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 $W_n(K)$.
{"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":null,"pages":null},"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}
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 $xin L^r_{infty}(I_{2pi})$.
{"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":null,"pages":null},"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}
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.
我们证明了左括号在$ * $-中心级数上幂零的一个判据，并讨论了Noetherian括号，得到了它们的一些基本性质。我们也证明了如果一个有限生成的括号$ a $是smoktunowicz -幂零的，那么$ a $的加性和乘性群同样是有限生成的。
{"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":null,"pages":null},"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}
We describe the algebra of derivations of some nilpotent Leibniz algebra, having dimensionality 3.
我们描述了维数为3的幂零莱布尼兹代数的导数代数。
{"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":null,"pages":null},"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}
We give specific examples of the spectral decomposition of self-adjoint operators in application to establish sharp inequalities for their powers.
我们给出了自伴随算子谱分解的具体例子，用于建立其幂的尖锐不等式。
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
V. Babenko, V. Babenko, O. Kovalenko, N. Parfinovych
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 $W^{1,infty}$.
{"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":null,"pages":null},"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}
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.
本文的主题是研究$(2n+1)$维cw -配合物$mathbb{C}Omega_n$的同调群。空间$mathbb{C}Omega_n$由复值函数组成，是近似理论中广为人知的空间$Omega_n$的类比。这些空间$mathbb{C}Omega_n$是A.M.在2015年推出的Pasko建立了空间的cw结构$mathbb{C}Omega_n$并使用这个cw结构建立了空间$mathbb{C}Omega_n$是单连通的。注意，上述空间的cw结构$mathbb{C}Omega_n$是鲁班构建的空间$Omega_n$的cw结构的类似物。上午更远。Pasko在维度$0, 1, ldots, n, 2n-1, 2n, 2n+1$中发现了空间$mathbb{C}Omega_n$的同调群。本文的目标是找到同源群$H_{n+1}left ( mathbb{C}Omega_n right )$。证明了$n$为奇数时为$H_{n+1} left ( mathbb{C}Omega_n right )=mathbb{Z}^frac{n+1}{2}$, $n$为偶数时为$H_{n+1} left ( mathbb{C}Omega_n right )=mathbb{Z}^frac{n+2}{2}$。
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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