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Free products of cyclic groups in groups of infinite unitriangular matrices 无穷单三角形矩阵群中循环群的自由积
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.28-33
A. Oliynyk
Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.
研究了结合酉环上的无限单三角形矩阵群。这些群自然作用于底层环上的无限维自由模。如果下环是有限的,它们就是无限的。利用它们与有限自动机所定义的群的联系,研究了用带无限单棱矩阵构造群的自由积的忠实表示问题。对于任意素数p,给出了特征为p的酉结合环上的带无限酉三角形矩阵有限集生成循环p群自由积的充分条件。这些条件是基于下环上有限维自由模作用的某些性质。结果表明,这些条件都是满足的。对于有限个循环p群的任意自由积,给出了特征p的唯一结合环上的无限单三角形矩阵集生成给定自由积的构造性实例。这些无限矩阵是由有限维的幂零约旦块构成的。关于所提出的例子和其他类型的忠实表示的性质的几个开放问题被制定。
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引用次数: 0
Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping 具有平面连接和一维Weingarten映射的余维二等仿射浸入
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.99-112
O. O. Shugailo
In the paper we study equiaffine immersions $fcolon (M^n,nabla) rightarrow {mathbb{R}}^{n+2}$ with flat connection $nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $fcolon ({M}^n,nabla)rightarrow({mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:$(i)$ $vec{r}=g(u^1,ldots,u^n) vec{a}_1+intvec{varphi}(u^1)du^1+sumlimits_{i=2}^n u^ivec{a}_i;$$(ii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{a}+int v(u^1) vec{eta}(u^1)du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)vec{eta}(u^1)du^1;$$(iii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{rho}(u^1)+int (v(u^1) - u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1.$
本文研究等仿射浸没 $fcolon (M^n,nabla) rightarrow {mathbb{R}}^{n+2}$ 带平连接 $nabla$ 和一维维因加滕映射。对于这种浸没,有两种类型的横向分布等仿框架。对于这两类等仿坐标系,我们给出了具有给定性质的子流形的参数化。本文的主要结果包含在定理1、定理2和推论1中 $fcolon ({M}^n,nabla)rightarrow({mathbb{R}}^{n+2},D)$ 是一种仿射浸渍,具有点余维2,等仿射结构,平面连接 $nabla$,则其参数化存在三种类型:$(i)$ $vec{r}=g(u^1,ldots,u^n) vec{a}_1+intvec{varphi}(u^1)du^1+sumlimits_{i=2}^n u^ivec{a}_i;$$(ii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{a}+int v(u^1) vec{eta}(u^1)du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)vec{eta}(u^1)du^1;$$(iii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{rho}(u^1)+int (v(u^1) - u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1.$
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引用次数: 0
On generalized homoderivations of prime rings 关于素环的广义同导
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.12-27
N. Rehman, E. K. Sogutcu, H. M. Alnoghashi
Let $mathscr{A}$ be a ring with its center $mathscr{Z}(mathscr{A}).$ An additive mapping $xicolon mathscr{A}to mathscr{A}$ is called a homoderivation on $mathscr{A}$ if $forall a,bin mathscr{A}colonquad xi(ab)=xi(a)xi(b)+xi(a)b+axi(b).$ An additive map $psicolon mathscr{A}to mathscr{A}$ is called a generalized homoderivation with associated homoderivation $xi$ on $mathscr{A}$ if $forall a,bin mathscr{A}colonquadpsi(ab)=psi(a)psi(b)+psi(a)b+axi(b).$ This study examines whether a prime ring $mathscr{A}$ with a generalized homoderivation $psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities: $psi(a)psi(b)+abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)+abin mathscr{Z}(mathscr{A}),$ $psi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(ab)+abin mathscr{Z}(mathscr{A}),quadpsi(ab)-abin mathscr{Z}(mathscr{A}),$ $psi(ab)+bain mathscr{Z}(mathscr{A}),quadpsi(ab)-bain mathscr{Z}(mathscr{A})quad (forall a, bin mathscr{A}).$ Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.
设$mathscr{A}$为一个以中心为中心的环$mathscr{Z}(mathscr{A}).$加性映射$xicolon mathscr{A}to mathscr{A}$称为$mathscr{A}$ if&#x0D上的同质导数;$forall a,bin mathscr{A}colonquad xi(ab)=xi(a)xi(b)+xi(a)b+axi(b).$ 
在$mathscr{A}$ if&#x0D上,一个可加映射$psicolon mathscr{A}to mathscr{A}$被称为具有关联同质导数$xi$的广义同质导数;$forall a,bin mathscr{A}colonquadpsi(ab)=psi(a)psi(b)+psi(a)b+axi(b).$ 
本文研究了一个满足特定代数恒等式的具有广义同导数$psi$的素环$mathscr{A}$是否可交换。确切地说,我们讨论下列恒等式:
$psi(a)psi(b)+abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)+abin mathscr{Z}(mathscr{A}),$ 
$psi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(ab)+abin mathscr{Z}(mathscr{A}),quadpsi(ab)-abin mathscr{Z}(mathscr{A}),$ 
$psi(ab)+bain mathscr{Z}(mathscr{A}),quadpsi(ab)-bain mathscr{Z}(mathscr{A})quad (forall a, bin mathscr{A}).$ 
并举例证明了对各定理的假设所加的限制并不是多余的。
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 $forall a,bin mathscr{A}colonquad xi(ab)=xi(a)xi(b)+xi(a)b+axi(b).$
 An additive map $psicolon mathscr{A}to mathscr{A}$ is called a generalized homoderivation with associated homoderivation $xi$ on $mathscr{A}$ if
 $forall a,bin mathscr{A}colonquadpsi(ab)=psi(a)psi(b)+psi(a)b+axi(b).$
 This study examines whether a prime ring $mathscr{A}$ with a generalized homoderivation $psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities:
 $psi(a)psi(b)+abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)+abin mathscr{Z}(mathscr{A}),$
 $psi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(ab)+abin mathscr{Z}(mathscr{A}),quadpsi(ab)-abin mathscr{Z}(mathscr{A}),$
 $psi(ab)+bain mathscr{Z}(mathscr{A}),quadpsi(ab)-bain mathscr{Z}(mathscr{A})quad (forall a, bin mathscr{A}).$
 Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099711","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
An exact estimate of the third Hankel determinants for functions inverse to convex functions 对逆凸函数的第三个汉克尔行列式的精确估计
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.34-39
B. Rath, K. S. Kumar, D. V. Krishna
Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions. Let $w=f(z)=z+a_{2}z^{2}+cdots$ be analytic in the unit disk $mathbb{D}={zinmathbb{C}:|z|<1}$, and $mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+cdots$ for some $|w|
解析一元函数的汉克尔行列式界的研究是20世纪初以来许多学者对解析一元函数几何性质研究的一个突出兴趣。许多作者得到了解析一元函数不同子类的第三汉克尔行列式的非尖锐上界,直到Kwon等人得到了卡拉多类第四系数的精确估计。最近作者利用第四系数的精确估计,得到了与解析一元函数子类相关的第三汉克尔行列式的锐界。让 $w=f(z)=z+a_{2}z^{2}+cdots$ 在单位圆盘上解析 $mathbb{D}={zinmathbb{C}:|z|<1}$,和 $mathcal{S}$ 的归一化一元函数的子类 $f(0)=0$,和 $f'(0)=1$. 让 $z=f^{-1}$ 是的反函数 $f$,由 $f^{-1}(w)=w+t_2w^2+cdots$ 对一些人来说 $|w|<r_o(f)$. 让 $mathcal{S}^csubsetmathcal{S}$ 中的凸函数的子集 $mathbb{D}$. 在本文中,我们估计了逆函数的第三汉克尔行列式的最佳可能上界 $z=f^{-1}$ 什么时候 $fin mathcal{S}^c$.让 $mathcal{S}^c$ 是一类凸函数。我们证明了下列命题(定理) $fin$ $mathcal{S}^c$那么,begin{equation*}big|H_{3,1}(f^{-1})big| leq frac{1}{36}end{equation*} 达到不平等是为了 $p_0(z)=(1+z^3)/(1-z^3).$
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引用次数: 0
Analytic in a unit polydisc functions of bounded $L$-index in direction 解析一元多盘函数的有界L -方向索引
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.55-78
A. Bandura, T. Salo
The concept of bounded $L$-index in a direction $mathbf{b}=(b_1,ldots,b_n)inmathbb{C}^nsetminus{mathbf{0}}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,ldots,z_n)inmathbb{D}^n$ one has $L(z)>betamax_{1le jle n}frac{|b_j|}{1-|z_j|},$ $beta=mathrm{const}>1,$ $mathbb{D}^n$ is the unit polydisc, i.e. $mathbb{D}^n={zinmathbb{C}^n: |z_j|le 1, jin{1,ldots,n}}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle ${z+tmathbf{b}: |t|=r/L(z)}$ by their values at the center of the circle, where $tinmathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs ${z^0+tmathbf{b}: |t|le r/L(z^0)}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.
有界的概念 $L$-一个方向上的指数 $mathbf{b}=(b_1,ldots,b_n)inmathbb{C}^nsetminus{mathbf{0}}$ 是对单位多盘上的一类解析函数的推广,在哪里 $L$ 是否有连续函数对每一个都满足 $z=(z_1,ldots,z_n)inmathbb{D}^n$ 一个是 $L(z)>betamax_{1le jle n}frac{|b_j|}{1-|z_j|},$ $beta=mathrm{const}>1,$ $mathbb{D}^n$ 是单位多盘,即。 $mathbb{D}^n={zinmathbb{C}^n: |z_j|le 1, jin{1,ldots,n}}.$ 对于该类的函数,我们得到了提供的有界性的充要条件 $L$-方向上的指数。他们描述了解析函数导数的最大模的局部性质 $F$ 在每个切片圆上 ${z+tmathbf{b}: |t|=r/L(z)}$ 通过它们在圆心的值,其中 $tinmathbb{C}.$ 其他准则通过这些函数的最大模量描述了类似的最小模量的局部行为。我们证明了用函数描述对数导数在某例外集外的估计的对数准则的一个类比 $L$. 集合由所有切片盘的并集生成 ${z^0+tmathbf{b}: |t|le r/L(z^0)}$,其中 $z^0$ 是函数的零点吗 $F$. 模拟还表明了函数的零分布 $F$ 在所有切片圆盘上是均匀的。在一维情况下,该断言在微分方程的解析理论和无穷积(即Blaschke积、Naftalevich-Tsuji积)中有许多应用。对单位多面体上的解析函数,也推导出了海曼定理的类比。这表明在有界的定义中 $L$在索引方向上,可以去掉分母中的阶乘。这允许研究方向微分方程解析解的性质。
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引用次数: 0
On the distribution of unique range sets and its elements over the extended complex plane 扩展复平面上唯一值域集及其元素的分布
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.40-54
S. Mallick
In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular,identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets.
本文从不同的角度讨论了唯一范围集及其元素在扩展复平面上的分布,得到了关于唯一范围集的结构和位置的一些新结果。这些新结果具有巨大的应用价值,如分类C的不同子集是否为唯一值域集,探索每一个双线性变换为亚纯函数保留唯一值域集的事实,提供一些唯一值域集存在的更简单和更短的证明,揭示任何亚纯函数的零点或极点存在于唯一值域集的事实,特别是,将代数基本定理确定为更具体的领域和更多的应用。我们还提出了一些开放的问题来揭示唯一范围集元素的神秘排列。
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引用次数: 0
Transformation operators for impedance Sturm–Liouville operators on the line 变换算子为阻抗Sturm-Liouville算子
Q3 Mathematics Pub Date : 2023-09-22 DOI: 10.30970/ms.60.1.79-98
M. Kazanivskiy, Ya. Mykytyuk, N. Sushchyk
In the Hilbert space $H:=L_2(mathbb{R})$, we consider the impedance Sturm--Liouville operator $T:Hto H$ generated by the differential expression $ -pfrac{d}{dx}{frac1{p^2}}frac{d}{dx}p$, where the function $p:mathbb{R}tomathbb{R}_+$ is of bounded variation on $mathbb{R}$ and $inf_{xinmathbb{R}} p(x)>0$. Existence of the transformation operator for the operator $T$ and its properties are studied. In the paper, we suggest an efficient parametrization of the impedance function p in term of a real-valued bounded measure $muin boldsymbol M$ via$p_mu(x):= e^{mu([x,infty))}, xinmathbb{R}.$For a measure $muin boldsymbol M$, we establish existence of the transformation operator for the Sturm--Liouville operator $T_mu$, which is constructed with the function $p_mu$. Continuous dependence of the operator $T_mu$ on $mu$ is also proved. As a consequence, we deduce that the operator $T_mu$ is unitarily equivalent to the operator $T_0:=-d^2/dx^2$.
在Hilbert空间$H:=L_2(mathbb{R})$中,我们考虑由微分表达式$ -pfrac{d}{dx}{frac1{p^2}}frac{d}{dx}p$产生的阻抗Sturm—Liouville算子$T:Hto H$,其中函数$p:mathbb{R}tomathbb{R}_+$在$mathbb{R}$和$inf_{xinmathbb{R}} p(x)>0$上有界变化。研究了算子$T$的变换算子的存在性及其性质。&#x0D;本文通过$p_mu(x):= e^{mu([x,infty))}, xinmathbb{R}.$提出了阻抗函数p用实值有界测度$muin boldsymbol M$有效参数化的方法。对于测度$muin boldsymbol M$,我们建立了Sturm—Liouville算子$T_mu$的变换算子的存在性,该变换算子由函数$p_mu$构造。证明了算子$T_mu$对$mu$的连续依赖。因此,我们推导出运算符$T_mu$与运算符$T_0:=-d^2/dx^2$是一元等价的。
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引用次数: 0
Spaces of non-additive measures generated by triangular norms 由三角模生成的非加性测度的空间
Q3 Mathematics Pub Date : 2023-06-24 DOI: 10.30970/ms.59.2.215-224
Kh.O. Sukhorukova
We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.
我们考虑紧Hausdorff空间上的非加性测度,它是幂等测度和max-min测度的推广。这些测度与连续三角范数有关,它们被定义为从紧致豪斯多夫空间到单位区间的连续函数空间上的泛函。所获得的测度空间(称为*-测度,其中*是三角范数)被赋予弱*拓扑。这个构造确定了紧豪斯多夫空间范畴中的一个函子。特别证明了有限支撑的*-测度在*-测度的空间中是稠密的。本文的一个主要结果提供了紧致豪斯多夫空间X上*-测度的另一种描述,即X×[0,1]中某些子集的超空间。这是Brydun和Zarichnyi证明的最大-最小测度定理的类似。
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引用次数: 4
One class of continuous locally complicated functions related to infinite-symbol $Phi$-representation of numbers 一类与无穷符号$Phi$有关的连续局部复函数——数字表示
Q3 Mathematics Pub Date : 2023-06-24 DOI: 10.30970/ms.59.2.123-131
M. Pratsovytyi, O. Baranovskyi, O. Bondarenko, S. Ratushniak
In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ mathbb{Z}={0,pm 1, pm 2,...}$ and base $tau=frac{sqrt{5}-1}{2}$: $displaystyle x=b_{alpha_1}+sumlimits_{k=2}^{m}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_m(emptyset)},quadx=b_{alpha_1}+sumlimits_{k=2}^{infty}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_n...},$ where $alpha_nin mathbb{Z}$, $Theta_n=Theta_{-n}=tau^{3+|n|}$,$b_n=sumlimits_{i=-infty}^{n-1}Theta_i=begin{cases}tau^{2-n}, & mbox{if } nleq0, 1-tau^{n+1}, & mbox{if } ngeq 0.end{cases}$ The function $f$, which is the main object of the study, is defined by equalities$displaystylebegin{cases}f(x=Delta^{Phi}_{i_1...i_k...})=sigma_{i_11}+sumlimits_{k=2}^{infty}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_k...},f(x=Delta^{Phi}_{i_1...i_m(emptyset)})=sigma_{i_11}+sumlimits_{k=2}^{m}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_m(emptyset)},end{cases}$ where an infinite matrix $||p_{ik}||$ ($iin mathbb{Z}$, $kin mathbb N$) satisfies the conditions 1) $|p_{ik}|<1$ $forall iin mathbb{Z}$, $forall kin mathbb N;quad$2) $sumlimits_{iin mathbb{Z}}p_{ik}=1$ $forall kinmathbb N$; 3) $0
在本文中,我们引入并研究了一类在区间$(0;1)$上定义的连续函数,使用字母表$mathbb{Z}={0,pm1,pm2,…}$和基$tau=frac{sqrt的自变量的特殊编码(表示){5}-1}{2} $:$displaystyle x=b_{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2…alpha_n…},$其中$alpha_ninmathbb{Z}$,$Theta_n=Theta_{-n}=tau^{3+|n|}$,$b_n=sumlimits_{i=-infty}^{n-1}Theta_i=begin{cases}tau^{2-n},&&mbox{if}nleq0,1-tau ^{n+1},mbox{if}ngeq 0.end{casses}$函数$f$是研究的主要对象,由等式$displaystyleboot定义{cases}f(x=Δ^{Phi}_{i_1…i_k…}^{k-1}p_{i_jj}equivDelta_{i_1…i_k…},f(x=Delta^{Phi}_{i1…i_m(pemptyset)}^{k-1}p_{i_jj}equivDelta_{i_1…i_m(emptyset)},end{cases}$其中一个无限矩阵$|p_{ik}|$($iinmathbb{Z}$,$kinmath bb N$)满足条件1)$|p_{ik}|<1$ for all iinathbb{Z}$,$ for ll kinath bb N;quad$2)$sumlimits_{iinmathbb{Z}}p_{ik}=1$$对于所有kinmath bb N$;3) $0^{k-1}p_{i_jj}
{"title":"One class of continuous locally complicated functions related to infinite-symbol $Phi$-representation of numbers","authors":"M. Pratsovytyi, O. Baranovskyi, O. Bondarenko, S. Ratushniak","doi":"10.30970/ms.59.2.123-131","DOIUrl":"https://doi.org/10.30970/ms.59.2.123-131","url":null,"abstract":"In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ mathbb{Z}={0,pm 1, pm 2,...}$ and base $tau=frac{sqrt{5}-1}{2}$: $displaystyle x=b_{alpha_1}+sumlimits_{k=2}^{m}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_m(emptyset)},quadx=b_{alpha_1}+sumlimits_{k=2}^{infty}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_n...},$ \u0000where $alpha_nin mathbb{Z}$, $Theta_n=Theta_{-n}=tau^{3+|n|}$,$b_n=sumlimits_{i=-infty}^{n-1}Theta_i=begin{cases}tau^{2-n}, & mbox{if } nleq0, 1-tau^{n+1}, & mbox{if } ngeq 0.end{cases}$ \u0000The function $f$, which is the main object of the study, is defined by equalities$displaystylebegin{cases}f(x=Delta^{Phi}_{i_1...i_k...})=sigma_{i_11}+sumlimits_{k=2}^{infty}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_k...},f(x=Delta^{Phi}_{i_1...i_m(emptyset)})=sigma_{i_11}+sumlimits_{k=2}^{m}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_m(emptyset)},end{cases}$ where an infinite matrix $||p_{ik}||$ ($iin mathbb{Z}$, $kin mathbb N$) satisfies the conditions \u00001) $|p_{ik}|<1$ $forall iin mathbb{Z}$, $forall kin mathbb N;quad$2) $sumlimits_{iin mathbb{Z}}p_{ik}=1$ $forall kinmathbb N$; \u00003) $0<sumlimits_{k=2}^{infty}prodlimits_{j=1}^{k-1}p_{i_jj}<infty~~forall (i_j)in L;quad$4) $0<sigma_{ik}equivsumlimits_{j=-infty}^{i-1}p_{jk}<1$ $forall iin mathbb Z, forall kin mathbb N.$ \u0000This class of functions contains monotonic, non-monotonic, nowhere monotonic functions and functionswithout monotonicity intervals except for constancy intervals, Cantor-type andquasi-Cantor-type functions as well as functions of bounded and unbounded variation. The criteria for the function $f$ to be monotonic and to be a function of the Cantor type as well as the criterion of nowhere monotonicity are proved. Expressions for the Lebesgue measure of the set of non-constancy of the function and for the variation of the function are found. Necessary and sufficient conditions for thefunction to be of unbounded variation are established.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41805080","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
Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents 具有任意复指数的整个多重Dirichlet级数的Wiman型不等式
Q3 Mathematics Pub Date : 2023-06-24 DOI: 10.30970/ms.59.2.178-186
A. Kuryliak
It is proved analogues of the classical Wiman's inequality} for  the class $mathcal{D}$ of absolutely convergents in the whole complex plane $mathbb{C}^p$ (entire) Dirichlet series of the form $displaystyle F(z)=sumlimits_{|n|=0}^{+infty} a_ne^{(z,lambda_n)}$ with such a sequence of exponents $(lambda_n)$ that ${lambda_ncolon ninmathbb{Z}^p}subset mathbb{C}^p$ and $lambda_nnot=lambda_m$ for all $nnot= m$. For $Finmathcal{D}$ and $zinmathbb{C}^psetminus{0}$ we denote  $mathfrak{M}(z,F):=sumlimits_{|n|=0}^{+infty}|a_n|e^{Re(z,lambda_n)},quadmu(z,F):=sup{|a_n|e^{mathop{rm Re}(z,lambda_n)}colon ninmathbb{Z}^ p_+},$ $(m_k)_{kgeq 0}$ is $(mu_{k})_{kgeq 0}$ the sequence $(-ln|a_{n}|)_{ninmathbb{Z}^p_+}$ arranged by non-decreasing. The  main result of the paper: Let $Fin mathcal{D}.$ If $(exists alpha > 0)colon$ $intnolimits_{t_0}^{+infty}t^{-2}{(n_1(t))^{alpha}}dt<+infty,$  $n_1(t)overset{def}=sumnolimits_{mu_nleq t} 1,quad t_0>0,$ then there exists a set $Esubsetgamma_{+}(F),$ such that $tau_{2p}(Ecapgamma_{+}(F))=int_{Ecapgamma_{+}(F)}|z|^{-2p}dxdyleq C_p, z=x+iyinmathbb{C}^p,$  and relation $mathfrak{M}(z,F)= o(mu(z,F)ln^{1/alpha} mu(z,F))$ holds as $zto infty$ $(zin gamma_Rsetminus E)$ for each $R>0$, where $gamma_R=Big{zinmathbb{C}^psetminus{0}colon K_F(z)leq R Big},quad K_F(z)=supBig{frac1{Phi_z( t)}int^{ t}_0 frac {{Phi_z}(u)}{u} ducolon t geq t_0Big},$ $gamma(F)={zinmathbb{C}colon limlimits_{tto +infty}Phi_z(t)=+infty},quad gamma_+(F)=mathop{cup}_{R>0}gamma_R$, $Phi_z(t)=frac1{t}lnmu(tz,F)$. In general, under the specified conditions, the obtained inequality is exact.
它被证明是经典的维曼不等式的类似物 $mathcal{D}$ 在整个复平面上绝对收敛 $mathbb{C}^p$ (整个)狄利克雷级数的形式 $displaystyle F(z)=sumlimits_{|n|=0}^{+infty} a_ne^{(z,lambda_n)}$ 有这样一个指数序列 $(lambda_n)$ 那${lambda_ncolon ninmathbb{Z}^p}subset mathbb{C}^p$ 和 $lambda_nnot=lambda_m$ 对所有人 $nnot= m$. 因为 $Finmathcal{D}$ 和 $zinmathbb{C}^psetminus{0}$ 我们表示 $mathfrak{M}(z,F):=sumlimits_{|n|=0}^{+infty}|a_n|e^{Re(z,lambda_n)},quadmu(z,F):=sup{|a_n|e^{mathop{rm Re}(z,lambda_n)}colon ninmathbb{Z}^ p_+},$ $(m_k)_{kgeq 0}$ 是 $(mu_{k})_{kgeq 0}$ 顺序 $(-ln|a_{n}|)_{ninmathbb{Z}^p_+}$ 按非递减排列。本文的主要结论是:让 $Fin mathcal{D}.$ 如果 $(exists alpha > 0)colon$ $intnolimits_{t_0}^{+infty}t^{-2}{(n_1(t))^{alpha}}dt0,$ 那么就存在一个集合 $Esubsetgamma_{+}(F),$ 这样 $tau_{2p}(Ecapgamma_{+}(F))=int_{Ecapgamma_{+}(F)}|z|^{-2p}dxdyleq C_p, z=x+iyinmathbb{C}^p,$与关系 $mathfrak{M}(z,F)= o(mu(z,F)ln^{1/alpha} mu(z,F))$ 保持为 $zto infty$ $(zin gamma_Rsetminus E)$ 对于每一个 $R>0$,其中 $gamma_R=Big{zinmathbb{C}^psetminus{0}colon K_F(z)leq R Big},quad K_F(z)=supBig{frac1{Phi_z( t)}int^{ t}_0 frac {{Phi_z}(u)}{u} ducolon t geq t_0Big},$ $gamma(F)={zinmathbb{C}colon limlimits_{tto +infty}Phi_z(t)=+infty},quad gamma_+(F)=mathop{cup}_{R>0}gamma_R$, $Phi_z(t)=frac1{t}lnmu(tz,F)$. 一般来说,在一定条件下,所得不等式是精确的。
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引用次数: 0
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