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Generalized derivations of order $2$ on multilinear polynomials in prime rings 素环中多元线性多项式上$2$阶的广义导数
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.26-35
B. Prajapati, C. Gupta
Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,ldots,x_n)$ over $C$. Let $p,qin R$ be such that $pF^2(u)u+F^2(u)uq=0$ for all $uin S$. Then for all $xin R$ one of the followings holds:1) there exists $ain Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$,2) $p=-qin C$,3) $f(x_1,ldots,x_n)^2$ is central valued on $R$ and there exists $ain Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$.
设$R$是一个特征不同于$2$的素环,它有一个右Martindale商环$Q_r$和一个扩展质心$C$。设$F$是$R$的非零广义导数,$S$是一个非中心值的多元线性多项式$F (x_1,ldots,x_n)$ / $C$的求值集。设$p,q在R$中满足$pF^2(u)u+F^2(u)uq=0$对于所有$u在S$中。那么对于R$中的所有$x,下列条件之一成立:1)在Q_r$中存在$a使得$F(x)=ax$或$F(x)=xa$且$a^2=0$,2) $p=-q在C$中,3)$F(x_1,ldots,x_n)^2$是$R$上的中心值,并且在Q_r$中存在$a使得$F(x)=ax$且$pa^2+a^2q=0$。
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引用次数: 0
Continued $mathbf{A_2}$-fractions and singular functions 续$mathbf{A_2}$-分数和奇异函数
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.3-12
M. Pratsiovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak
In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2={frac12,1}$, $a_nin A_2$ and establish the normal property of numbers of the segment $I=[frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{sumlimits_{n=1}^{infty}(2a_n-1)v_n},$where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1in R$).
本文用两元字母$A_2={frac12,1}$, $a_nin A_2$深化了无限$A_2$ -连分数$[0;a_1,a_2,...,a_n,...]$理论的度量分量,并根据它们的$A_2$ -表示:$x=[0;a_1,a_2,...,a_n,...]$建立了区段$I=[frac12;1]$的数的正规性质。证明了几乎所有(在勒贝格测度的意义上)段$I$的数在它们的$A_2$ -表示中无限多次地使用任意长度的字母表元素的每一个元组作为该表示的连续数字。这个数字的正常性质被有效地用来证明函数$f(x=[0;a_1,a_2,...,a_n,...])=e^{sumlimits_{n=1}^{infty}(2a_n-1)v_n},$的奇异性,其中$v_1+v_2+...+v_n+...$是一个给定的绝对收敛级数,当函数$f$是连续的(这是只在$v_n=frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1in R$的情况下)。
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引用次数: 2
On linear sections of orthogonally additive operators 关于正交加法算子的线性截面
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.94-102
A. Gumenchuk, I. Krasikova, M. Popov
Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the $C$-compactness is equivalent to the $AM$-compactness. Next we prove that, under mild assumptions, every linear section of a $C$-compact orthogonally additive operator is $AM$-compact, and every linear section of a narrow orthogonally additive operator is narrow.
我们的第一个结果断言,对于从具有主投影性质的Riesz空间作用到具有阶连续范数的Banach格的线性正则算子,$C$-紧性等价于$AM$-紧度。接下来我们证明,在温和的假设下,$C$-紧正交可加算子的每个线性区间都是$AM$-紧的,并且窄正交可加运算符的每个线性区段都是窄的。
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引用次数: 0
Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order 具有非素数阶循环子群范数约束的非周期群
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.36-44
M. Drushlyak, T. Lukashova
One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $Sigma$-norms of a group. A $Sigma$-norm is the intersection of the normalizers of all subgroups of a system $Sigma$. The authors study non-periodic groups with the restrictions on such a $Sigma$-norm -- the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{bar{p}})$of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{bar{p}})$.Additionally the relations between the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order and the norm $N_{G}(C_{infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{infty})$ of infinite cyclic subgroups norms $N_{G}(C _{infty})$ and $N_{G}(C _{bar{p}})$ coincide if and only if $N_{G}(C _{infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.In this case $N_{G}(C_{bar {p}})=N_{G}(C_{infty})=G$.
群论的主要方向之一是研究特征子群对整个群结构的影响。这样的特征子群包括一个群的不同$Sigma$范数。$Sigma$范数是系统$Sigma$的所有子群的归一化器的交集。本文研究了非素阶循环子群的范数$N_{G}(C_{bar{p}})$,它是$G$的复合或无限阶循环子群正规化子的交集。证明了如果$G$是混合非周期群,则其非素数阶循环子群的范数$N_。此外,非周期群$G$具有非素数阶循环子群的非阿贝尔范数$N_,以及$G=N_{G}(C_{bar{p}})$。此外,还研究了非素数阶循环子群的范数$N_{G}。发现在具有无限循环子群的非阿贝尔范数$N_[infty})=G$。
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引用次数: 0
Essential spectra in non-Archimedean fields 非阿基米德场的基本光谱
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.82-93
A. Ammar, F. Boutaf, A. Jeribi
In the paper we extend some aspects of the essential spectra theory of linear operators acting in non-Archimedean (or p-adic) Banach spaces. In particular, we establish sufficient conditions for the relations between the essential spectra of the sum of two bounded linear operators and the union of their essential spectra. Moreover, we give essential prerequisites by studying the duality between p-adic upper and p-adic lower semi-Fredholm operators. We close this paper by giving some properties of the essential spectra.
本文推广了作用在非阿基米德(或p-adic)Banach空间中的线性算子的本质谱理论的一些方面。特别地,我们建立了两个有界线性算子之和的本质谱与其本质谱并集之间关系的充分条件。此外,我们通过研究p-adic上半Fredholm算子和p-adic下半Fredhol姆算子之间的对偶性,给出了必要的前提条件。我们通过给出本质谱的一些性质来结束本文。
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引用次数: 0
On modulus inequality of the order $p$ for the inner dilatation 关于内扩张的$p$阶模不等式
Q3 Mathematics Pub Date : 2022-04-16 DOI: 10.30970/ms.59.2.141-155
R. Salimov, E. Sevost’yanov, V. Targonskii
The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation of the order~$p.$ Weseparately considered the situations of homeomorphisms and mappingswith branch points. In particular, we have established thathomeomorphisms of the Sobolev class satisfy the upper estimate ofthe distortion of the modulus at the inner and boundary points ofthe domain. In addition, we have proved that similar estimates ofcapacity distortion occur at the inner points of the domain for opendiscrete mappings. Also, we have shown that open discrete and closedmappings satisfy some estimates of the distortion of the modulus offamilies of paths at the boundary points. The results of themanuscript are obtained mainly under the condition that theso-called inner dilatation of mappings is locally integrable. Themain approach used in the proofs is the choice of admissiblefunctions, using the relations between the modulus and capacity, andconnections between different modulus of families of paths (similarto Hesse, Ziemer and Shlyk equalities). In this context, we haveobtained some lower estimate of the modulus of families of paths inSobolev classes. The manuscript contains some examples related toapplications of obtained results to specific mappings.
本文研究了平面域上具有有界和有限畸变的映射。我们的研究引出了Sobolev类的映射与路径族模畸变上界之间的联系。对于这一类,我们已经证明了关于阶~$p.$的所谓内部扩张的Poletsky型不等式我们分别考虑了同胚和分支点映射的情形。特别地,我们已经建立了Sobolev类的同胚满足域的内点和边界点处的模的畸变的上估计。此外,我们还证明了对于开离散映射,类似的电容失真估计发生在域的内点。此外,我们还证明了开离散映射和闭映射满足对边界点处路径环境模失真的一些估计。本文的结果主要是在所谓映射的内扩张是局部可积的条件下得到的。证明中使用的主要方法是使用模和容量之间的关系,以及不同模的路径族之间的联系(类似于Hesse、Ziemer和Shlyk等式)来选择可容许函数。在这种情况下,我们得到了Sobolev类中路径族模的一些较低估计。手稿中包含了一些与所得结果应用于特定映射有关的例子。
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引用次数: 1
Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces l_p$-空间上的一类数值半径峰$n$-线性映射
Q3 Mathematics Pub Date : 2022-03-31 DOI: 10.30970/ms.57.1.10-15
S. Kim
For $ngeq 2$ and a real Banach space $E,$ ${mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$Pi(E)=Big{[x^*, (x_1, ldots, x_n)]: x^{*}(x_j)=|x^{*}|=|x_j|=1~mbox{for}~{j=1, ldots, n}Big}.$$For $Tin {mathcal L}(^n E:E),$ we define $$qopnamerelax o{Nr}({T})=Big{[x^*, (x_1, ldots, x_n)]in Pi(E): |x^{*}(T(x_1, ldots, x_n))|=v(T)Big},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {em numerical radius peak mapping} if there is $[x^{*}, (x_1, ldots, x_n)]in Pi(E)$ such that $qopnamerelax o{Nr}({T})={pm [x^{*}, (x_1, ldots, x_n)]}.$In this paper, we investigate some class of numerical radius peak mappings in ${mathcalL}(^n l_p:l_p)$ for $1leq p0.$Define $Tin {mathcal L}(^n l_p:l_p)$ by$$TBig(sum_{iin mathbb{N}}x_i^{(1)}e_i, cdots, sum_{iin mathbb{N}}x_i^{(n)}e_i Big)=sum_{jin mathbb{N}}a_{j}~x_{j}^{(1)}cdots x_{j}^{(n)}~e_j.qquadeqno(*)$$In particular is proved the following statements:$1.$ If $1< p<+infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0in mathbb{N}$ such that$$|a_{j_0}|>|a_{j}|~mbox{for every}~jin mathbb{N}backslash{j_0}.$$ $2.$ If $p=1$ then $T$ is not a numerical radius peak mapping in ${mathcal L}(^n l_1:l_1).$
对于$ngeq 2$和实巴拿赫空间$E,$ ${数学L}(^n E:E)$表示从$E$到自身的所有连续$n$线性映射的空间。让$ $ 大π(E) = {[x ^ * (x_1、 ldots x_n)]: x ^ {*} (x_j) = | x ^ {*} | = | x_j | = 1 ~ mbox {} ~ { ldots j = 1, n} 大}。$ $ $ T { mathcal L} (^ n E: E),我们定义美元美元 qopname 放松o {Nr} ({T}) =大 {[x ^ * (x_1、 ldots x_n)] π(E): | x ^ {*} (T (x_1、 ldots x_n)) | = v (T)大 },在v (T)美元美元表示的数值半径$ T $。T美元被称为{ em数值半径峰映射}如果有美元[x ^ {*}, (x_1、 ldots x_n)] π(E),这样美元 qopname 放松o {Nr} ({T}) = {下午 [x ^ {*}, (x_1、 ldots x_n)] }。本文研究了${mathcalL}(^n l_p:l_p)$中$1leq p0的一类数值半径峰映射。定义T美元在{ mathcal L} (^ n l_p: l_p) $ $ $ T 大( sum_{我 mathbb {n}} x_i ^ {(1)} e_i, cdots sum_{我 mathbb {n}} x_i ^ {(n)} e_i 大)= sum_ {j mathbb {n}}现代{j} ~间{j} ^ {(1)} cdots间{j} ^ {(n)} ~ e_j。 qquad eqno(*) $ $特别是证明下列语句: 1美元。如果美元1 < p |现代{j} | ~ mbox{每}~ j mathbb {N} 反斜杠 {j_0 }。$ $ $ 2。$如果$p=1$,则$T$不是${mathcal L}(^n l_1:l_1)中的数值半径峰映射
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引用次数: 0
Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties 单位球上的切片全纯函数:L -指标在一个方向上的有界性及相关性质
Q3 Mathematics Pub Date : 2022-03-31 DOI: 10.30970/ms.57.1.68-78
Andriy Ivanovych Bandura, T. Salo, O. Skaskiv
Let $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice ${z^0+tmathbf{b}: tinmathbb{C}}$ with the unit ball $mathbb{B}^n={zinmathbb{C}^: |z|:=sqrt{|z|_1^2+ldots+|z_n|^2}<1}$ for any $z^0inmathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $mathbf{b},$ where $mathbf{L}: mathbb{B}^ntomathbb{R}_+$ is a positive continuous function such that $L(z)>frac{beta|mathbf{b}|}{1-|z|}$ and $beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.
设$mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$为固定方向。我们考虑单位球中几个复变量的切片全纯函数,即我们研究在每个切片${z^0+tmathbf{b}:tinmathbb{C}}$与单位球$mathbb{b}^n={z inmathbb{C}^:|z|:=sqrt{|z|_1^2+ldots+|z_n|^2}frac{β|mathbf{b}{1-|z|}$的交集上解析的函数,并且$beta>1$是一些常数。对于这一类的函数,我们推导出海曼定理的类似式。它在微分方程的应用中是有用的判据。对于单位球中的切片全纯函数,我们引入了在方向上具有有界值$L$-分布的函数的概念。证明了单位球中的切片全纯函数在一个方向上具有有界值$L$-分布,当且仅当其方向导数在同一方向上有界$L$。其他命题涉及存在性定理。我们证明了对于在固定方向上的任何切片上具有有界零乘性的任何切片全纯函数$F$,存在这样一个正连续函数$L$,使得函数$F美元在该方向上具有有边界的$L$-索引。
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引用次数: 0
Bounds on the extent of a topological space 拓扑空间范围的界
Q3 Mathematics Pub Date : 2022-03-31 DOI: 10.30970/ms.57.1.62-67
A. Ravsky, T. Banakh
The extent $e(X)$ of a topological space $X$ is the supremum of sizes of closed discrete subspaces of $X$. Assuming that $X$ belongs to some class of topological spaces, we bound $e(X)$ byother cardinal characteristics of $X$, for instance Lindel"of number, spread or density.
拓扑空间$X$的区段$e(X)$是$X$的闭离散子空间的大小的极值。假设$X$属于某一类拓扑空间,我们用$X$的其他基本特征(如Lindel number, spread或density)来约束$e(X)$。
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引用次数: 0
An exact constant on the estimation of the approximation of classes of periodic functions of two variables by Ceśaro means 用Ceśaro方法估计两个变量的周期函数类的近似的一个精确常数
Q3 Mathematics Pub Date : 2022-03-31 DOI: 10.30970/ms.57.1.3-9
O. Rovenska
In the present work, we study problem related to the approximation of continuous $2pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others. The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.
本文研究了用傅里叶级数线性逼近连续$2pi$周期函数的问题。周期函数线性逼近的最简单例子是用傅里叶级数的部分和逼近这个函数。然而,众所周知,部分傅立叶和序列在连续$2 $周期函数上不是一致收敛的。因此,大量的论文致力于研究不同近似方法的近似性质,这些近似方法是由傅里叶级数的部分和的某些变换产生的。该方法允许我们构造三角多项式序列,该序列对C$中的所有函数$f 一致收敛。特别是Ceśaro均值和Fejer和在过去几十年中得到了广泛的研究。这一领域的一个重要问题是研究固定类周期函数的傅里叶和的线性平均偏差上界不等式中的精确常数。研究三角多项式偏差的积分表示的方法是由傅里叶级数的线性求和方法产生的。它们是在Nikolsky, Stechkin, Nagy和其他人的论文中发展起来的。本文给出了用傅里叶和线性逼近连续函数类的已知结果,以及在某些特殊情况下得到的新事实。本文研究了Lipschitz类中傅里叶和的Ceśaro逼近方法。在某些情况下,我们找到了二阶矩形统一度规的偏差上界的精确不等式Ceśaro在两个变量的Lipschitz周期函数类上的均值。
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引用次数: 0
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Matematychni Studii
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