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Normal Properties of Numbers in Terms of their Representation by the Perron Series 用Perron级数表示数的正规性质
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s11253-023-02246-y
Mykola Moroz
{"title":"Normal Properties of Numbers in Terms of their Representation by the Perron Series","authors":"Mykola Moroz","doi":"10.1007/s11253-023-02246-y","DOIUrl":"https://doi.org/10.1007/s11253-023-02246-y","url":null,"abstract":"<p>We study the representation of real numbers by Perron series (<i>P</i>-representation) given by\u0000</p><span>$$left(left.0;1right]ni x=sum_{n=0}^{infty }frac{{r}_{0}{r}_{1}dots {r}_{n}}{left({p}_{1}-1right){p}_{1}dots left({p}_{n}-1right){p}_{n}{p}_{n+1}}={Delta }_{{p}_{1}{p}_{2}dots }^{P}right.,$$</span><p>where <i>r</i><sub><i>n</i></sub>, <i>p</i><sub><i>n</i></sub> ∈ ℕ, <i>p</i><sub><i>n</i>+1</sub> ≥ <i>r</i><sub><i>n</i></sub> + 1, and its transcoding (<span>(overline{P })</span>-representation)\u0000</p><span>$${x=Delta }_{{g}_{1}{g}_{2}dots }^{overline{P} },$$</span><p>where <i>g</i><sub><i>n</i></sub> = <i>p</i><sub><i>n</i></sub> − <i>r</i><sub><i>n−</i>1</sub><i>.</i> We establish the properties of <span>(overline{P })</span>-representations typical of almost all numbers with respect to the Lebesgue measure (normal properties of the representations of numbers). We also examine the conditions of existence of the frequency of a digit <i>i</i> in the <span>(overline{P })</span>-representation of a number <span>({x=Delta }_{{g}_{1}{g}_{2}dots {g}_{2}dots }^{overline{P} })</span> defined by the equality\u0000</p><span>$${nu }_{i}^{overline{P} }left(xright)=underset{kto infty }{mathrm{lim}}frac{{N}_{i}^{overline{P} }left(x,kright)}{k},$$</span><p>where <span>({N}_{i}^{overline{P} }left(x,kright))</span> denotes the amount of numbers <i>n</i> such that <i>g</i><sub><i>n</i></sub> = <i>i</i> and <i>n</i> ≤ <i>k.</i> In particular, we establish conditions under which the frequency <span>({nu }_{i}^{overline{P} }left(xright))</span> exists and is constant for almost all <i>x</i> ∈ (0; 1]<i>.</i> In addition, we also determine the conditions under which the digits in <span>(overline{P })</span>-representations are encountered finitely or infinitely many times for almost all numbers from (0; 1]<i>.</i></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516381","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
Almost Everywhere Convergence of T Means with Respect to the Vilenkin System of Integrable Functions 对于可积函数的维伦金系统，T几乎处处收敛
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-25 DOI: 10.1007/s11253-023-02247-x
N. Nadirashvili
{"title":"Almost Everywhere Convergence of T Means with Respect to the Vilenkin System of Integrable Functions","authors":"N. Nadirashvili","doi":"10.1007/s11253-023-02247-x","DOIUrl":"https://doi.org/10.1007/s11253-023-02247-x","url":null,"abstract":"<p>We prove and discuss some new weak-type (1,1) inequalities for the maximal operators of <i>T</i> means with respect to the Vilenkin system generated by monotonic coefficients. We also apply the accumulated results to prove that these <i>T</i> means are almost everywhere convergent. As applications, we present both some well-known and new results.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"20 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516375","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
Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative 具有Riemann-Liouville导数的分数阶Schrödinger方程的时变源识别问题
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-25 DOI: 10.1007/s11253-023-02243-1
Ravshan Ashurov, Marjona Shakarova
{"title":"Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative","authors":"Ravshan Ashurov, Marjona Shakarova","doi":"10.1007/s11253-023-02243-1","DOIUrl":"https://doi.org/10.1007/s11253-023-02243-1","url":null,"abstract":"<p>We consider a Schrödinger equation <span>(i{partial }_{t}^{rho }uleft(x,tright)-{u}_{xx}left(x,tright)=pleft(tright)qleft(xright)+fleft(x,tright),0&lt;tle T,0&lt;rho &lt;1,)</span> with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with <i>u</i>(<i>x, t</i>)<i>,</i> a time-dependent factor <i>p</i>(<i>t</i>) of the source function is also unknown. To solve this inverse problem, we use an additional condition <i>B</i>[<i>u</i>(<i>∙, t</i>)] =<i>ψ</i>(<i>t</i>) with an arbitrary bounded linear functional <i>B.</i> The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method makes it possible to study a similar problem by taking, instead of <i>d</i><sup>2</sup><i>/dx</i><sup>2</sup><i>,</i> an arbitrary elliptic differential operator <i>A</i>(<i>x,D</i>) with compact inverse.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"9 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516376","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: 4
Univalence Criteria for Locally Univalent Analytic Functions 局部一元解析函数的一元准则
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-25 DOI: 10.1007/s11253-023-02250-2
Zhenyong Hu, Jinhua Fan, Xiaoyuan Wang
{"title":"Univalence Criteria for Locally Univalent Analytic Functions","authors":"Zhenyong Hu, Jinhua Fan, Xiaoyuan Wang","doi":"10.1007/s11253-023-02250-2","DOIUrl":"https://doi.org/10.1007/s11253-023-02250-2","url":null,"abstract":"<p>Suppose that <i>p</i>(<i>z</i>) = 1 + <i>zϕ″</i>(<i>z</i>)<i>/ϕ′</i>(<i>z</i>), where <i>ϕ</i>(<i>z</i>) is a locally univalent analytic function in the unit disk <b>D</b> with <i>ϕ</i>(0) = <i>ϕ′</i>(1) <i>−</i> 1 = 0<i>.</i> We establish the lower and upper bounds for the best constants <i>σ</i><sub>0</sub> and <i>σ</i><sub>1</sub> such that <span>({e}^{{-sigma }_{0}/2}&lt;left|pleft(zright)right|&lt;{e}^{{sigma }_{0}/2})</span> and |<i>p</i>(<i>w</i>)/<i>p</i>(<i>z</i>)| &lt; <span>({e}^{{sigma }_{1}})</span> for <i>z</i>, <i>w</i> ∈ <b>D</b>, respectively, imply the univalence of <i>ϕ</i>(<i>z</i>) in <b>D.</b></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516373","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 of Generalized Poisson Integrals by Interpolating Trigonometric Polynomials 用插值三角多项式逼近广义泊松积分
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-25 DOI: 10.1007/s11253-023-02248-w
Anatolii Serdyuk, Tetyana Stepanyuk
{"title":"Approximation of Generalized Poisson Integrals by Interpolating Trigonometric Polynomials","authors":"Anatolii Serdyuk, Tetyana Stepanyuk","doi":"10.1007/s11253-023-02248-w","DOIUrl":"https://doi.org/10.1007/s11253-023-02248-w","url":null,"abstract":"<p>We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for 2<i>π</i>-periodic functions <i>f</i> that can be represented in the form of generalized Poisson integrals of functions <i>φ</i> from the space <i>L</i><sub><i>p</i></sub>, 1 ≤ <i>p</i> ≤ ∞<i>.</i> In these inequalities, the moduli of deviations of the interpolation Lagrange polynomials <span>(left|fleft(xright)-{widetilde{S}}_{n-1}left(f;xright)right|)</span> for every <i>x</i> ∈ ℝ are expressed via the best approximations <span>({E}_{n}{left(varphi right)}_{{L}_{p}})</span> of the functions <i>φ</i> by trigonometric polynomials in the <i>L</i><sub><i>p</i></sub>-metrics. We also deduce asymptotic equalities for the exact upper bounds of pointwise approximations of the generalized Poisson integrals of functions that belong to the unit balls in the spaces <i>L</i><sub><i>p</i></sub>, 1 ≤ <i>p</i> ≤ ∞, by interpolating trigonometric polynomials on the classes <span>({C}_{beta ,p}^{alpha ,r})</span>.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"48 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516377","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
Monotone Generalized α-Nonexpansive Mappings on CAT_p(0) Spaces CAT_p(0) 空间上的单调广义 α-无穷映射
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-25 DOI: 10.1007/s11253-023-02249-9
Emirhan Hacıoğlu, Faik Gürsoy, Abdul Rahim Khan
{"title":"Monotone Generalized α-Nonexpansive Mappings on CAT_p(0) Spaces","authors":"Emirhan Hacıoğlu, Faik Gürsoy, Abdul Rahim Khan","doi":"10.1007/s11253-023-02249-9","DOIUrl":"https://doi.org/10.1007/s11253-023-02249-9","url":null,"abstract":"","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"59 5","pages":"1108-1127"},"PeriodicalIF":0.5,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139237715","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 the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum 广义Dedekind和及若干由Kloosterman和加权的广义Hardy和的均值
4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-11 DOI: 10.1007/s11253-023-02234-2
Muhammet Cihat Dağlı, Hamit Sever
{"title":"On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum","authors":"Muhammet Cihat Dağlı, Hamit Sever","doi":"10.1007/s11253-023-02234-2","DOIUrl":"https://doi.org/10.1007/s11253-023-02234-2","url":null,"abstract":"","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"17 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135042890","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
Evolutionary Pseudodifferential Equations with Smooth Symbols in S-Type Spaces s型空间中具有光滑符号的进化伪微分方程
4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-11 DOI: 10.1007/s11253-023-02233-3
Vasyl Horodets’kyi, Roman Petryshyn, Olha Martynyuk
{"title":"Evolutionary Pseudodifferential Equations with Smooth Symbols in S-Type Spaces","authors":"Vasyl Horodets’kyi, Roman Petryshyn, Olha Martynyuk","doi":"10.1007/s11253-023-02233-3","DOIUrl":"https://doi.org/10.1007/s11253-023-02233-3","url":null,"abstract":"","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"53 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135041605","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
Impulsive Dirac System on Time Scales 时间尺度上的脉冲狄拉克系统
4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-11 DOI: 10.1007/s11253-023-02231-5
Bilender P. Allahverdiev, Hüseyin Tuna
{"title":"Impulsive Dirac System on Time Scales","authors":"Bilender P. Allahverdiev, Hüseyin Tuna","doi":"10.1007/s11253-023-02231-5","DOIUrl":"https://doi.org/10.1007/s11253-023-02231-5","url":null,"abstract":"","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"52 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135043133","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: 1
Some Refinements of the Hermite–Hadamard Inequality with the Help of Weighted Integrals 利用加权积分对Hermite-Hadamard不等式的一些改进
4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-11 DOI: 10.1007/s11253-023-02232-4
B. Bayraktar, J. E. Nápoles, F. Rabossi
{"title":"Some Refinements of the Hermite–Hadamard Inequality with the Help of Weighted Integrals","authors":"B. Bayraktar, J. E. Nápoles, F. Rabossi","doi":"10.1007/s11253-023-02232-4","DOIUrl":"https://doi.org/10.1007/s11253-023-02232-4","url":null,"abstract":"","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"49 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135042959","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: 1
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