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Fractal functions of exponential type that is generated by the $mathbf{Q_2^*}$-representation of argument 由参数的$mathbf{Q_2^*}$-表示生成的指数型分形函数
Matematychni Studii Pub Date : 2021-12-27 DOI: 10.30970/ms.56.2.133-143
M. Pratsovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak
{"title":"Fractal functions of exponential type that is generated by the $mathbf{Q_2^*}$-representation of argument","authors":"M. Pratsovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak","doi":"10.30970/ms.56.2.133-143","DOIUrl":"https://doi.org/10.30970/ms.56.2.133-143","url":null,"abstract":"We consider function $f$ which is depended on the parameters $0<ain R$, $q_{0n}in (0;1)$, $nin N$ and convergent positive series $v_1+v_2+...+v_n+...$, defined by equality $f(x=Delta^{Q_2^*}_{alpha_1alpha_2...alpha_n...})=a^{varphi(x)}$, where $alpha_nin {0,1}$, $varphi(x=Delta^{Q_2^*}_{alpha_1alpha_2...alpha_n...})=alpha_1v_1+...+alpha_nv_n+...$, $q_{1n}=1-q_{0n}$, $Delta^{Q_2^*}_{alpha_1...alpha_n...}=alpha_1q_{1-alpha_1,1}+sumlimits_{n=2}^{infty}big(alpha_nq_{1-alpha_n,n}prodlimits_{i=1}^{n-1}q_{alpha_i,i}big)$.In the paper we study structural, variational, integral, differential and fractal properties of the function $f$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43163171","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
Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane 关于Dirichlet级数在半平面上绝对收敛的极大项的性质的注记
Matematychni Studii Pub Date : 2021-12-27 DOI: 10.30970/ms.56.2.144-148
M. Sheremeta
{"title":"Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane","authors":"M. Sheremeta","doi":"10.30970/ms.56.2.144-148","DOIUrl":"https://doi.org/10.30970/ms.56.2.144-148","url":null,"abstract":"By $S_0(Lambda)$ denote a class of Dirichlet series $F(s)=sum_{n=0}^{infty}a_nexp{slambda_n} (s=sigma+it)$ withan increasing to $+infty$ sequence $Lambda=(lambda_n)$ of exponents ($lambda_0=0$) and the abscissa of absolute convergence $sigma_a=0$.We say that $Fin S_0^*(Lambda)$ if $Fin S_0(Lambda)$ and $ln lambda_n=o(ln |a_n|)$ $(ntoinfty)$. Let$mu(sigma,F)=max{|a_n|exp{(sigmalambda_n)}colon nge 0}$ be the maximal term of Dirichlet series. It is proved that in order that $ln (1/|sigma|)=o(ln mu(sigma))$ $(sigmauparrow 0)$ for every function $Fin S_0^*(Lambda)$ it is necessary and sufficient that $displaystyle varlimsuplimits_{ntoinfty}frac{ln lambda_{n+1}}{ln lambda_n}<+infty. $For an analytic in the disk ${zcolon |z|<1}$ function $f(z)=sum_{n=0}^{infty}a_n z^n$ and $rin (0, 1)$ we put $M_f(r)=max{|f(z)|colon |z|=r<1}$ and $mu_f(r)=max{|a_n|r^ncolon nge 0}$. Then from hence we get the following statement: {sl if there exists a sequence $(n_j)$ such that $ln n_{j+1}=O(ln n_{j})$ and $ln n_{j}=o(ln |a_{n_{j}}|)$ as $jtoinfty$,  then the functions $ln mu_f(r)$ and $ln M_f(r)$ are or not are slowly increasing simultaneously.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42074305","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
Repdigits as difference of two Fibonacci or Lucas numbers 将数字表示为两个斐波那契数或卢卡斯数之差
Matematychni Studii Pub Date : 2021-12-26 DOI: 10.30970/ms.56.2.124-132
P. Ray, K. Bhoi
{"title":"Repdigits as difference of two Fibonacci or Lucas numbers","authors":"P. Ray, K. Bhoi","doi":"10.30970/ms.56.2.124-132","DOIUrl":"https://doi.org/10.30970/ms.56.2.124-132","url":null,"abstract":"In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)in {(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)in{(6,4),(7,4),(7,6),(8,2)},$ where $n>m.$Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},33=F_{9}-F_{1}=F_{9}-F_{2},55=F_{11}-F_{9}=F_{12}-F_{11},88=F_{11}-F_{1}=F_{11}-F_{2},555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6}, 22=L_{7}-L_{4},4=L_{8}-L_{2}$ (Theorem 3).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45082100","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
Singularly perturbed rank one linear operators 奇摄动秩一线性算子
Matematychni Studii Pub Date : 2021-12-26 DOI: 10.30970/ms.56.2.162-175
M. Dudkin, O. Dyuzhenkova
{"title":"Singularly perturbed rank one linear operators","authors":"M. Dudkin, O. Dyuzhenkova","doi":"10.30970/ms.56.2.162-175","DOIUrl":"https://doi.org/10.30970/ms.56.2.162-175","url":null,"abstract":"The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${mathcal H}_{-1}$-class and ${mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${mathcal H}_{-2}$ operator can be conveniently described by methods of class ${mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45709876","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
Fermat and Mersenne numbers in $k$-Pell sequence $k$-Pell序列中的Fermat和Mersenne数
Matematychni Studii Pub Date : 2021-12-26 DOI: 10.30970/ms.56.2.115-123
B. Normenyo, S. Rihane, A. Togbé
{"title":"Fermat and Mersenne numbers in $k$-Pell sequence","authors":"B. Normenyo, S. Rihane, A. Togbé","doi":"10.30970/ms.56.2.115-123","DOIUrl":"https://doi.org/10.30970/ms.56.2.115-123","url":null,"abstract":"For an integer $kgeq 2$, let $(P_n^{(k)})_{ngeq 2-k}$ be the $k$-generalized Pell sequence, which starts with $0,ldots,0,1$ ($k$ terms) and each term afterwards is defined by the recurrence$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+cdots +P_{n-k}^{(k)},quad text{for all }n geq 2.$For any positive integer $n$, a number of the form $2^n+1$ is referred to as a Fermat number, while a number of the form $2^n-1$ is referred to as a Mersenne number. The goal of this paper is to determine Fermat and Mersenne numbers which are members of the $k$-generalized Pell sequence. More precisely, we solve the Diophantine equation $P^{(k)}_n=2^apm 1$ in positive integers $n, k, a$ with $k geq 2$, $ageq 1$. We prove a theorem which asserts that, if the Diophantine equation $P^{(k)}_n=2^apm 1$ has a solution $(n,a,k)$ in positive integers $n, k, a$ with $k geq 2$, $ageq 1$, then we must have that $(n,a,k)in {(1,1,k),(3,2,k),(5,5,3)}$. As a result of our theorem, we deduce that the number $1$ is the only Mersenne number and the number $5$ is the only Fermat number in the $k$-Pell sequence.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43910415","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
On pseudobounded and premeage paratopological groups 关于伪有界和准拓扑群
Matematychni Studii Pub Date : 2021-10-27 DOI: 10.30970/ms.56.1.20-27
A. Ravsky, T. Banakh
{"title":"On pseudobounded and premeage paratopological groups","authors":"A. Ravsky, T. Banakh","doi":"10.30970/ms.56.1.20-27","DOIUrl":"https://doi.org/10.30970/ms.56.1.20-27","url":null,"abstract":"Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$ is a natural number.The group $G$ is premeager, if $Gne N^n$ for any nowhere dense subset $N$ of$G$ and any positive integer $n$.In this paper we investigate relations between the above classes of groups andanswer some questions posed by F. Lin, S. Lin, and S'anchez.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41844990","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
Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients 一类指数系数微分方程的拟星解和拟凸解
Matematychni Studii Pub Date : 2021-10-27 DOI: 10.30970/ms.56.1.39-47
M. Sheremeta
{"title":"Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients","authors":"M. Sheremeta","doi":"10.30970/ms.56.1.39-47","DOIUrl":"https://doi.org/10.30970/ms.56.1.39-47","url":null,"abstract":"Dirichlet series $F(s)=e^{s}+sum_{k=1}^{infty}f_ke^{slambda_k}$ with the exponents $1<lambda_kuparrow+infty$ and the abscissa of absolute convergence $sigma_a[F]ge 0$ is said to be pseudostarlike of order $alphain [0,,1)$ and type $beta in (0,,1]$ if$left|dfrac{F'(s)}{F(s)}-1right|<betaleft|dfrac{F'(s)}{F(s)}-(2alpha-1)right|$ for all $sin Pi_0={scolon ,text{Re},s<0}$. Similarly, the function $F$ is said to be pseudoconvex of order $alphain [0,,1)$ and type $beta in (0,,1]$ if$left|dfrac{F''(s)}{F'(s)}-1right|<betaleft|dfrac{F''(s)}{F'(s)}-(2alpha-1)right|$ for all $sin Pi_0$. Some conditions are found on the parameters $b_0,,b_1,,c_0,,c_1,,,c_2$ and the coefficients $a_n$, under which the differential equation $dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=sumlimits_{n=1}^{infty}a_ne^{ns}$has an entire solution which is pseudostarlike or pseudoconvex of order $alphain [0,,1)$ and type $beta in (0,,1]$. It is proved that by some conditions for such solution the asymptotic equality holds  $ln,max{|F(sigma+it)|colon tin {mathbb R}}=dfrac{1+o(1)}{2}left(|b_0|+sqrt{|b_0|^2+4|c_0|}right)$ as $sigma to+infty$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43522455","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
Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations 一类非线性方程组七阶无导数算法的扩展球收敛性
Matematychni Studii Pub Date : 2021-10-23 DOI: 10.30970/ms.56.1.72-82
I. Argyros, Debasis Sharma, C. Argyros, S. K. Parhi, S. K. Sunanda, M. Argyros
{"title":"Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations","authors":"I. Argyros, Debasis Sharma, C. Argyros, S. K. Parhi, S. K. Sunanda, M. Argyros","doi":"10.30970/ms.56.1.72-82","DOIUrl":"https://doi.org/10.30970/ms.56.1.72-82","url":null,"abstract":"In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighthorder have been utilized to establish the convergence of a derivative free class of seventh orderiterative algorithms. Moreover, no error distances or results on uniqueness of the solution weregiven. In this study, extended ball convergence analysis is derived for this class by imposingconditions on the first derivative. Additionally, we offer error distances and convergence radiustogether with the region of uniqueness for the solution. Therefore, we enlarge the practicalutility of these algorithms. Also, convergence regions of a specific member of this class are displayedfor solving complex polynomial equations. At the end, standard numerical applicationsare provided to illustrate the efficacy of our theoretical findings.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45537462","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
Induced mappings on $C_n(X)/{C_n}_K(X)$ $C_n(X)/{C_n}_K(X)$上的诱导映射
Matematychni Studii Pub Date : 2021-10-23 DOI: 10.30970/ms.56.1.83-95
E. Castañeda-Alvarado, J. G. Anaya, J. A. Martínez-Cortéz
{"title":"Induced mappings on $C_n(X)/{C_n}_K(X)$","authors":"E. Castañeda-Alvarado, J. G. Anaya, J. A. Martínez-Cortéz","doi":"10.30970/ms.56.1.83-95","DOIUrl":"https://doi.org/10.30970/ms.56.1.83-95","url":null,"abstract":"Given a continuum $X$ and $ninmathbb{N}$. Let $C_n(X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components. Let ${C_n}_K(X)$ be the hyperspace of all elements in $C_n(X)$ containing $K$ where $K$ is a compact subset of $X$. $C^n_K(X)$ denotes the quotient space $C_n(X)/{C_n}_K(X)$. Given a mapping $f:Xto Y$ between continua, let $C_n(f):C_n(X)to C_n(Y)$ be the induced mapping by $f$, defined by $C_n(f)(A)=f(A)$. We denote the natural induced mapping between $C^n_K(X)$ and $C^n_{f(K)}(Y)$ by $C^n_K(f)$. In this paper, we study relationships among the mappings $f$, $C_n(f)$ and $C^n_K(f)$ for the following classes of mappings: almost monotone, atriodic, confluent, joining, light, monotone, open, OM, pseudo-confluent, quasi-monotone, semi-confluent, strongly freely decomposable, weakly confluent, and weakly monotone.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69301843","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
The renewal equation in nonlinear approximation 非线性近似中的更新方程
Matematychni Studii Pub Date : 2021-10-23 DOI: 10.30970/ms.56.1.103-106
O. Yarova, Y. Yeleyko
{"title":"The renewal equation in nonlinear approximation","authors":"O. Yarova, Y. Yeleyko","doi":"10.30970/ms.56.1.103-106","DOIUrl":"https://doi.org/10.30970/ms.56.1.103-106","url":null,"abstract":"The family of Markov processes are considered in the article. We study the multidimensional renewal equation in nonlinear approximation. The purpose of the work is to find the limit of renewal function.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44005053","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
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