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.
{"title":"Permutation codes over Sylow 2-subgroups $Syl_2(S_{2^n})$ of symmetric groups $S_{2^n}$","authors":"V. Olshevska","doi":"10.15421/242107","DOIUrl":"https://doi.org/10.15421/242107","url":null,"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.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78118904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Criterion of the best non-symmetric approximant for multivariable functions in space $L_{1, p_2,...,p_n}$","authors":"M. Tkachenko, V. M. Traktynska","doi":"10.15421/242109","DOIUrl":"https://doi.org/10.15421/242109","url":null,"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$$|f|_{1,p_2,...,p_n;alpha,beta}=left[intlimits_{a_n}^{b_n}cdotsleft[intlimits_{a_2}^{b_2}left[intlimits_{a_1}^{b_1} |f(x)|_{alpha,beta} dx_1right]^{p_2} dx_2right]^{frac{p_3}{p_2}}cdots dx_nright]^{frac{1}{p_n}},$$where $0<alpha,beta<infty$, $ f_{+}(x)=max{f(x),0}, f_{-}(x)=max{-f(x),0},$ $mathrm{sgn}_{alpha,beta}f(x)=alphacdotmathrm{sgn}f_{+}(x)-betacdotmathrm{sgn}f_{-}(x),$ $|f|_{alpha,beta}=alpha cdot f_{+}+beta cdot f_{-} =f(x)cdot mathrm{sgn}_{alpha,beta}f(x)$, is obtained in the article.It is proved that if $P_m=sumlimits_{k=1}^{m}c_kvarphi_k$, 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$$$int limits_K P_mcdot F_0^{ast}dx leq int limits_{a_n}^{b_n}...int limits_{a_2}^{b_2}int limits_{e_{x_2,...,x_n}}|P_m|_{beta , alpha}dx_1 cdot operatorname *{ess ,sup}_ {x_1 in [a_1,b_1]} |F_0^{ast}|_{frac{1}{alpha },frac{1}{beta }} dx_2...dx_n,$$where $K=[a_1,b_1]times ldotstimes [a_n,b_n],$ $e_{x_2,...,x_n}={ x_1in [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[intlimits_{a_i}^{b_i}ldotsleft[ intlimits_{a_{k+1}}^{b_{k+1}}left[intlimits_{a_k}^{b_k}|f|_{alpha,beta}^{p_k}dx_kright]^{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}},$$($1leq k<ileq n$), $R_m^{ast}=f-P_m^{ast}$.This criterion is a generalization of the known Smirnov's criterion for functions of two variables, when $alpha =beta =1$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80563269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Methods of group theory in Leibniz algebras: some compelling results","authors":"I. Subbotin","doi":"10.15421/242108","DOIUrl":"https://doi.org/10.15421/242108","url":null,"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.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79174968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
对于一类由分布梯度限制定义的多元函数，我们找到了一种加权积分的渐近最优恢复方法。
{"title":"On asymptotically optimal cubatures for multidimensional Sobolev spaces","authors":"V. Babenko, Y. Babenko, O. Kovalenko","doi":"10.15421/242106","DOIUrl":"https://doi.org/10.15421/242106","url":null,"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.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80144636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We have established the estimation of deviation of continuous $2pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.
{"title":"Estimation of deviation of continuous $2pi$-periodic functions from corresponding trigonometric polynomials of S.N. Bernstein's type","authors":"N.Ya. Yatsenko","doi":"10.15421/247712","DOIUrl":"https://doi.org/10.15421/247712","url":null,"abstract":"We have established the estimation of deviation of continuous $2pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84874470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct the approximate solution and stability map of one set of differential equations with periodic coefficients.
构造了一类周期系数微分方程的近似解和稳定性映射。
{"title":"Construction of approximate solution and stability map of one set of differential equations with periodic coefficients, which can be reduced to canonical form","authors":"V. Shevchenko, V. Semenov","doi":"10.15421/247735","DOIUrl":"https://doi.org/10.15421/247735","url":null,"abstract":"We construct the approximate solution and stability map of one set of differential equations with periodic coefficients.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77120255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose the matrix statement of the method of computation of stress-deformed state of structure that is assembled from two or more rotation hulls that are connected through circular ring. It is assumed that the connection ring is affected by concentrated force factors in radial, circular, and axial directions, and the hulls are loaded with pressure.
{"title":"Statics of conjugate rotation hulls","authors":"V. P. Privarnikova","doi":"10.15421/247733","DOIUrl":"https://doi.org/10.15421/247733","url":null,"abstract":"We propose the matrix statement of the method of computation of stress-deformed state of structure that is assembled from two or more rotation hulls that are connected through circular ring. It is assumed that the connection ring is affected by concentrated force factors in radial, circular, and axial directions, and the hulls are loaded with pressure.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79895014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.
{"title":"On some properties of normally-inflective complexes with simple inflective center in $E_3$","authors":"Ye.N. Ishchenko","doi":"10.15421/247727","DOIUrl":"https://doi.org/10.15421/247727","url":null,"abstract":"In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82519600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we have pointed out the conditions under which the subspaces of dimensionality $n^2$ ($n=2,3,ldots$), extremal for $H_{omega}$ classes of continuous functions of two variables, do not exist.
{"title":"Extremal subspaces in the problem about widths of $H_{omega}$ classes in the space of continuous functions of two variables","authors":"L. B. Khodak","doi":"10.15421/247709","DOIUrl":"https://doi.org/10.15421/247709","url":null,"abstract":"In the paper, we have pointed out the conditions under which the subspaces of dimensionality $n^2$ ($n=2,3,ldots$), extremal for $H_{omega}$ classes of continuous functions of two variables, do not exist.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76561664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we have found the A.N. Kolmogorov's width of the class $W^r L^+_p$ ($r=1,2,ldots$, $1 leqslant p leqslant infty$) of all $2pi$-periodic functions $f(x)$ whose $(r-1)$-th derivative $f^{(r-1)}(x)$ is absolutely continuous and $| f^{(r)}_+ |_p leqslant 1$.
{"title":"On widths of one class of periodic functions","authors":"V. G. Doronin, A. Ligun","doi":"10.15421/247704","DOIUrl":"https://doi.org/10.15421/247704","url":null,"abstract":"In the paper, we have found the A.N. Kolmogorov's width of the class $W^r L^+_p$ ($r=1,2,ldots$, $1 leqslant p leqslant infty$) of all $2pi$-periodic functions $f(x)$ whose $(r-1)$-th derivative $f^{(r-1)}(x)$ is absolutely continuous and $| f^{(r)}_+ |_p leqslant 1$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90325872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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