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On the Variational Statement of One Boundary-Value Problem with Free Interface 论自由界面一个边值问题的变式表述
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-12-08 DOI: 10.1007/s11253-023-02260-0
Aleksander Timokha
{"title":"On the Variational Statement of One Boundary-Value Problem with Free Interface","authors":"Aleksander Timokha","doi":"10.1007/s11253-023-02260-0","DOIUrl":"https://doi.org/10.1007/s11253-023-02260-0","url":null,"abstract":"<p>With the help of Clebsch’s potentials, we propose a Bateman–Luke-type variational principle for a boundary- value problem with a free (unknown) interface between two ideal compressible barotropic fluids (liquid and gas) admitting rotational flows.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138554121","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
A Tangent Inequality Over Primes 质数上的正切不等式
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s11253-023-02245-z
S. I. Dimitrov
{"title":"A Tangent Inequality Over Primes","authors":"S. I. Dimitrov","doi":"10.1007/s11253-023-02245-z","DOIUrl":"https://doi.org/10.1007/s11253-023-02245-z","url":null,"abstract":"<p>We introduce a new Diophantine inequality with prime numbers. Let <span>(1&lt;c&lt;frac{10}{9}.)</span> We show that, for any fixed <i>θ &gt;</i> 1<i>,</i> every sufficiently large positive number <i>N,</i> and a small constant <i>ε &gt;</i> 0<i>,</i> the tangent inequality\u0000</p><span>$$left|{p}_{1}^{c} {mathrm{tan}}^{theta }left(mathrm{log}{p}_{1}right)+{p}_{2}^{c} {mathrm{tan}}^{theta }left(mathrm{log}{p}_{2}right)+{p}_{3}^{c} {mathrm{tan}}^{theta }left(mathrm{log}{p}_{3}right)-Nright|&lt;varepsilon $$</span><p>has a solution in prime numbers <i>p</i><sub>1</sub><i>, p</i><sub>2</sub><i>,</i> and <i>p</i><sub>3</sub><i>.</i></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516370","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
Determination of Some Properties of Starlike and Close-to-Convex Functions According to Subordinate Conditions with Convexity of a Certain Analytic Function 根据具有一定解析函数凸性的从属条件确定星形和近凸函数的一些性质
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s11253-023-02251-1
Hasan Şahin, İsmet Yildiz
{"title":"Determination of Some Properties of Starlike and Close-to-Convex Functions According to Subordinate Conditions with Convexity of a Certain Analytic Function","authors":"Hasan Şahin, İsmet Yildiz","doi":"10.1007/s11253-023-02251-1","DOIUrl":"https://doi.org/10.1007/s11253-023-02251-1","url":null,"abstract":"<p>Investigation of the theory of complex functions is one of the most fascinating aspects of the theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Numerous mathematical concepts are explained when viewed through the theory of complex functions. Let <span>(fleft(zright)in A, fleft(zright)=z+{sum }_{nge 2}^{infty }{a}_{n}{z}^{n},)</span> be an analytic function in an open unit disc <i>U</i> = {<i>z</i> : <i>|z| &lt;</i> 1<i>, z</i> ∈ ℂ} normalized by <i>f</i>(0) = 0 and <i>f</i>′(0) = 1<i>.</i> For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where <i>r</i> is a positive integer of order <span>({2}^{-r}left(0&lt;{2}^{-r}le frac{1}{2}right).)</span> By using subordination, we propose a criterion for <i>f</i>(<i>z</i>) ∈ <i>S</i><sup>*</sup>[<i>a</i><sup><i>r</i></sup><i>, b</i><sup><i>r</i></sup>]<i>.</i> The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity are examined for <i>f</i>(<i>z</i>) ∈ <i>A</i>.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516371","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
Weighted Discrete Hardy’s Inequalities 加权离散Hardy不等式
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s11253-023-02252-0
Pascal Lefèvre
{"title":"Weighted Discrete Hardy’s Inequalities","authors":"Pascal Lefèvre","doi":"10.1007/s11253-023-02252-0","DOIUrl":"https://doi.org/10.1007/s11253-023-02252-0","url":null,"abstract":"<p>We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, <i>Arch. Math. (Basel)</i>, <b>114</b>, No. 2, 195–198 (2020)].</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516374","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: 6
Uncertainty Principles for the q-Hankel–Stockwell Transform q-Hankel-Stockwell变换的不确定性原理
IF 0.5 4区 数学
Ukrainian Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s11253-023-02244-0
Kamel Brahim, Hédi Ben Elmonser
{"title":"Uncertainty Principles for the q-Hankel–Stockwell Transform","authors":"Kamel Brahim, Hédi Ben Elmonser","doi":"10.1007/s11253-023-02244-0","DOIUrl":"https://doi.org/10.1007/s11253-023-02244-0","url":null,"abstract":"<p>By using the <i>q</i>-Jackson integral and some elements of the <i>q</i>-harmonic analysis associated with the <i>q</i>-Hankel transform, we introduce and study a <i>q</i>-analog of the Hankel–Stockwell transform. We present some properties from harmonic analysis (Plancherel formula, inversion formula, reproducing kernel, etc.). Furthermore, we establish a version of Heisenberg’s uncertainty principles. Finally, we study the <i>q</i>-Hankel–Stockwell transform on a subset of finite measure.</p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516378","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
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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