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Global well-posedness for the higher order non-linear Schrödinger equation in modulation spaces 调制空间中高阶非线性薛定谔方程的全局拟合性
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-23 DOI: 10.1016/j.jmaa.2024.128985
X. Carvajal , P. Gamboa , R. Santos
{"title":"Global well-posedness for the higher order non-linear Schrödinger equation in modulation spaces","authors":"X. Carvajal ,&nbsp;P. Gamboa ,&nbsp;R. Santos","doi":"10.1016/j.jmaa.2024.128985","DOIUrl":"10.1016/j.jmaa.2024.128985","url":null,"abstract":"<div><div>We consider the initial value problem (IVP) associated with a higher order non-linear Schrödinger (h-NLS) equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>+</mo><mi>b</mi><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>u</mi><mo>=</mo><mn>2</mn><mi>i</mi><mi>a</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mn>6</mn><mi>b</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo>,</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>R</mi><mo>,</mo></math></span></span></span> with given data in the modulation space <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. Using ideas from Killip, Visan, Zhang, Oh and Wang, we prove that the IVP associated with the h-NLS equation is globally well-posed in the modulation spaces <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for <span><math><mi>s</mi><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> and <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators 通过邻接线性算子求解具有混合边界条件的静态线性弹性的应力解
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-23 DOI: 10.1016/j.jmaa.2024.128986
Ivan Gudoshnikov, Michal Křížek
{"title":"Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators","authors":"Ivan Gudoshnikov,&nbsp;Michal Křížek","doi":"10.1016/j.jmaa.2024.128986","DOIUrl":"10.1016/j.jmaa.2024.128986","url":null,"abstract":"<div><div>We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion 与从属布朗运动相关的薛定谔算子热含量的估计值
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128992
Luis Acuña Valverde
{"title":"Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion","authors":"Luis Acuña Valverde","doi":"10.1016/j.jmaa.2024.128992","DOIUrl":"10.1016/j.jmaa.2024.128992","url":null,"abstract":"<div><div>In this paper, we study the heat content for the Schrödinger operator related to a subordinate Brownian motion and we also establish its small time asymptotic behavior for suitable potentials <em>V</em>. The case <span><math><mi>V</mi><mo>=</mo><mi>c</mi><msub><mrow><mn>1</mn></mrow><mrow><mi>Ω</mi></mrow></msub></math></span> for <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> and Ω a Borel set of finite measure is investigated in detail.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution 随机 Kuramoto-Sivashinsky 方程的后向问题：条件稳定性和数值解
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128988
Zewen Wang , Weili Zhu , Bin Wu , Bin Hu
{"title":"A backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution","authors":"Zewen Wang ,&nbsp;Weili Zhu ,&nbsp;Bin Wu ,&nbsp;Bin Hu","doi":"10.1016/j.jmaa.2024.128988","DOIUrl":"10.1016/j.jmaa.2024.128988","url":null,"abstract":"<div><div>In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable <em>x</em> for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&lt;</mo><mi>T</mi></math></span> and when <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Optimal domain of generalized Volterra operators 广义 Volterra 算子的最优域
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128978
C. Bellavita , V. Daskalogiannis , G. Nikolaidis , G. Stylogiannis
{"title":"Optimal domain of generalized Volterra operators","authors":"C. Bellavita ,&nbsp;V. Daskalogiannis ,&nbsp;G. Nikolaidis ,&nbsp;G. Stylogiannis","doi":"10.1016/j.jmaa.2024.128978","DOIUrl":"10.1016/j.jmaa.2024.128978","url":null,"abstract":"<div><div>For <em>g</em> in BMOA, we consider the generalized Volterra operator <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> acting on Hardy spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. This article aims to study the largest space of analytic functions, which is mapped by <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> into the Hardy space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. We call this space the optimal domain of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> and we describe its structural properties. Motivation for this comes from the work of G. Curbera and W. Ricker <span><span>[7]</span></span> who studied the optimal domain of the classical Cesáro operator.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 a generalized Möbius invariant function space 关于广义莫比乌斯不变函数空间
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128979
Xiaojing Zhou
{"title":"On a generalized Möbius invariant function space","authors":"Xiaojing Zhou","doi":"10.1016/j.jmaa.2024.128979","DOIUrl":"10.1016/j.jmaa.2024.128979","url":null,"abstract":"<div><div>In this paper, we introduce a new class of generalized Möbius function space of analytic functions in the unit disk, that contains <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>, <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>, <span><math><msubsup><mrow><mtext>BMOA</mtext></mrow><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span>, and <span><math><mi>F</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>p</mi><mi>α</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>s</mi><mo>)</mo></math></span> as particular cases. We study several basic properties of such new spaces and also characterize these spaces via Carleson-type measures. As for some applications, we study the corresponding little-o spaces, as well as establish several embedding relations of these new spaces with Bloch-type spaces. Our result generalizes an early work of Zhu in 2007.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients 关于具有非全局 Lipschitz 连续和超线性增长漂移与扩散系数的 Lévy 驱动型 McKean-Vlasov SDEs 的无限时间跨度近似问题
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128982
Ngoc Khue Tran , Trung-Thuy Kieu , Duc-Trong Luong , Hoang-Long Ngo
{"title":"On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients","authors":"Ngoc Khue Tran ,&nbsp;Trung-Thuy Kieu ,&nbsp;Duc-Trong Luong ,&nbsp;Hoang-Long Ngo","doi":"10.1016/j.jmaa.2024.128982","DOIUrl":"10.1016/j.jmaa.2024.128982","url":null,"abstract":"<div><div>This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 fractional Orlicz-Hardy inequalities 关于分数奥立兹-哈代不等式
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-22 DOI: 10.1016/j.jmaa.2024.128980
T.V. Anoop , Prosenjit Roy , Subhajit Roy
{"title":"On fractional Orlicz-Hardy inequalities","authors":"T.V. Anoop ,&nbsp;Prosenjit Roy ,&nbsp;Subhajit Roy","doi":"10.1016/j.jmaa.2024.128980","DOIUrl":"10.1016/j.jmaa.2024.128980","url":null,"abstract":"<div><div>We establish the weighted fractional Orlicz-Hardy inequalities for various Young functions satisfying the <span><math><msub><mrow><mo>△</mo></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition. Further, we identify the critical cases for such Young function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Young function Φ satisfying the <span><math><msub><mrow><mo>△</mo></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition and for any <span><math><mi>Λ</mi><mo>&gt;</mo><mn>1</mn></math></span>, the following inequality is established<span><span><span><math><mi>Φ</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>≤</mo><mi>λ</mi><mi>Φ</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>+</mo><mfrac><mrow><mi>C</mi><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></mrow><mrow><msup><mrow><mo>(</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mi>Φ</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mo>∀</mo><mspace></mspace><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>,</mo><mspace></mspace><mo>∀</mo><mspace></mspace><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>Λ</mi><mo>]</mo><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>:</mo><mo>=</mo><mi>sup</mi><mo>⁡</mo><mo>{</mo><mi>t</mi><mi>φ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>/</mo><mi>Φ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>:</mo><mi>t</mi><mo>&gt;</mo><mn>0</mn><mo>}</mo></math></span>, <em>φ</em> is the right derivatives of Φ and <span><math><mi>C</mi><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span> is a positive constant that depends only on Φ and Λ.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Shadowing of the induced map for contracting homeomorphisms 收缩同构的诱导映射阴影
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-21 DOI: 10.1016/j.jmaa.2024.128983
W. Jung , M. Lee , C.A. Morales
{"title":"Shadowing of the induced map for contracting homeomorphisms","authors":"W. Jung ,&nbsp;M. Lee ,&nbsp;C.A. Morales","doi":"10.1016/j.jmaa.2024.128983","DOIUrl":"10.1016/j.jmaa.2024.128983","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map <span><math><msub><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if <em>X</em> is Polish, then the restriction of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora <span><span>[4]</span></span>, concerning the Kantorovich-Rubinstein and Prokhorov metrics.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
The generalized Łojasiewicz inequality for definable and subanalytic multifunctions 可定义多重函数和亚解析多重函数的广义罗亚斯维兹不等式
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-18 DOI: 10.1016/j.jmaa.2024.128977
Michał Kosiba
{"title":"The generalized Łojasiewicz inequality for definable and subanalytic multifunctions","authors":"Michał Kosiba","doi":"10.1016/j.jmaa.2024.128977","DOIUrl":"10.1016/j.jmaa.2024.128977","url":null,"abstract":"<div><div>This paper is devoted to obtaining the Łojasiewicz inequality (version for two functions), in both the definable and subanalytic cases, under the most relaxed assumptions. It means that we drop the usual continuity and compactness assumptions. In the second part of the paper we concentrate on the Łojasiewicz inequality for multifunctions and apply it to the natural multifunctions related to the medial axis of a set (basic notion in pattern recognition).</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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