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On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series 论泰勒-德里赫利数列芬顿型定理中例外集的 h 度量
Q3 Mathematics Pub Date : 2024-03-27 DOI: 10.30970/ms.61.1.109-112
Andrii Bodnarchuk, Yu.M. Gal', O. Skaskiv
We consider the class $S(lambda,beta,tau)$ of convergent for all  $xge0$ Taylor-Dirichlet type series of the form $$F(x) =sum_{n=0}^{+infty}{b_ne^{xlambda_n+tau(x)beta_n}},  b_ngeq 0 (ngeq 0),$$  where  $taucolon [0,+infty)to (0,+infty)$ is a continuously differentiable non-decreasing function, $lambda=(lambda_n)$ and $beta=(beta_n)$ are such that $lambda_ngeq 0, beta_ngeq 0$ $(ngeq 0)$. In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference  ``Complex Analysis and Related Topics'' (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function  $h(x)colon [0,+infty)to (0,+infty)$, $h'(x)nearrow +infty$ $ (xto +infty)$, every sequence  $lambda=(lambda_n)$ such that  $displaystylesum_{n=0}^{+infty}frac1{lambda_{n+1}-lambda_n}<+infty$ and for any non-decreasing sequence  $beta=(beta_n)$ such that $beta_{n+1}-beta_nlelambda_{n+1}-lambda_n$ $(ngeq 0)$  there exist a function  $tau(x)$ such that $tau'(x)ge 1$ $(xgeq x_0)$, a function  $Fin S(alpha, beta, tau)$, a set  $E$ and  a constant $d>0$ such that $h-mathop{meas} E:=int_E dh(x)=+infty$ and $(forall xin E)colon F(x)>(1+d)mu(x,F),$ where $mu(x,F)=max{|a_n|e^{xlambda_n+tau(x)beta_n}colon nge 0}$ is the maximal term of the series.   At the same time, we also pose some open questions and formulate one conjecture.
我们考虑了$S(lambda,beta,tau)$这一类对于所有$xge0$都收敛的泰勒-德里赫特类型序列,其形式为$F(x) =sum_{n=0}^{+infty}{b_ne^{xlambda_n+tau(x)beta_n}},b_ngeq 0 (ngeq 0)、其中 $taucolon [0,+infty)to (0,+infty)$ 是一个连续可微的非递减函数,$lambda=(lambda_n)$ 和 $beta=(beta_n)$ 都使得 $lambda_ngeq 0, beta_ngeq 0$ (ngeq 0)$。本文部分回答了萨洛-T.M.、斯卡斯基夫-O.B.、特鲁塞维奇-O.M.在 "复杂分析及相关主题 "国际会议(利沃夫,2013 年 9 月 23-28 日)上提出的问题([2])。我们证明了以下陈述:对于每个递增函数 $h(x)colon [0,+infty)to (0,+infty)$, $h'(x)nearrow +infty$ $ (xto +infty)$、每一个序列 $lambda=(lambda_n)$ 这样 $displaystylesum_{n=0}^{+infty}frac1{lambda_{n+1}-lambda_n}0$ 这样 $h-mathop{meas} E:=int_E dh(x)=+infty$ 并且 $(forall xin E)colon F(x)>(1+d)mu(x,F),$ 其中 $mu(x,F)=max{|a_n|e^{xlambda_n+tau(x)beta_n}colon nge 0}$ 是数列的最大项。 同时,我们还提出了一些开放性问题和一个猜想。
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引用次数: 0
Almost periodic distributions and crystalline measures 几乎周期性分布和晶体测量
Q3 Mathematics Pub Date : 2024-03-20 DOI: 10.30970/ms.61.1.97-108
V. MatematychniStudii., No 61, S. Favorov
We study temperate distributions and measures with discrete support in Euclidean space and their Fourier transformswith special attention to almost periodic distributions. In particular, we prove that if distances between points of the support of a measure do not quickly approach 0 at infinity, then this measure is a Fourier quasicrystal (Theorem 1). We also introduce a new class of almost periodicity of distributions,close to the previous one, and study its properties.Actually, we introduce the concept of s-almost periodicity of temperate distributions. We establish the conditions for a measure $mu$ to be s-almost periodic (Theorem 2), a connection between s-almost periodicityand usual almost periodicity of distributions (Theorem 3). We also prove that the Fourier transform of an almost periodic distribution with locally finite support is a measure (Theorem 4),and prove a necessary and sufficient condition on a locally finite set $E$ for each measure with support on $E$ to have s-almost periodic Fourier transform (Theorem 5).
我们研究欧几里得空间中具有离散支持的温带分布和度量及其傅里叶变换,并特别关注几乎周期性的分布。特别是,我们证明了如果一个度量的支持点之间的距离在无穷远处不会迅速接近 0,那么这个度量就是傅里叶准晶体(定理 1)。实际上,我们引入了温带分布的 s 近似周期性的概念。我们建立了一个度量 $mu$ 几乎是 s-almost 周期性的条件(定理 2),以及 s-almost 周期性与通常分布的几乎周期性之间的联系(定理 3)。我们还证明了具有局部有限支持的几乎周期性分布的傅里叶变换是一种度量(定理 4),并证明了在局部有限集合 $E$ 上每个具有支持的度量具有 s 几乎周期性傅里叶变换的必要条件和充分条件(定理 5)。
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引用次数: 0
Real univariate polynomials with given signs of coefficients and simple real roots 具有给定系数符号和简单实数根的实数单变量多项式
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.22-34
V. MatematychniStudii., No 61, V. P. Kostov
We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $ell ^+$ and $ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $mathcal{P}^d:={ P:=x^d+a_1x^{d-1}+dots +a_d}$, $a_jin mathbb{R}^*=mathbb{R}setminus { 0}$, the corresponding sign patterns $sigma=(sigma_1,sigma_2,dots, sigma_d)$, where $sigma_j=$sign$(a_j)$, and the sets $mathcal{P}^d_{sigma ,(ell ^+,ell ^-)}subset mathcal{P}^d$ of polynomials with given triples $(sigma ,(ell ^+,ell ^-))$.We prove that for degree $dleq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $sigma =(-,-,-,+)$, $ell^+=0$, $ell ^-=2$, with $d=5$, $sigma =(-,-,-,-,+)$, $ell^+=0$, $ell ^-=3$, and the ones obtained from them under the $mathbb{Z}_2times mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting  involutions: $i_mcolon P(x)mapsto (-1)^dP(-x)$ and $i_rcolon P(x)mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex  conjugate pairs of roots ($2nleq d$);2) for $1leq sleq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2nleq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $dgeq 6$, we give an example of a set $mathcal{P}^d_{sigma ,(ell^+,ell^-)}$  having $Lambda (d)$ connected compo-nents, where $Lambda (d)rightarrow infty$ as $drightarrow infty$.
我们继续研究笛卡尔符号规则的不同方面,讨论实数度 $d$ 单变量一元多项式(即前导系数为 $1$)的集合的连通性,这些集合具有给定数 $ell ^+$ 和 $ell ^-$ 的正实根和负实根,以及给定的系数符号;实根假定都是简单的,系数假定都是非范数。也就是说,我们考虑的空间是 $mathcal{P}^d:={ P:=x^d+a_1x^{d-1}+dots +a_d}$, $a_jin mathbb{R}^*=mathbb{R}setminus { 0}$, 相应的符号模式 $sigma=(sigma_1,sigma_2,dots,sigma_d)$、其中 $sigma_j=$sign$(a_j)$, 以及具有给定三元组 $(sigma ,(ell ^+,ell ^-))$ 的多项式的集合 $mathcal{P}^d_{sigma ,(ell ^+,ell ^-)} 子集 mathcal{P}^d$.我们证明,对于度数 $dleq 5$,所有这样的集合都是连通的或空的。大多数连通集合是可收缩的,即能够通过连续变形还原为其中一点。空集恰恰是具有 $d=4$,$sigma =(-,-,-,+)$,$ell^+=0$,$ell ^-=2$,具有 $d=5$,$sigma =(-,-,-,-,+)$,$ell^+=0$,$ell ^-=3$的集合、以及它们在$mathbb{Z}_2times mathbb{Z}_2$-action下得到的结果,该action定义在阶数为$d$的一元多项式集合上,其两个生成器是两个交换渐开线:$i_mcolon P(x)mapsto (-1)^dP(-x)$ 和 $i_rcolon P(x)mapsto x^dP(1/x)/P(0)$.我们证明,对于任意的 $d$,以下两个集合是可收缩的:1)所有系数都为正且有恰好 $n$ 复共轭根对(2nleq d$)的度为 $d$ 的实数一元多项式集合;2)对于 $1leq sleq d$,有恰好 $n$ 共轭根对(2nleq d$)的度为 $d$ 的实数一元多项式集合,其前 $s$ 系数为正,后 $d+1-s$ 系数为负。对于任意的度 $dgeq 6$,我们举例说明一个集合 $mathcal{P}^d_{sigma ,(ell^+,ell^-)}$ 具有 $Lambda (d)$ 连接成分,其中 $Lambda (d)rightarrow infty$ 为 $drightarrow infty$。
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引用次数: 0
On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane 关于在半平面内绝对收敛的某类实系数狄利克列数列
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.35-50
M. Sheremeta
 For $h>0$, $alphain [0,h)$ and $muin {mathbb R}$  denote by   $SD_h(mu, alpha)$ a class of absolutely convergent in the half-plane $Pi_0={s:, text{Re},s<0}$ Dirichlet series $F(s)=e^{sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ such that   smallskipcenterline{$text{Re}left{frac{(mu-1)F'(s)-mu F''(s)/h}{(mu-1)F(s)-mu F'(s)/h}right}>alpha$ for all $sin Pi_0$,}   smallskipnoi and let  $Sigma D_h(mu, alpha)$ be a class of absolutely convergent in half-plane $Pi_0$ Dirichlet series $F(s)=e^{-sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ such that   smallskipcenterline{$text{Re}left{frac{(mu-1)F'(s)+mu F''(s)/h}{(mu-1)F(s)+mu F'(s)/h}right}<-alpha$ for all $sin Pi_0$.}   smallskipnoi Then $SD_h(0, alpha)$ consists of pseudostarlike functions of order $alpha$ and $SD_h(1, alpha)$ consists of pseudoconvex functions of order $alpha$.   For functions from the classes  $SD_h(mu, alpha)$ and  $Sigma D_h(mu, alpha)$, estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements:  1) In order that function $F(s)=e^{sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ belongs to $SD_h(mu, alpha)$, it is sufficient, and in the case when $f_k(mulambda_k/h-mu+1)le 0$ for all $kge 1$, it is necessary that}   smallskipcenterline{$ sumlimits_{k=1}^{infty}big|f_kbig(frac{mulambda_k}{h}-mu+1big)big|(lambda_k-alpha)le h-alpha,$}   noi {where $h>0, alphain [0, h)$ (Theorem 1).}   noi 2) {In order that function $F(s)=e^{-sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ belongs to $Sigma D_h(mu, alpha)$, it is sufficient, and in the case when $f_k(mulambda_k/h+mu-1)le 0$ for all $kge 1$, it is necessary that   smallskipcenterline{$sumlimits_{k=1}^{infty}big|f_kbig(frac{mulambda_k}{h}+mu-1big)big|(lambda_k+alpha)le h-alpha,$}   noi where $h>0,  alphain [0, h)$ (Theorem~2).}  Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus $m$ were also studied.
对于 $h>0$, $alphain [0,h)$ 和 $muin {mathbb R}$,用 $SD_h(mu, alpha)$ 表示在半平面 $Pi_0={s:对于所有 $sin Pi_0$,} smallskipnoi 并让 $Sigma D_h(mu、讓 $Sigma D_h(mu, alpha)$ 是一類絕對收斂的半平面 $Pi_0$ Dirichlet 數列 $F(s)=e^{-sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ such that smallskipcenterline{$text{Re}left{frac{(mu-1)F'(s)+mu F''(s)/h}{(mu-1)F(s)+mu F'(s)/h}right}0,在 [0, h]$(定理 1)。} 2) {为了使函数$F(s)=e^{-sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ 属于$Sigma D_h(mu,alpha)$,这就足够了,并且在所有$kge 1$的情况下,当$f_k(mulambda_k/h+mu-1)le 0$时、it is necessary that smallskipcenterline{$sumlimits_{k=1}^{infty}big|f_kbig(frac{mulambda_k}{h}+mu-1big)big|(lambda_k+alpha)le h-alpha,$}. 其中 $h>0, alphain [0, h)$ (定理~2)}。 对这些函数的邻域进行了研究。还研究了普通哈达玛组合和属$m$的哈达玛组合。
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引用次数: 0
Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$ 霍恩超几何函数 $H_4$ 的分枝续分展开的数值稳定性
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.51-60
R. Dmytryshyn, C. Cesarano, I.-A.V. Lutsiv, M. Dmytryshyn
In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.
在本文中,我们考虑了支链续分数的一些数值方面的问题,支链续分数是表示和扩展多个复变函数的解析函数的特殊函数族,包括超几何函数的广义。后向递推算法是计算支化连续分数近似值的基本工具之一。与大多数递推过程一样,它容易受到误差增长的影响。递归过程的每个循环不仅会产生自己的舍入误差,还会继承之前所有循环的舍入误差。另一方面,一般来说,分支续分数是一种非线性研究对象(两个分数线性映射之和并不总是分数线性映射)。在这项工作中,我们处理的是汇合支链续分数,它在形式上是一种续分数。这里的本质区别在于,续分数的近似值就是所谓的支链续分数的图近似值。在计算霍恩超几何函数 H4 的支链续分数展开的 n 次近似值时,建立了对后向递推算法所产生的相对舍入误差的估计。推导过程使用了支链续分数理论的方法,这些方法对于制定收敛标准至关重要。数值示例说明了后向递推算法的数值稳定性。
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引用次数: 0
Existence of basic solutions of first order linear homogeneous set-valued differential equations 一阶线性均质集值微分方程基本解的存在性
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.61-78
A. Plotnikov, T. A. Komleva, N. Skripnik
The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.
本文介绍了各种集值映射的导数、它们的主要性质以及它们之间的关系。接下来,我们考虑了具有不同类型导数(Hukuhara 导数、PS-导数和 BG-导数)的线性同构集值微分方程的 Cauchy 问题。众所周知,这种具有 PS 衍射和 BG 衍射的初值问题有无穷多个解。其中两个解称为基本解,即解部分的直径函数为单调递增函数(第一个基本解)或单调递减函数(第二个基本解)。然而,第二基本解并不总是存在。我们为此类初值问题基本解的存在提供了条件。结果表明,它们的存在取决于导数的类型、右侧系数矩阵和初始集的类型。考虑了模型实例。
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引用次数: 0
Reflectionless Schrodinger operators and Marchenko parametrization 无反射薛定谔算子和马琴科参数化
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.79-83
Ya. Mykytyuk, N. Sushchyk
Let $T_q=-d^2/dx^2 +q$ be a Schr"odinger operator in the space $L_2(mathbb{R})$. A potential $q$ is called reflectionless if the operator $T_q$ is reflectionless. Let $mathcal{Q}$ be the set of all reflectionless potentials of the Schr"odinger operator, and let $mathcal{M}$ be the set of nonnegative Borel measures on $mathbb{R}$ with compact support. As shown by Marchenko, each potential $qinmathcal{Q}$ can be associated with a unique measure $muinmathcal{M}$. As a result, we get the bijection $Thetacolon mathcal{Q}to mathcal{M}$. In this paper, we show that one can define topologies on $mathcal{Q}$ and $mathcal{M}$, under which the mapping $Theta$ is a homeomorphism.
设 $T_q=-d^2/dx^2 +q$ 是空间 $L_2(mathbb{R})$中的一个薛定谔算子。如果算子 $T_q$ 是无反射的,则称为无反射势 $q$。让 $mathcal{Q}$ 成为薛定谔算子的所有无反射势的集合,让 $mathcal{M}$ 成为$mathbb{R}$上具有紧凑支持的非负博雷尔量的集合。正如马琴科所证明的,每个势 $qinmathcal{Q}$ 都可以与唯一的量 $muinmathcal{M}$ 相关联。因此,我们得到了 $Thetacolon mathcal{Q}to mathcal{M}$ 的双射。在本文中,我们证明了可以在 $mathcal{Q}$ 和 $mathcal{M}$ 上定义拓扑,在拓扑下映射 $Theta$ 是同态的。
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引用次数: 0
Monogenic free inverse semigroups and partial automorphisms of regular rooted trees 规则有根树的单原自由逆半群和部分自动形态
Q3 Mathematics Pub Date : 2024-03-19 DOI: 10.30970/ms.61.1.3-9
E. Kochubinska, A. Oliynyk
For a one-to-one partial mapping on an infinite set, we present a criterion in terms of its cycle-chain decomposition that the inverse subsemigroup generated by this mapping is monogenic free inverse. We also give a sufficient condition for a regular rooted tree partial automorphism to extend to a partial automorphism of another regular rooted tree so that the inverse semigroup gene-ra-ted by this extended partial automorphism is monogenic free inverse. The extension procedure we develop is then applied to $n$-ary adding machines.
对于无穷集上的一对一局部映射,我们根据其循环链分解提出了一个标准,即该映射产生的逆子半群是单源自由逆。我们还给出了一个充分条件,即有规则有根树的部分自动形可以扩展为另一个有规则有根树的部分自动形,从而由这个扩展的部分自动形生成的逆子半群是单源自由逆。我们开发的扩展程序随后被应用于 $n$ary 加法器。
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引用次数: 0
On locally compact shift continuous topologies on the semigroup $boldsymbol{B}_{[0,infty)}$ with an adjoined compact ideal 关于有邻接紧凑理想的半群 $boldsymbol{B}_{[0,infty)}$ 上的局部紧凑移位连续拓扑学
Q3 Mathematics Pub Date : 2024-01-12 DOI: 10.30970/ms.61.1.10-21
O. Gutik, Markian Khylynskyi
Let $[0,infty)$ be the set of all non-negative real numbers. The set $boldsymbol{B}_{[0,infty)}=[0,infty)times [0,infty)$ with the following binary operation $(a,b)(c,d)=(a+c-min{b,c},b+d-min{b,c})$ is a bisimple inverse semigroup.In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup $boldsymbol{B}_{[0,infty)}$ with an adjoined compact ideal of the following tree types.The semigroup $boldsymbol{B}_{[0,infty)}$ with the induced usual topology $tau_u$ from $mathbb{R}^2$, with the topology $tau_L$ which is generated by the natural partial order on the inverse semigroup $boldsymbol{B}_{[0,infty)}$, and the discrete topology are denoted by $boldsymbol{B}^1_{[0,infty)}$, $boldsymbol{B}^2_{[0,infty)}$, and $boldsymbol{B}^{mathfrak{d}}_{[0,infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $boldsymbol{B}^1_{[0,infty)}$ ($boldsymbol{B}^2_{[0,infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the topological space $S_1^I$ ($S_2^I$) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on $S^1_{boldsymbol{0}}=boldsymbol{B}^1_{[0,infty)}cup{boldsymbol{0}}$ (resp. $S^2_{boldsymbol{0}}=boldsymbol{B}^2_{[0,infty)}cup{boldsymbol{0}}$) with an adjoined zero $boldsymbol{0}$ is either homeomorphic to the one-point Alexandroff compactification of the topological space $boldsymbol{B}^1_{[0,infty)}$ (resp. $boldsymbol{B}^2_{[0,infty)}$) or zero is an isolated point of $S^1_{boldsymbol{0}}$ (resp. $S^2_{boldsymbol{0}}$).Also, we proved that if $S_{mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $boldsymbol{B}^{mathfrak{d}}_{[0,infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{mathfrak{d}}^I$.
让 $[0,infty)$ 是所有非负实数的集合。集合 $boldsymbol{B}_{[0,infty)}=[0,infty)times[0,infty)$具有如下二元运算 $(a,b)(c,d)=(a+c-min{b,c},b+d-min{b,c})$ 是一个双简单逆半群。本文将研究具有以下树型的邻接紧凑理想的半群 $boldsymbol{B}_{[0,infty)}$ 上的 Hausdorff 局部紧凑移位连续拓扑。半群 $boldsymbol{B}_{[0,infty)}$ 具有来自 $mathbb{R}^2$ 的诱导通常拓扑 $tau_u$,拓扑 $tau_L$是由逆半群 $boldsymbol{B}_{[0、和离散拓扑分别用 $boldsymbol{B}^1_{[0,infty)}$, $boldsymbol{B}^2_{[0,infty)}$ 和 $boldsymbol{B}^{mathfrak{d}}_{[0,infty)}$ 表示。我们证明,如果 $S_1^I$ ($S_2^I$) 是一个 Hausdorff 局部紧凑半拓扑半群 $boldsymbol{B}^1_{[0,infty)}$ ($boldsymbol{B}^2_{[0、infty)}$)有一个邻接的紧凑理想 $I$,那么要么 $I$ 是 $S_1^I$ ($S_2^I$) 的开放子集,要么拓扑空间 $S_1^I$ ($S_2^I$) 是紧凑的。作为推论,我们可以得到在 $S^1_{boldsymbol{0}}=boldsymbol{B}^1_{[0,infty)}cup{boldsymbol{0}}$(res.$S^2_{boldsymbol{0}}=boldsymbol{B}^2_{[0,infty)}/cup{boldsymbol{0}/}$)有一个邻接零$boldsymbol{0}$要么与拓扑空间$boldsymbol{B}^1_{[0,infty)}$的一点亚历山德罗夫压缩同构(res.或零是 $S^1_{boldsymbol{0}}$ (即 $S^2_{boldsymbol{0}}$)的孤立点。此外,我们还证明了如果 $S_{mathfrak{d}}^I$ 是一个具有邻接紧凑理想 $I$ 的 Hausdorff 局部紧凑半拓扑半群 $boldsymbol{B}^{mathfrak{d}}_{[0,infty)}$,那么 $I$ 是 $S_{mathfrak{d}}^I$ 的一个开放子集。
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引用次数: 0
On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem 论从迪里希特边界问题的频谱中恢复量子树的形状
Q3 Mathematics Pub Date : 2023-12-18 DOI: 10.30970/ms.60.2.162-172
O. Boyko, O. Martynyuk, V. Pivovarchik
Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. It is proved that there are no co-spectral (i.e., having the same spectrum of such problem) among equilateral trees of $leq 8$ vertices. All co-spectral trees of $9$ vertices are presented.
考虑了等边树上的 Sturm-Liouville 方程所产生的谱问题,在垂顶处有 Dirichlet 边界条件,在内侧顶点处有连续性和 Kirchhoff 条件。研究证明,顶点为 $leq 8$ 的等边树之间不存在共谱(即具有相同的问题谱)。提出了所有顶点为 $9$ 的共谱树。
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引用次数: 0
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Matematychni Studii
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