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Researches in Mathematics最新文献

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On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$ 论列代数 $W_n(K)$ 的一些可解和局部零能子布拉的最大性
Researches in Mathematics Pub Date : 2023-10-08 DOI: 10.15421/242312
D. Efimov, M. Sydorov, K. Sysak
{"title":"On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$","authors":"D. Efimov, M. Sydorov, K. Sysak","doi":"10.15421/242312","DOIUrl":"https://doi.org/10.15421/242312","url":null,"abstract":"Let $K$ be an algebraically closed field of characteristic zero,  $P_n=K[x_1,ldots ,x_n]$  the polynomial ring, and  $W_n(K)$  the Lie algebra of all $K$-derivations on $P_n$.   One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0partial_1+cdots+P_{n-1}partial_n$, where $partial_i:=partial/partial x_i$ are partial derivatives on $P_n$ and $P_0=K.$ This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations  define automorphisms of the ring $P_n$ and were studied by many authors. The  subalgebra $u_n(K) $ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)partial_1+cdots +(P_{n-1}+x_nP_{n-1})partial_n,$ which  is solvable of the derived length $ 2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$  is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K)$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139321820","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}
引用次数: 0
A sharp Remez type inequalities for the functions with asymmetric restrictions on the oldest derivative 最老导数上具有非对称限制的函数的尖锐Remez型不等式
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242304
V. Kofanov, A. V. Zhuravel
{"title":"A sharp Remez type inequalities for the functions with asymmetric restrictions on the oldest derivative","authors":"V. Kofanov, A. V. Zhuravel","doi":"10.15421/242304","DOIUrl":"https://doi.org/10.15421/242304","url":null,"abstract":"For odd $rin mathbb{N}$; $alpha, beta >0$; $pin [1, infty]$; $delta in (0, 2 pi)$, any $2pi$-periodic function $xin L^r_{infty}(I_{2pi})$, $I_{2pi}:=[0, 2pi]$, and arbitrary measurable set $B subset I_{2pi},$ $mu B leqslant delta/lambda,$ where $lambda=$ $left({left|varphi_{r}^{alpha, beta}right|_{infty} left| {alpha^{-1}}{x_+^{(r)}} + {beta^{-1}}{x_-^{(r)}}right|_infty}{E^{-1}_0(x)_infty}right)^{1/r}$, we obtain sharp Remez type inequality $$E_0(x)_infty leqslant frac{|varphi_r^{alpha, beta}|_infty}{E_0(varphi_r^{alpha, beta})^{gamma}_{L_p(I_{2pi} setminus B_delta)}} left|x right|^{gamma}_{{L_p} left(I_{2pi} setminus B right)}left| {alpha^{-1}}{x_+^{(r)}} + {beta^{-1}}{x_-^{(r)}}right|_infty^{1-gamma},$$ where $gamma=frac{r}{r+1/p},$ $varphi_r^{alpha, beta}$ is non-symmetric ideal Euler spline of order $r$, $B_delta:= left[M- delta_2, M+ delta_1 right]$, $M$ is the point of local maximum of spline $varphi_r^{alpha, beta}$ and $delta_1 > 0$, $delta_2 > 0$ are such that $varphi_r^{alpha, beta}(M+ delta_1) = varphi_r^{alpha, beta}(M- delta_2), ;; delta_1 + delta_2 = delta .$In particular, we prove the sharp inequality of Hörmander-Remez type for the norms of intermediate derivatives of the functions $xin L^r_{infty}(I_{2pi})$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89134766","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}
引用次数: 0
On the structure of some nilpotent braces 关于一些幂零大括号的结构
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242303
M. Dixon, L. A. Kurdachenko
{"title":"On the structure of some nilpotent braces","authors":"M. Dixon, L. A. Kurdachenko","doi":"10.15421/242303","DOIUrl":"https://doi.org/10.15421/242303","url":null,"abstract":"We prove a criteria for nilpotency of left braces in terms of the $star$-central series and also discuss Noetherian braces, obtaining some of their elementary properties. We also show that if a finitely generated brace $A$ is Smoktunowicz-nilpotent, then the additive and multiplicative groups of $A$ are likewise finitely generated.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83754677","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}
引用次数: 0
On the algebra of derivations of some nilpotent Leibniz algebras 一些幂零莱布尼兹代数的导数代数
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242306
L. A. Kurdachenko, M. Semko, V. Yashchuk
{"title":"On the algebra of derivations of some nilpotent Leibniz algebras","authors":"L. A. Kurdachenko, M. Semko, V. Yashchuk","doi":"10.15421/242306","DOIUrl":"https://doi.org/10.15421/242306","url":null,"abstract":"We describe the algebra of derivations of some nilpotent Leibniz algebra, having dimensionality 3.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78008615","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}
引用次数: 1
Application of spectral decomposition to establish inequalities for operators 应用谱分解建立算子不等式
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242302
R. Bilichenko, S. Zhir
{"title":"Application of spectral decomposition to establish inequalities for operators","authors":"R. Bilichenko, S. Zhir","doi":"10.15421/242302","DOIUrl":"https://doi.org/10.15421/242302","url":null,"abstract":"We give specific examples of the spectral decomposition of self-adjoint operators in application to establish sharp inequalities for their powers.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78867461","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}
引用次数: 0
On the structure of groups admitting faithful modules with certain conditions of primitivity 论具有一定原性条件的群的结构
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242307
A. Tushev
{"title":"On the structure of groups admitting faithful modules with certain conditions of primitivity","authors":"A. Tushev","doi":"10.15421/242307","DOIUrl":"https://doi.org/10.15421/242307","url":null,"abstract":"In the paper we study structure of soluble-by-finite groups of finite torsion-free rank which admit faithful modules with conditions of primitivity. In particular, we prove that under some additional conditions if an infinite finitely generated linear group $G$ of finite rank admits a fully primitive fully faithful module then $G$ has infinite $FC$-centre.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73883493","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}
引用次数: 0
Description of the automorphism groups of some Leibniz algebras 一些莱布尼兹代数的自同构群的描述
Researches in Mathematics Pub Date : 2023-06-19 DOI: 10.15421/242305
L. A. Kurdachenko, O. Pypka, M. Semko
{"title":"Description of the automorphism groups of some Leibniz algebras","authors":"L. A. Kurdachenko, O. Pypka, M. Semko","doi":"10.15421/242305","DOIUrl":"https://doi.org/10.15421/242305","url":null,"abstract":"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 elements $a,b,cin L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,bin L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the 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 nilpotent three-dimensional Leibniz algebras.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"225 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86686330","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}
引用次数: 1
On Landau-Kolmogorov type inequalities for charges and their applications 关于电荷的Landau-Kolmogorov型不等式及其应用
Researches in Mathematics Pub Date : 2023-04-20 DOI: 10.15421/242301
V. Babenko, V. Babenko, O. Kovalenko, N. Parfinovych
{"title":"On Landau-Kolmogorov type inequalities for charges and their applications","authors":"V. Babenko, V. Babenko, O. Kovalenko, N. Parfinovych","doi":"10.15421/242301","DOIUrl":"https://doi.org/10.15421/242301","url":null,"abstract":"In this article we prove sharp Landau-Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $mathbb{R}^d$, $dgeqslant 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon-Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $frac{partial ^d f}{partial x_1ldotspartial x_d}$ belongs to the Sobolev space $W^{1,infty}$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77000788","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}
引用次数: 1
The homology groups $H_{n+1} left( mathbb{C}Omega_n right)$ 同源基 $H_{n+1} left( mathbb{C}Omega_n right)$
Researches in Mathematics Pub Date : 2022-12-31 DOI: 10.15421/242210
A. Paśko
{"title":"The homology groups $H_{n+1} left( mathbb{C}Omega_n right)$","authors":"A. Paśko","doi":"10.15421/242210","DOIUrl":"https://doi.org/10.15421/242210","url":null,"abstract":"The topic of the paper is the investigation of the homology groups of the $(2n+1)$-dimensional CW-complex $mathbb{C}Omega_n$. The spaces $mathbb{C}Omega_n$ consist of complex-valued functions and are the analogue of the spaces  $Omega_n$, widely known in the approximation theory. The spaces $mathbb{C}Omega_n$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $mathbb{C}Omega_n$ and using this CW-structure established that the spaces $mathbb{C}Omega_n$ are simply connected. Note that the mentioned CW-structure of the spaces $mathbb{C}Omega_n$ is the analogue of the CW-structure of the spaces $Omega_n$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $mathbb{C}Omega_n$ in the dimensionalities $0, 1, ldots, n, 2n-1, 2n, 2n+1$.  The goal of the present paper is to find the homology group $H_{n+1}left ( mathbb{C}Omega_n right )$. It is proved that $H_{n+1} left ( mathbb{C}Omega_n right )=mathbb{Z}^frac{n+1}{2}$ if $n$ is odd and $H_{n+1} left ( mathbb{C}Omega_n right )=mathbb{Z}^frac{n+2}{2}$ if $n$ is even.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"2012 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86375059","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}
引用次数: 0
A Lambda Analogue of the Gamma Function and its Properties 函数的类似函数及其性质
Researches in Mathematics Pub Date : 2022-12-31 DOI: 10.15421/242209
K. Nantomah, I. Ege
{"title":"A Lambda Analogue of the Gamma Function and its Properties","authors":"K. Nantomah, I. Ege","doi":"10.15421/242209","DOIUrl":"https://doi.org/10.15421/242209","url":null,"abstract":"We consider a generalization of the gamma function which we term as lambda analogue of the gamma function or $lambda$-gamma function and further, we establish some of its accompanying properties. For the particular case when $lambda=1$, the results established reduce to results involving the classical gamma function. The techniques employed in proving our results are analytical in nature.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83812019","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}
引用次数: 1
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