Researches in Mathematics
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<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><em>Attention:</em> The template of the submission (.tex and .cls files) was updated on 05.07.2021. 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/245
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2021-07-052021-07-0529112In memoriam: Lilia Georgiivna Boitsun, a mathematician and bright person
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/246
The article is devoted to the talented mathematician, candidate of physical and mathematical sciences Boitsun Lilia Georgiivna. The article describes her life and career, scientific activity.V.F. BabenkoR.O. BilichenkoM.B. VakarchukO.V. KovalenkoS.V. KonarevaV.O. KofanovT.Yu. LeskevychV.P. MotornyiN.V. ParfinovychA.M. PaskoO.V. PolyakovO.O. RudenkoT.I. RybnikovaD.S. SkorokhodovM.Ye. TkachenkoV.M. Traktynska
Copyright (c) 2021 V.F. Babenko, R.O. Bilichenko, M.B. Vakarchuk, O.V. Kovalenko, S.V. Konareva, V.O. Kofanov, T.Yu. Leskevych, N.V. Parfinovych, A.M. Pasko, O.V. Polyakov, O.O. Rudenko, T.I. Rybnikova, D.S. Skorokhodov, M.Ye. Tkachenko, V.M. Traktynska
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2021-07-052021-07-0529131010.15421/242101Sharp inequalities of various metrics on the classes of functions with given comparison function
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/247
For any $$$q > p > 0$$$, $$$\omega > 0,$$$ $$$d \ge 2 \omega,$$$ we obtain the following sharp inequality of various metrics<br />$$<br />\|x\|_{L_q(I_{d})} \le \frac{\|\varphi +<br />c\|_{L_q(I_{2\omega})}}{\|\varphi + c \|_{L_p(I_{2\omega})}}<br />\|x\|_{L_p(I_{d})}<br />$$<br />on the set $$$S_{\varphi}(\omega)$$$ of $$$d$$$-periodic functions $$$x$$$ having zeros with given the sine-shaped $$$2\omega$$$-periodic comparison function $$$\varphi$$$, where $$$c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$$$ is such that<br />$$<br />\|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi +<br />c)_{\pm}\|_{L_p(I_{2\omega})}\,.<br />$$<br />In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $$$\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$$$.T.V. AlexandrovaV.A. Kofanov
Copyright (c) 2021 T.V. Alexandrova, V.A. Kofanov
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2021-07-052021-07-05291112310.15421/242102On the homology groups $$$H_k(\mathbb{C}\Omega_n)$$$, $$$k=1, ..., n$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/248
In the paper the homology groups of the $$$(2n+1)$$$-dimensional CW-complex $$$\mathbb{C}\Omega_n$$$ are investigated. The spaces $$$\mathbb{C}\Omega_n$$$ consist of complex-valued functions and generalize the widely known in the approximation theory spaces $$$\Omega_n$$$. The research of the homotopy properties of the spaces $$$\Omega_n$$$ has been started by V.I. Ruban who in 1985 found the n-dimensional homology group of the space $$$\Omega_n$$$ and in 1999 found all the cohomology groups of this space. The spaces $$$\mathbb{C}\Omega_n$$$ have been introduced by A.M. Pasko who in 2015 has built the structure of CW-complex on these spaces. This CW-structure is analogue of the CW-structure of the space $$$\Omega_n$$$ introduced by V.I. Ruban. In present paper in order to investigate the homology groups of the spaces $$$\mathbb{C}\Omega_n$$$ we calculate the relative homology groups $$$H_k(\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1})$$$, it turned out that the groups $$$H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )$$$ are trivial if $$$1\leq k < n$$$ and $$$H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}^{C^{k-n}_{n+1}}$$$ if $$$n \leq k \leq 2n+1$$$, in particular $$$H_n \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}$$$. Further we consider the exact homology sequence of the pair $$$\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$$$ and prove that its inclusion operator $$$i_*: H_k(\mathbb{C}\Omega_n) \rightarrow H_k(\mathbb{C}\Omega_{n+1})$$$ is zero. Taking into account that the relative homology groups $$$H_k \left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$$$ are zero if $$$1\leq k \leq n$$$ and the inclusion operator $$$i_*=0$$$ we have derived from the exact homology sequence of the pair $$$\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$$$ that the homology groups $$$H_k \left ( \mathbb{C}\Omega_n \right ), 1\leq k<n,$$$ are trivial. The similar considerations made it possible to calculate the group $$$H_n(\mathbb{C}\Omega_n)$$$. So the homology groups $$$H_k(\mathbb{C}\Omega_n), n \geq 2, k=1,...,n,$$$ have been found.A.M. Pasko
Copyright (c) 2021 A.M. Pasko
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2021-07-052021-07-05291243010.15421/242103The uniqueness of the best non-symmetric $$$L_1$$$-approximant for continuous functions with values in $$$\mathbb{R}^m_p$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/249
The article considers the questions of the uniqueness of the best non-symmetric $$$L_1$$$-approximations of continuous functions with values in $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$ by elements of the two-dimensional subspace $$$H_2= \mathrm{span} \{1, g_{a,b}\}$$$, where <br />$$<br />g_{a,b}(x)=\left\{ \begin{matrix} <br />-b\cdot (x-1)^2, & x\in [0;1), & \\<br />0, & x\in [1;a-1), & (a\geq 2, b>0),\\<br />(x-a+1)^2,& x\in [a-1,a],&<br />\end{matrix} \right.<br />$$<br />It is obtained that when $$$b\in (0;1)\cup (1;+\infty), a\geq 2$$$, the subspace $$$H_2$$$ is a unicity space of the best $$$(\alpha ,\beta )$$$-approximations for continuous on the $$$[0;a]$$$ functions with values in the space $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$. In case $$$b=1$$$, $$$a\geq 4$$$ it is proved that the subspace $$$H_2$$$ is not a unicity subspace of the best non-symmetric approximations for these functions.<br />Received results summarize the previously obtained Strauss results for the real functions in the case $$$\alpha = \beta = 1$$$, as well as the results of Babenko and Glushko for the the best $$$(\alpha ,\beta )$$$-approximation for continuous functions on a segment with values in the space $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$.M.Ye. TkachenkoV.M. Traktynska
Copyright (c) 2021 M.Ye. Tkachenko, V.M. Traktynska
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2021-07-052021-07-05291314210.15421/242104