M}$ are found. A necessary and sufficient condition for a matrix to belong to the class $\mathfrak{M}$ is also given.
Copyright (c) 2019 D. Ya. Trebenko, O. O. Trebenko http://creativecommons.org/licenses/by/4.0
Ніде не монотонні функції, пов'язані з зображеннями чисел рядами Кантора
In the paper, we study properties of two continuous nowhere monotonic functions such that their argument is represented in Cantor numeral system with a sequence of natural bases $(s_k)$, where $s_k=2k+1$, and value of the function is determined by a chain dependence of digits of $Q_s$-representation of a number on digits of representation of the argument.
Про розподіл значень однієї сингулярної функції, пов'язаної з зображенням чисел рядами Люрота
In the article we study the distribution of the random variable $Y = F (X)$, where $X$ --- is evenly distributed on $[0; 1]$ random variable, $F$ --- distribution function, the digits of which are positive by the Lurot's rows are independent and equally distributed random variables. The problem of the Lebesgue structure of the distribution $Y$ is solved, the criterion of discreteness (and continuity) of the distribution $Y$ is proved, the conditions under which the distribution is not pure are found.
Канторівська двійково-фібоначчієва система числення у задачах теорії функцій
In the article we study a class of continuous functions with locally complicated structure defined in terms of the representation of numbers in the Cantor number system: \[[0;1]\ni x=\frac{\alpha_1}{s_1}+\frac{\alpha_2}{s_1s_2}+\cdots+\frac{\alpha_n}{s_1s_2 \ldots s_n}+\cdots \equiv\Delta_{\alpha_1\alpha_2 \ldots \alpha_n \ldots},\] where $s_n=2^{\varphi_n}$, $(\varphi_n)$ is a classical Fibonacci sequence: $\varphi_1=1=\varphi_2$, \mbox{$\varphi_{n+2}=\varphi_n+\varphi_{n+1}$,} $\alpha_n\in \{0, 1, \ldots, s_n-1\}$. Singular, nowhere monotonic and nondifferentiable functions, functions with bounded and unbounded variation are among the studied functions.
Copyright (c) 2019 О. І. Бондаренко, Н. М. Василенко, М. В. Працьовитий http://creativecommons.org/licenses/by/4.0