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>In January 2022, the journal was accepted and now is in the process of being added to Scopus.</p><p><em>Attention:</em> The template of the submission (.tex and .cls files) was updated on 20.03.2022. 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>en-USAuthors 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>vestmath@mmf.dnu.edu.ua (Editorial Mediator)dnuvestmath@gmail.com (Honcharov S.V.)Thu, 30 Dec 2021 00:00:00 +0000OJS 2.4.8.0http://blogs.law.harvard.edu/tech/rss60Preamble
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/375
Editors Editors
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/375Thu, 30 Dec 2021 00:00:00 +0000On the Construction of Chaotic Dynamical Systems on the Box Fractal
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/376
In this paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($$$B$$$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $$$B$$$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.N. Aslan, M. Saltan
Copyright (c) 2021 N. Aslan, M. Saltan
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/376Thu, 30 Dec 2021 00:00:00 +0000On asymptotically optimal cubatures for multidimensional Sobolev spaces
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/377
We find an asymptotically optimal method of recovery of the weighted integral for the classes of multivariate functions that are defined via restrictions on their (distributional) gradient.V.F. Babenko, Yu.V. Babenko, O.V. Kovalenko
Copyright (c) 2021 V.F. Babenko, Yu.V. Babenko, O.V. Kovalenko
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/377Thu, 30 Dec 2021 00:00:00 +0000Permutation codes over Sylow 2-subgroups $$$Syl_2(S_{2^n})$$$ of symmetric groups $$$S_{2^n}$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/378
The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances.<br />In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $$$Syl_2(S_{2^n})$$$ of symmetric group $$$S_{2^n}$$$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $$$O(2^n)$$$. <br />The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $$$Syl_2(S_{2^n})$$$ of symmetric group $$$S_{2^n}$$$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.V.A. Olshevska
Copyright (c) 2021 V.A. Olshevska
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/378Thu, 30 Dec 2021 00:00:00 +0000Methods of group theory in Leibniz algebras: some compelling results
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The theory of Leibniz algebras has been developing quite intensively. Most of the results on the structural features of Leibniz algebras were obtained for finite-dimensional algebras and many of them over fields of characteristic zero. A number of these results are analogues of the corresponding theorems from the theory of Lie algebras. The specifics of Leibniz algebras, the features that distinguish them from Lie algebras, can be seen from the description of Leibniz algebras of small dimensions. However, this description concerns algebras over fields of characteristic zero. Some reminiscences of the theory of groups are immediately striking, precisely with its period when the theory of finite groups was already quite developed, and the theory of infinite groups only arose, i.e., with the time when the formation of the general theory of groups took place. Therefore, the idea of using this experience naturally arises. It is clear that we cannot talk about some kind of similarity of results; we can talk about approaches and problems, about application of group theory philosophy. Moreover, every theory has several natural problems that arise in the process of its development, and these problems quite often have analogues in other disciplines. In the current survey, we want to focus on such issues: our goal is to observe which parts of the picture involving a general structure of Leibniz algebras have already been drawn, and which parts of this picture should be developed further.I.Ya. Subbotin
Copyright (c) 2021 I.Ya. Subbotin
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/379Thu, 30 Dec 2021 00:00:00 +0000Criterion of the best non-symmetric approximant for multivariable functions in space $$$L_{1, p_2,...,p_n}$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/380
The criterion of the best non-symmetric approximant for $$$n$$$-variable functions in the space $$$L_{1, p_2,...,p_n}$$$ $$$(1<p_i<+\infty , i=2,3,...,n)$$$ with $$$(\alpha ,\beta )$$$-norm<br />$$\|f\|_{1,p_2,...,p_n;\alpha,\beta}=\left[\int\limits_{a_n}^{b_n}\cdots\left[\int\limits_{a_2}^{b_2}\left[\int\limits_{a_1}^{b_1} |f(x)|_{\alpha,\beta} dx_1\right]^{p_2} dx_2\right]^{\frac{p_3}{p_2}}\cdots dx_n\right]^{\frac{1}{p_n}},$$<br />where $$$0<\alpha,\beta<\infty$$$, $$$\ f_{+}(x)=\max\{f(x),0\},\ f_{-}(x)=\max\{-f(x),0\},$$$ $$$\mathrm{sgn}_{\alpha,\beta}f(x)=\alpha\cdot\mathrm{sgn}f_{+}(x)-\beta\cdot\mathrm{sgn}f_{-}(x),$$$ $$$|f|_{\alpha,\beta}=\alpha \cdot f_{+}+\beta \cdot f_{-} =f(x)\cdot \mathrm{sgn}_{\alpha,\beta}f(x)$$$, is obtained in the article.<br />It is proved that if $$$P_m=\sum\limits_{k=1}^{m}c_k\varphi_k$$$, where $$$\{\varphi_k\}_{k=1}^m$$$ is a linearly independent system functions of $$$L_{1,p_2,...,p_n}$$$, $$$c_k$$$ are real numbers, then the polynomial $$$P_m^{\ast}$$$ is the best $$$(\alpha ,\beta )$$$-approximant for $$$f$$$ in the space $$$L_{1,p_2,...,p_n}$$$ $$$(1<p_i<\infty $$$, $$$i=2,3,...,n)$$$, if and only if, for any polynomial $$$P_m$$$<br />$$\int \limits_K P_m\cdot F_0^{\ast}dx \leq <br />\int \limits_{a_n}^{b_n}...\int \limits_{a_2}^{b_2}\int \limits_{e_{x_2,...,x_n}}|P_m|_{\beta , \alpha}dx_1 \cdot<br /> \operatorname *{ess \,sup}_ {x_1 \in [a_1,b_1]} |F_0^{\ast}|_{\frac{1}{\alpha },\frac{1}{\beta }} dx_2...dx_n,$$<br />where $$$K=[a_1,b_1]\times \ldots\times [a_n,b_n],$$$ $$$e_{x_2,...,x_n}=\{ x_1\in [a_1,b_1] : f-P_m^{\ast}=0\},$$$<br />$$F_0^{\ast}=\frac{|R_m^{\ast}|_{1; \alpha ,\beta }^{p_2-1}|R_m^{\ast}|_{1,p_2; \alpha ,\beta }^{p_3-p_2}\cdot ... \cdot |R_m^{\ast}|_{1,p_2,...,p_{n-1}; \alpha ,\beta }^{p_n-p_{n-1}}\mathrm{sgn}_{\alpha ,\beta} R_m^{\ast}}{||R_m^{\ast}||_{1,p_2,...,p_n; \alpha ,\beta}^{p_n-1}},$$<br />$$|f|_{p_k,\ldots,p_i;\alpha,\beta}=\left[\int\limits_{a_i}^{b_i}\ldots\left[ \int\limits_{a_{k+1}}^{b_{k+1}}\left[<br />\int\limits_{a_k}^{b_k}|f|_{\alpha,\beta}^{p_k}dx_k\right]^{\frac{p_{k+1}}{p_k}}dx_{k+1} \right]^{\frac{p_{k+2}}{p_{k+1}}}\ldots dx_i \right]^{\frac{1}{p_i}},$$<br />($$$1\leq k<i\leq n$$$), $$$R_m^{\ast}=f-P_m^{\ast}$$$.<br />This criterion is a generalization of the known Smirnov's criterion for functions of two variables, when $$$\alpha =\beta =1$$$.M.Ye. Tkachenko, V.M. Traktynska
Copyright (c) 2021 M.Ye. Tkachenko, V.M. Traktynska
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https://vestnmath.dnu.dp.ua/index.php/rim/article/view/380Thu, 30 Dec 2021 00:00:00 +0000