Pub Date : 2024-03-27DOI: 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.
{"title":"On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series","authors":"Andrii Bodnarchuk, Yu.M. Gal', O. Skaskiv","doi":"10.30970/ms.61.1.109-112","DOIUrl":"https://doi.org/10.30970/ms.61.1.109-112","url":null,"abstract":"We consider the class $S(lambda,beta,tau)$ of convergent for all $xge0$ \u0000Taylor-Dirichlet type series of the form \u0000$$F(x) =sum_{n=0}^{+infty}{b_ne^{xlambda_n+tau(x)beta_n}}, \u0000b_ngeq 0 (ngeq 0),$$ \u0000 where $taucolon [0,+infty)to \u0000(0,+infty)$ is a continuously differentiable non-decreasing function, \u0000$lambda=(lambda_n)$ and $beta=(beta_n)$ are such that $lambda_ngeq 0, beta_ngeq 0$ $(ngeq 0)$. \u0000In 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 \u0000$displaystylesum_{n=0}^{+infty}frac1{lambda_{n+1}-lambda_n}<+infty$ \u0000and for any non-decreasing sequence $beta=(beta_n)$ such that \u0000$beta_{n+1}-beta_nlelambda_{n+1}-lambda_n$ $(ngeq 0)$ \u0000there 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 \u0000the maximal term of the series. \u0000 \u0000At the same time, we also pose some open questions and formulate one conjecture.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140377471","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 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）。
{"title":"Almost periodic distributions and crystalline measures","authors":"V. MatematychniStudii., No 61, S. Favorov","doi":"10.30970/ms.61.1.97-108","DOIUrl":"https://doi.org/10.30970/ms.61.1.97-108","url":null,"abstract":"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). \u0000We 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).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140388602","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 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 $drightarrow infty$.
{"title":"Real univariate polynomials with given signs of coefficients and simple real roots","authors":"V. MatematychniStudii., No 61, V. P. Kostov","doi":"10.30970/ms.61.1.22-34","DOIUrl":"https://doi.org/10.30970/ms.61.1.22-34","url":null,"abstract":"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 $drightarrow infty$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389700","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 $h>0$, $alphain [0,h)$ 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.
{"title":"On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane","authors":"M. Sheremeta","doi":"10.30970/ms.61.1.35-50","DOIUrl":"https://doi.org/10.30970/ms.61.1.35-50","url":null,"abstract":" For $h>0$, $alphain [0,h)$ and $muin {mathbb R}$ denote by $SD_h(mu, alpha)$ a class \u0000of absolutely convergent in the half-plane $Pi_0={s:, text{Re},s<0}$ Dirichlet series \u0000$F(s)=e^{sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ such that \u0000 \u0000smallskipcenterline{$text{Re}left{frac{(mu-1)F'(s)-mu F''(s)/h}{(mu-1)F(s)-mu F'(s)/h}right}>alpha$ for all $sin Pi_0$,} \u0000 \u0000smallskipnoi and \u0000let $Sigma D_h(mu, alpha)$ be a class of absolutely convergent in half-plane $Pi_0$ Dirichlet series \u0000$F(s)=e^{-sh}+sum_{k=1}^{infty}f_kexp{slambda_k}$ such that \u0000 \u0000smallskipcenterline{$text{Re}left{frac{(mu-1)F'(s)+mu F''(s)/h}{(mu-1)F(s)+mu F'(s)/h}right}<-alpha$ for all $sin Pi_0$.} \u0000 \u0000smallskipnoi \u0000Then $SD_h(0, alpha)$ consists of pseudostarlike functions of order $alpha$ and $SD_h(1, alpha)$ consists of pseudoconvex functions of order $alpha$. \u0000 \u0000For 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 \u0000$SD_h(mu, alpha)$, it is \u0000sufficient, and in the case when $f_k(mulambda_k/h-mu+1)le 0$ for all $kge 1$, it is necessary that} \u0000 \u0000smallskipcenterline{$ \u0000sumlimits_{k=1}^{infty}big|f_kbig(frac{mulambda_k}{h}-mu+1big)big|(lambda_k-alpha)le h-alpha,$} \u0000 \u0000noi {where $h>0, alphain [0, h)$ (Theorem 1).} \u0000 \u0000noi 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 \u0000sufficient, and in the case when $f_k(mulambda_k/h+mu-1)le 0$ for all $kge 1$, it is necessary that \u0000 \u0000smallskipcenterline{$sumlimits_{k=1}^{infty}big|f_kbig(frac{mulambda_k}{h}+mu-1big)big|(lambda_k+alpha)le h-alpha,$} \u0000 \u0000noi 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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389869","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}
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。
{"title":"Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$","authors":"R. Dmytryshyn, C. Cesarano, I.-A.V. Lutsiv, M. Dmytryshyn","doi":"10.30970/ms.61.1.51-60","DOIUrl":"https://doi.org/10.30970/ms.61.1.51-60","url":null,"abstract":"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389868","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 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.
{"title":"Existence of basic solutions of first order linear homogeneous set-valued differential equations","authors":"A. Plotnikov, T. A. Komleva, N. Skripnik","doi":"10.30970/ms.61.1.61-78","DOIUrl":"https://doi.org/10.30970/ms.61.1.61-78","url":null,"abstract":"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389551","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 $T_q=-d^2/dx^2 +q$ 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.
{"title":"Reflectionless Schrodinger operators and Marchenko parametrization","authors":"Ya. Mykytyuk, N. Sushchyk","doi":"10.30970/ms.61.1.79-83","DOIUrl":"https://doi.org/10.30970/ms.61.1.79-83","url":null,"abstract":"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389470","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 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.
{"title":"Monogenic free inverse semigroups and partial automorphisms of regular rooted trees","authors":"E. Kochubinska, A. Oliynyk","doi":"10.30970/ms.61.1.3-9","DOIUrl":"https://doi.org/10.30970/ms.61.1.3-9","url":null,"abstract":"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. \u0000We 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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140389718","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 $[0,infty)$ 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 $S_{mathfrak{d}}^I$.
{"title":"On locally compact shift continuous topologies on the semigroup $boldsymbol{B}_{[0,infty)}$ with an adjoined compact ideal","authors":"O. Gutik, Markian Khylynskyi","doi":"10.30970/ms.61.1.10-21","DOIUrl":"https://doi.org/10.30970/ms.61.1.10-21","url":null,"abstract":"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 $S_{mathfrak{d}}^I$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140509159","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}
Pub Date : 2023-12-18DOI: 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.
{"title":"On recovering the shape of a quantum tree from the spectrum of the Dirichlet boundary problem","authors":"O. Boyko, O. Martynyuk, V. Pivovarchik","doi":"10.30970/ms.60.2.162-172","DOIUrl":"https://doi.org/10.30970/ms.60.2.162-172","url":null,"abstract":"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138965346","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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