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The Norming Sets of $$mathcal{L}left({}^{m}{l}_{1}^{n}right)$$ $$mathcal{L}left({}^{m}{l}_{1}^{n}right)$$ 的规范集
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-09-06 DOI: 10.1007/s11253-024-02329-4
Sung Guen Kim
{"title":"The Norming Sets of $$mathcal{L}left({}^{m}{l}_{1}^{n}right)$$","authors":"Sung Guen Kim","doi":"10.1007/s11253-024-02329-4","DOIUrl":"https://doi.org/10.1007/s11253-024-02329-4","url":null,"abstract":"<p>Let <i>n</i> ∈ ℕ, <i>n</i> ≥ 2<i>.</i> An element (<i>x</i><sub>1</sub>,…,<i>x</i><sub><i>n</i></sub>) ∈ <i>E</i><sub><i>n</i></sub> is called a <i>norming point</i> of <i>T</i> ∈ <span>(mathcal{L}left({}^{n}Eright))</span> if ||<i>x</i><sub>1</sub>|| = <i>…</i> = ||<i>x</i><sub><i>n</i></sub>|| = 1 and <i>|T</i>(<i>x</i><sub>1</sub>,<i>…</i>,<i>x</i><sub><i>n</i></sub>)<i>|</i> = ||<i>T</i>||, where ℒ(<sup><i>n</i></sup><i>E</i>) denotes the space of all continuous <i>n</i>-linear forms on <i>E.</i> For <i>T</i> ∈ ℒ (<sup><i>n</i></sup><i>E</i>), we define\u0000</p><span>$$text{Norm}left(Tright)=left{left({x}_{1},dots ,{x}_{n}right)in {E}^{n}:left({x}_{1},dots ,{x}_{n}right)text{ is a norming point of }Tright}.$$</span><p>The set Norm(<i>T</i>) is called the <i>norming set</i> of <i>T.</i> For <i>m</i> ∈ ℕ<i>, m</i> ≥ 2, we characterize Norm(<i>T</i>) for any <i>T</i> ∈ <span>(mathcal{L}left({}^{m}{l}_{1}^{n}right))</span>, where <span>({l}_{1}^{n}={mathbb{R}}^{n})</span> with the <i>l</i><sub>1</sub>-norm. As applications, we classify Norm(<i>T</i>) for every <i>T</i> ∈ <span>(mathcal{L}left({}^{m}{l}_{1}^{n}right))</span> with <i>n</i> = 2, 3 and <i>m</i> = 2<i>.</i></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"12 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some New Cesàro Sequence Spaces of Order α 一些新的α阶塞萨罗序列空间
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-09-06 DOI: 10.1007/s11253-024-02333-8
Medine Yeşilkayagil Savaşcı, Feyzi Başar
{"title":"Some New Cesàro Sequence Spaces of Order α","authors":"Medine Yeşilkayagil Savaşcı, Feyzi Başar","doi":"10.1007/s11253-024-02333-8","DOIUrl":"https://doi.org/10.1007/s11253-024-02333-8","url":null,"abstract":"<p>We introduce the spaces ℓ<sub>∞</sub>(𝒞<sub>α</sub>), <i>f</i>(𝒞<sub>α</sub>), and <i>f</i><sub>0</sub>(𝒞<sub>α</sub>) of Cesàro bounded, Cesàro almost convergent, and Cesàro almost null sequences of order α<i>,</i> respectively. Moreover, we establish some inclusion relations for these spaces and determine the α -, <i>β</i>- and <i>γ</i>-duals of the spaces ℓ<sub>∞</sub> (𝒞<sub>α</sub>) and <i>f</i>(𝒞<sub>α</sub>)<i>.</i> Finally, we characterize the classes of matrix transformations from the space <i>f</i>(𝒞<sub>α</sub>) to any sequence space <i>Y</i> and from any sequence space <i>Y</i> to the space <i>f</i>(𝒞<sub>α</sub>)<i>.</i></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence and Uniqueness of Solution for a Parabolic-Hyperbolic Equation with Multiplicative Control and Nonlocal Boundary Conditions 具有乘法控制和非局部边界条件的抛物线-超双曲方程的解的存在性和唯一性
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-17 DOI: 10.1007/s11253-024-02320-z
Volodymyr Kapustyan, Ivan Pyshnograiev
{"title":"Existence and Uniqueness of Solution for a Parabolic-Hyperbolic Equation with Multiplicative Control and Nonlocal Boundary Conditions","authors":"Volodymyr Kapustyan, Ivan Pyshnograiev","doi":"10.1007/s11253-024-02320-z","DOIUrl":"https://doi.org/10.1007/s11253-024-02320-z","url":null,"abstract":"<p>We consider a parabolic-hyperbolic equation with multiplicative control and nonlocal boundary conditions. By using the Riesz biorthogonal basis, the problem is reduced to a sequence of one-dimensional problems with alternative representations of their solutions. Conditions guaranteeing the existence and uniqueness of the solution to the analyzed problem are established.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"264 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Primes of the form [nc] with Square-Free n 无平方 n 的 [nc] 形式的素数
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-17 DOI: 10.1007/s11253-024-02318-7
S. I. Dimitrov
{"title":"Primes of the form [nc] with Square-Free n","authors":"S. I. Dimitrov","doi":"10.1007/s11253-024-02318-7","DOIUrl":"https://doi.org/10.1007/s11253-024-02318-7","url":null,"abstract":"<p>Let [·] be the floor function. We show that if 1 &lt;<i> c </i>&lt; <span>(frac{3849}{3334})</span><i>,</i> then there exist infinitely many prime numbers of the form [<i>n</i><sup><i>c</i></sup>]<i>,</i> where <i>n</i> is square free.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"51 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Turán-Type Inequalities for Generalized k-Bessel Functions 广义 k-贝塞尔函数的图兰式不等式
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-17 DOI: 10.1007/s11253-024-02319-6
Hanaa M. Zayed
{"title":"Turán-Type Inequalities for Generalized k-Bessel Functions","authors":"Hanaa M. Zayed","doi":"10.1007/s11253-024-02319-6","DOIUrl":"https://doi.org/10.1007/s11253-024-02319-6","url":null,"abstract":"<p>We propose an approach to the generalized k-Bessel function defined by</p><p><span>({text{U}}_{p,q,r}^{text{k}}left(zright)=sum_{n=0}^{infty }frac{{left(-rright)}^{n}}{{Gamma }_{k}left(nk+p+frac{q+1}{2}text{k}right)n!}{left(frac{z}{2}right)}^{2n+frac{p}{text{k}}},)</span></p><p>where k <i>&gt;</i> 0 and <i>p, q, r</i> ∈ <span>({mathbb{C}})</span>. We discuss the uniform convergence of <span>({text{U}}_{p,q,r}^{text{k}})</span> (<i>z</i>)<i>.</i> Moreover, we prove that the analyzed function is entire and determine its growth order and type. We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane. The integral representation for <span>({text{U}}_{p,q,r}^{text{k}})</span> (<i>z</i>) is found by using the representation for k-beta functions. We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions. In addition, some interesting properties, such as recurrence and differential relations, are demonstrated. Some of these properties can be used to establish Turán-type inequalities for this function. Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified k-Bessel function <span>({text{T}}_{p,q,1}^{text{k}})</span> defined by <span>({text{T}}_{p,q,1}^{text{k}})</span> (<i>z</i>) = <span>(i{-}^frac{p}{k}{text{U}}_{p,q,1}^{text{k}})</span> (<i>iz</i>)<i>,</i> as well as the quotient of the modified k-Bessel function, exponential, and k-hypergeometric functions. In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"33 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exact Solutions with Generalized Separation of Variables of the Nonlinear Heat Equation with a Source 有源非线性热方程的广义变量分离精确解
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-16 DOI: 10.1007/s11253-024-02316-9
Anatolii Barannyk, Tetyana Barannyk, Ivan Yuryk
{"title":"Exact Solutions with Generalized Separation of Variables of the Nonlinear Heat Equation with a Source","authors":"Anatolii Barannyk, Tetyana Barannyk, Ivan Yuryk","doi":"10.1007/s11253-024-02316-9","DOIUrl":"https://doi.org/10.1007/s11253-024-02316-9","url":null,"abstract":"<p>We propose a method for the construction of exact solutions to the nonlinear heat equation with a source based on the classical method of separation of variables, its generalization, and the method of reduction. We consider substitutions reducing the nonlinear heat equation to ordinary differential equations and to a system of two ordinary differential equations. The classes of exact solutions of the analyzed equation are constructed by the method of generalized separation of variables.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"13 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Multiplicative (Generalized)-(α, β)-Derivations in Prime Rings 论素数环中的乘（广义）-（α，β）-衍生
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-16 DOI: 10.1007/s11253-024-02322-x
Chirag Garg, R. K. Sharma
{"title":"On Multiplicative (Generalized)-(α, β)-Derivations in Prime Rings","authors":"Chirag Garg, R. K. Sharma","doi":"10.1007/s11253-024-02322-x","DOIUrl":"https://doi.org/10.1007/s11253-024-02322-x","url":null,"abstract":"<p>We discuss some algebraic identities related to multiplicative (generalized) derivations and multiplicative (generalized)-(<i>α</i>, <i>β</i>)-derivations on appropriate subsets in prime rings.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Parametric 2-Decompositions in Complete Linear Groups of Small Order Over a Field 域上小阶完全线性群中的参数二分解
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-16 DOI: 10.1007/s11253-024-02323-w
Volodymyr Shchedryk
{"title":"Parametric 2-Decompositions in Complete Linear Groups of Small Order Over a Field","authors":"Volodymyr Shchedryk","doi":"10.1007/s11253-024-02323-w","DOIUrl":"https://doi.org/10.1007/s11253-024-02323-w","url":null,"abstract":"<p>We obtain a parametric description of elements of complete linear groups of the second and third orders over an arbitrary field. It is based on their canonical (single-valued) representation as a product of elements from the commutators of certain Jordan matrices and representatives of the left cosets of these groups.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximation in the Mean for the Classes Of Functions in the Space L2[(0, 1); x] by The Fourier–Bessel Sums And Estimation of the Values of Their n-Widths 用傅里叶-贝塞尔和对空间 L2[(0, 1); x] 中的函数类的平均值进行逼近并估算其 n 宽值
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-16 DOI: 10.1007/s11253-024-02317-8
Sergii Vakarchuk, Mykhailo Vakarchuk
{"title":"Approximation in the Mean for the Classes Of Functions in the Space L2[(0, 1); x] by The Fourier–Bessel Sums And Estimation of the Values of Their n-Widths","authors":"Sergii Vakarchuk, Mykhailo Vakarchuk","doi":"10.1007/s11253-024-02317-8","DOIUrl":"https://doi.org/10.1007/s11253-024-02317-8","url":null,"abstract":"<p>In the space <i>L</i><sub>2</sub>[(0, 1); <i>x</i>], by using a system of functions <span>({left{{widehat{J}}_{v}left({mu }_{k,v}xright)right}}_{kin {mathbb{N}}}, vge 0,)</span> orthonormal with weight <i>x</i> and formed by a Bessel function of the first kind of index <i>v</i> and its positive roots, we construct generalized finite differences of the <i>m</i>th order <span>({Delta }_{gamma left(hright)}^{m}left(fright),)</span> <i>m</i> ∈ ℕ, <i>h</i> ∈ (0, 1), and the generalized characteristics of smoothness <span>({Phi }_{gamma left(hright)}^{left(gamma right)}left(f,tright)=left(1/tright)underset{0}{overset{t}{int }}Vert {Delta }_{gamma left(tau right)}^{m}left(fright)Vert dtau .)</span> For the classes <span>({mathcal{W}}_{2}^{r,v}{Phi }_{m}^{left(gamma right)},left(uppsi right))</span> defined by using the differential operator <span>({D}_{v}^{r},)</span> the function <span>({Phi }_{m}^{left(gamma right)}left(fright),)</span> and the majorant ψ, we establish lower and upper estimates for the values of a series of <i>n</i>-widths. We established the condition for ψ, which enables us to compute the exact values of <i>n</i>-widths. To illustrate our exact results, we present several specific examples. We also consider the problems of absolute and uniform convergence of Fourier–Bessel series on the interval (0, 1)<i>.</i></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"20 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some New Estimates for Integral Inequalities and Their Applications 积分不等式的一些新估计及其应用
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2024-08-16 DOI: 10.1007/s11253-024-02315-w
B. Bayraktar, S. I. Butt, J. E. Nápoles, F. Rabossi
{"title":"Some New Estimates for Integral Inequalities and Their Applications","authors":"B. Bayraktar, S. I. Butt, J. E. Nápoles, F. Rabossi","doi":"10.1007/s11253-024-02315-w","DOIUrl":"https://doi.org/10.1007/s11253-024-02315-w","url":null,"abstract":"<p>We obtain several new integral inequalities in terms of fractional integral operators for the functions whose first derivatives satisfy either the conditions of the Lagrange theorem or the Lipschitz condition. In some special cases, the obtained results provide better upper estimates than the results known in the literature for the Bullen-type inequality and the Hadamard-type right-hand side inequality. Finally, some error estimates for the trapezoidal formula are discussed.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"2 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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