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Researches in Mathematics
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Open Access journal
ISSN (Print) 2664-4991 - ISSN (Online) 2664-5009
Published by Oles Honchar Dnipropetrovsk National University

[7 journals]

Preamble
- Authors: Editors Editors
  PubDate: Thu, 30 Dec 2021 00:00:00 +000
On the Construction of Chaotic Dynamical Systems on the Box Fractal
- Authors: N. Aslan; M. Saltan
  Abstract: 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.
  PubDate: Thu, 30 Dec 2021 00:00:00 +000
On asymptotically optimal cubatures for multidimensional Sobolev spaces
- Authors: V.F. Babenko; Yu.V. Babenko, O.V. Kovalenko
  Abstract: 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.
  PubDate: Thu, 30 Dec 2021 00:00:00 +000
Permutation codes over Sylow 2-subgroups $$$Syl_2(S_{2^n})$$$ of symmetric
groups $$$S_{2^n}$$$
- Authors: V.A. Olshevska
  Abstract: 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.
  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 $$$O(2^n)$$$.
  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.
  PubDate: Thu, 30 Dec 2021 00:00:00 +000
Methods of group theory in Leibniz algebras: some compelling results
- Authors: I.Ya. Subbotin
  Abstract: 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.
  PubDate: Thu, 30 Dec 2021 00:00:00 +000
Criterion of the best non-symmetric approximant for multivariable
functions in space $$$L_{1, p_2,...,p_n}$$$
- Authors: M.Ye. Tkachenko; V.M. Traktynska
  Abstract: 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
  $$\ f\ _{1,p_2,...,p_n;\alpha,\beta}=\left[\int\limits_{a_n}^{b_n}\cdots\left[\int\limits_{a_2}^{b_2}\left[\int\limits_{a_1}^{b_1} f(x) _{\alpha,\beta} dx_1\right]^{p_2} dx_2\right]^{\frac{p_3}{p_2}}\cdots dx_n\right]^{\frac{1}{p_n}},$$
  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.
  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 $$$P_m$$$
  $$\int \limits_K P_m\cdot F_0^{\ast}dx \leq
  \int \limits_{a_n}^{b_n}...\int \limits_{a_2}^{b_2}\int \limits_{e_{x_2,...,x_n}} P_m _{\beta , \alpha}dx_1 \cdot
  \operatorname *{ess \,sup}_ {x_1 \in [a_1,b_1]} F_0^{\ast} _{\frac{1}{\alpha },\frac{1}{\beta }} dx_2...dx_n,$$
  where $$$K=[a_1,b_1]\times \ldots\times [a_n,b_n],$$$ $$$e_{x_2,...,x_n}=\{ x_1\in [a_1,b_1] : f-P_m^{\ast}=0\},$$$
  $$F_0^{\ast}=\frac{ R_m^{\ast} _{1; \alpha ,\beta }^{p_2-1} R_m^{\ast} _{1,p_2; \alpha ,\beta }^{p_3-p_2}\cdot ... \cdot R_m^{\ast} _{1,p_2,...,p_{n-1}; \alpha ,\beta }^{p_n-p_{n-1}}\mathrm{sgn}_{\alpha ,\beta} R_m^{\ast}}{ R_m^{\ast} _{1,p_2,...,p_n; \alpha ,\beta}^{p_n-1}},$$
  $$ f _{p_k,\ldots,p_i;\alpha,\beta}=\left[\int\limits_{a_i}^{b_i}\ldots\left[ \int\limits_{a_{k+1}}^{b_{k+1}}\left[
  \int\limits_{a_k}^{b_k} f _{\alpha,\beta}^{p_k}dx_k\right]^{\frac{p_{k+1}}{p_k}}dx_{k+1} \right]^{\frac{p_{k+2}}{p_{k+1}}}\ldots dx_i \right]^{\frac{1}{p_i}},$$
  ($$$1\leq k<i\leq n$$$), $$$R_m^{\ast}=f-P_m^{\ast}$$$.
  This criterion is a generalization of the known Smirnov's criterion for functions of two variables, when $$$\alpha =\beta =1$$$.
  PubDate: Thu, 30 Dec 2021 00:00:00 +000

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