Researches in Mathematics
https://vestnmath.dnu.dp.ua/index.php/rim
<p><strong><em>Announcement, April 2024:</em> The editorial board of the scientific journal Researches in Mathematics announces that in 2024, on the occasion of the 75th anniversary of outstanding mathematicians, members of the editorial board of this journal, <a href="https://www.scopus.com/authid/detail.uri?authorId=8906550500" target="_blank">Prof. V.F. Babenko</a> and <a href="https://www.scopus.com/authid/detail.uri?authorId=7004381008" target="_blank">Prof. V.O. Kofanov</a>, an additional special issue of the journal Researches in Mathematics on Approximation Theory is planned (Vol. 32, No. 2, 2024).</strong></p><table><tbody><tr><td><img src="/files/pics/gen_issue_cover.png" alt="" width="100px" height="150px" /></td><td style="padding-left: 16px; vertical-align: top;"><p>The journal publishes, semiannually, research articles in basic areas of theoretical and applied mathematics, including, but not limited to: analysis, algebra, geometry, topology, probability theory and mathematical statistics, differential equations and mathematical physics.</p><p>The journal was published annually from 1963 to 1997 under the title “Researches on modern problems of summation and approximation of functions and their applications” and from 1998 to 2018 under the title “Dnipro University Mathematics Bulletin”.</p><p>Since 2022, the journal has been included to Scopus.</p><p><em>Attention:</em> The template of the submission (.tex and .cls files) was updated on 07.07.2024. Please proceed to <a href="/index.php/rim/about/submissions#authorGuidelines">Author Guidelines</a> to download and use the latest version.</p></td></tr></tbody></table>Oles Honchar Dnipro National Universityen-USResearches in Mathematics2664-4991Authors who publish with this journal agree to the following terms:<ol><li>Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a <a href="http://creativecommons.org/licenses/by/3.0/" target="_new">Creative Commons Attribution License</a> that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.</li><li>Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.</li><li>Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See <a href="http://opcit.eprints.org/oacitation-biblio.html" target="_new">The Effect of Open Access</a>).</li></ol>Preamble
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/412
Editors Editors
Copyright (c) 2024 Editors Editors
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2024-07-082024-07-0832112Virtual endomorphisms of the group $$$pg$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/413
A virtual endomorphism of a group $$$G$$$ is a homomorphism of the form $$$\phi:H\rightarrow G$$$, where $$$H<G$$$ is a subgroup of finite index. A virtual endomorphism $$$\phi:H\rightarrow G$$$ is called simple if there are no nontrivial normal $$$\phi$$$-invariant subgroups, that is, the $$$\phi$$$-core is trivial. We describe all virtual endomorphisms of the plane group $$$pg$$$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $$$pg$$$ admits a transitive self-similar (as well as finite-state) action of degree $$$d$$$ if and only if $$$d\geq 2$$$ is not an odd prime, and admits a self-replicating action of degree $$$d$$$ if and only if $$$d\geq 6$$$ is not a prime or a power of $$$2$$$.I. BondarenkoD. Zashkolny
Copyright (c) 2024 I. Bondarenko, D. Zashkolny
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2024-07-082024-07-0832131510.15421/242401A study on unification of generalized hypergeometric function and Mittag-Leffler function with certain integral transforms of generalized basic hypergeometric function
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/414
This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between $$${}_{2}{{R}_{1}}^{\upsilon }(\mathfrak{z})$$$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by $$${}_{2}{{R}_{1}}^{\upsilon ,q}(\mathfrak{z})$$$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.K.K. ChaudharyS.B. Rao
Copyright (c) 2024 K.K. Chaudhary, S.B. Rao
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2024-07-082024-07-08321163210.15421/242402Projective tensor products of approximation spaces associated with positive operators
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/415
In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $$$K$$$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.M.I. DmytryshynL.I. Dmytryshyn
Copyright (c) 2024 M.I. Dmytryshyn, L.I. Dmytryshyn
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2024-07-082024-07-08321334410.15421/242403Some results on ultrametric 2-normed spaces
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/416
In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $$$b$$$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $$$b$$$-linear functionals in ultrametric 2-normed spaces are given.J. Ettayb
Copyright (c) 2024 J. Ettayb
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2024-07-082024-07-08321455910.15421/242404On the analytic extension of three ratios of Horn's confluent hypergeometric function $$$\mathrm{H}_7$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/417
In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $$$\mathrm{H}_7$$$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.V. HladunR. RusynM. Dmytryshyn
Copyright (c) 2024 V. Hladun, R. Rusyn, M. Dmytryshyn
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2024-07-082024-07-08321607010.15421/242405The norming set of a bilinear form on a certain normed space $$$\mathbb{R}^2$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/418
In this paper we classify the norming set of a bilinear form on the plane with a certain norm whose unit ball has only four extreme points. We obtain the results of [6, 8] as corollary.S.G. Kim
Copyright (c) 2024 S.G. Kim
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2024-07-082024-07-08321718210.15421/242406A Note on Some Properties of Unbounded Bilinear Forms Associated with Skew-Symmetric $$$L^q(\Omega)$$$-Matrices
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/419
We study the bilinear forms on the space of measurable $$$p$$$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $$$L^q$$$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $$$p$$$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.P.I. Kogut
Copyright (c) 2024 P.I. Kogut
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2024-07-082024-07-08321839210.15421/242407Action of derivations on polynomials and on Jacobian derivations
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/420
Let $$$\mathbb K$$$ be a field of characteristic zero, $$$A := \mathbb K[x_{1}, x_{2}]$$$ the polynomial ring and $$$W_2(\mathbb K)$$$ the Lie algebra of all $$$\mathbb K$$$-derivations on $$$A$$$. Every polynomial $$$f \in A$$$ defines a Jacobian derivation $$$D_f\in W_2(\mathbb K)$$$ by the rule $$$D_f(h)=\det J(f, h)$$$ for any $$$h\in A$$$, where $$$J(f, h)$$$ is the Jacobi matrix for $$$f, h$$$. The Lie algebra $$$W_2(\mathbb K)$$$ acts naturally on $$$A$$$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $$$W_2(\mathbb K)$$$. It is proved that for a Jordan chain $$$T(f_1)=\lambda f_1+f_2$$$, ..., $$$T(f_{k-1})=\lambda f_{k-1}+f_k$$$, $$$T(f_k)=\lambda f_k$$$ for a derivation $$$T\in W_2(\mathbb K)$$$ on $$$A$$$ there exists an analogous chain $$$[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$$$, ..., $$$[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$$$ in $$$W_2(\mathbb K)$$$. In case $$$A:=\mathbb K[x_1, \ldots , x_n]$$$, the action of normalizers of elements $$$f$$$ from $$$A$$$ in $$$W_n(\mathbb K)$$$ on the principal ideals $$$(f)$$$ is considered.O.Ya. KozachokA.P. Petravchuk
Copyright (c) 2024 O.Ya. Kozachok, A.P. Petravchuk
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2024-07-082024-07-083219310010.15421/242408Automorphism groups of some non-nilpotent Leibniz algebras
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/421
Let $$$L$$$ be an algebra over a field $$$F$$$ with the binary operations $$$+$$$ and $$$[,]$$$. Then $$$L$$$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $$$[a,[b,c]]=[[a,b],c]+[b,[a,c]]$$$ for all $$$a,b,c\in L$$$. A linear transformation $$$f$$$ of $$$L$$$ is called an endomorphism of $$$L$$$, if $$$f([a,b])=[f(a),f(b)]$$$ for all elements $$$a,b\in L$$$. A bijective endomorphism of $$$L$$$ is called an automorphism of $$$L$$$. It is easy to show that the set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of non-nilpotent three-dimensional Leibniz algebras.L.A. KurdachenkoP.Ye. MinaievO.O. Pypka
Copyright (c) 2024 L.A. Kurdachenko, P.Ye. Minaiev, O.O. Pypka
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2024-07-082024-07-0832110110910.15421/242409Torsion Groups with the Norm of pd-Subgroup of Finite Index
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/422
The authors study the relations between the properties of torsion groups and their norms of $$$pd$$$-subgroups. The norm $$$N_G^{pdI}$$$ of $$$pd$$$-subgroups of a group $$$G$$$ is the intersection of the normalizers of all its $$$pd$$$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $$$pd$$$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $$$pd$$$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $$$pd$$$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.T.D. LukashovaM.G. DrushlyakA.V. Pidopryhora
Copyright (c) 2024 T.D. Lukashova, M.G. Drushlyak, A.V. Pidopryhora
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2024-07-082024-07-0832111011710.15421/242410On Poisson (2-3)-algebras which are finite-dimensional over the center
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/423
One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $$$G/\zeta(G)$$$ of a group $$$G$$$ is finite, then its derived subgroup $$$[G,G]$$$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $$$n$$$-groups, associative algebras, Lie algebras, Lie $$$n$$$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $$$P$$$ has finite codimension, then $$$P$$$ includes an ideal $$$K$$$ of finite dimension such that $$$P/K$$$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.P.Ye. MinaievO.O. PypkaI.V. Shyshenko
Copyright (c) 2024 P.Ye. Minaiev, O.O. Pypka, I.V. Shyshenko
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2024-07-082024-07-0832111813310.15421/242411Simple proofs of certain inequalities with logarithmic coefficients of univalent functions
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/424
In this paper, we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent functions.M. ObradovićN. Tuneski
Copyright (c) 2024 M. Obradović, N. Tuneski
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2024-07-082024-07-0832113413810.15421/242412Construction of a non-linear analytical model for the rotation parts building up process using regression analysis
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/425
We show how with the help of nonlinear paired regression, namely cubic regression, having experimental data, it is possible to investigate the relationship of tangential stresses between different layers of a cylindrical part during build-up. Six pairs of dependencies (models) between tangential stresses are considered. For each pair of dependencies, the accuracy of the built model was assessed using the average error of approximation and Fisher's F-criterion, and a comparative analysis was conducted. Despite some contradictions that arose during the evaluation of the models according to various regression parameters it was established that at least four of the six models are optimal and allow to adequately model the process of the formation of the stress-strain state in the details and elements of structures with significantly lower costs even before the stage of manufacturing finished products.A.V. SiasievR.O. Bilichenko
Copyright (c) 2024 A.V. Siasiev, R.O. Bilichenko
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2024-07-082024-07-0832113915010.15421/242413