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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":"543 2","pages":"Article 128982"},"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":"543 2","pages":"Article 128980"},"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":"543 2","pages":"Article 128983"},"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":"543 2","pages":"Article 128977"},"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
Local rigidity of constant mean curvature hypersurfaces in space forms 空间形式中恒定平均曲率超曲面的局部刚性
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-18 DOI: 10.1016/j.jmaa.2024.128974
Yayun Chen , Tongzhu Li
{"title":"Local rigidity of constant mean curvature hypersurfaces in space forms","authors":"Yayun Chen ,&nbsp;Tongzhu Li","doi":"10.1016/j.jmaa.2024.128974","DOIUrl":"10.1016/j.jmaa.2024.128974","url":null,"abstract":"<div><div>In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let <span><math><mi>x</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, be a piece of immersed constant mean curvature hypersurface in the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional space form <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span>. We prove that if the scalar curvature <em>R</em> is constant and the number <em>g</em> of the distinct principal curvatures satisfies <span><math><mi>g</mi><mo>≤</mo><mn>3</mn></math></span>, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is an isoparametric hypersurface. Further, if <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a minimal hypersurface, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a totally geodesic hypersurface for <span><math><mi>c</mi><mo>≤</mo><mn>0</mn></math></span>, and <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span>, which solves the high dimensional version of Bryant Conjecture.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128974"},"PeriodicalIF":1.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527624","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
Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators 高阶薛定谔算子的无穷小相对有界性特征
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-18 DOI: 10.1016/j.jmaa.2024.128975
Jun Cao , Mengyao Gao , Yongyang Jin , Chao Wang
{"title":"Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators","authors":"Jun Cao ,&nbsp;Mengyao Gao ,&nbsp;Yongyang Jin ,&nbsp;Chao Wang","doi":"10.1016/j.jmaa.2024.128975","DOIUrl":"10.1016/j.jmaa.2024.128975","url":null,"abstract":"<div><div>Let <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><mi>H</mi><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi></math></span> be a higher order Schrödinger operators in the Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>V</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. In this paper, the authors characterize the infinitesimal relative boundedness and Trudinger's subordination for <em>H</em> on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> with <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, via the limit behavior of the family of operators <span><math><msub><mrow><mo>{</mo><mi>V</mi><msup><mrow><mo>(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>m</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>}</mo></mrow><mrow><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> and a generalized Kato-type class condition. The latter is weaker than the classical Kato class condition corresponding to the case <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span>. All these characterizations are new even when <span><math><mi>H</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi></math></span> is a second order Schrödinger operator.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128975"},"PeriodicalIF":1.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526968","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
Interlacing of zeros of odd period polynomials 奇周期多项式零点的交错
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-18 DOI: 10.1016/j.jmaa.2024.128976
Grace Ko , Jennifer Mackenzie , Hui Xue
{"title":"Interlacing of zeros of odd period polynomials","authors":"Grace Ko ,&nbsp;Jennifer Mackenzie ,&nbsp;Hui Xue","doi":"10.1016/j.jmaa.2024.128976","DOIUrl":"10.1016/j.jmaa.2024.128976","url":null,"abstract":"<div><div>The work of Conrey, Farmer and Imamoglu shows that all nontrivial zeros of the odd period polynomial of a Hecke eigenform of level one lie on the unit circle. We further show that nontrivial zeros of odd period polynomials of Hecke eigenforms of different weights share certain interlacing property.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128976"},"PeriodicalIF":1.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526969","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
Global existence of solutions to a nonlocal equation with degenerate anisotropic diffusion 具有退化各向异性扩散的非局部方程的全局存在解
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-17 DOI: 10.1016/j.jmaa.2024.128971
Maria Eckardt , Anna Zhigun
{"title":"Global existence of solutions to a nonlocal equation with degenerate anisotropic diffusion","authors":"Maria Eckardt ,&nbsp;Anna Zhigun","doi":"10.1016/j.jmaa.2024.128971","DOIUrl":"10.1016/j.jmaa.2024.128971","url":null,"abstract":"<div><div>Global existence of very weak solutions to a non-local diffusion-advection-reaction equation is established under no-flux boundary conditions in higher dimensions. The equation features degenerate myopic diffusion and nonlocal adhesion and is an extension of a mass-conserving model recently derived in <span><span>[21]</span></span>. The admissible degeneracy of the diffusion tensor is characterised in terms of the upper box fractal dimension.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128971"},"PeriodicalIF":1.2,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings 最佳排列涂层热方程有效边界条件的误差估计
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-16 DOI: 10.1016/j.jmaa.2024.128972
Lixin Meng , Zhitong Zhou
{"title":"Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings","authors":"Lixin Meng ,&nbsp;Zhitong Zhou","doi":"10.1016/j.jmaa.2024.128972","DOIUrl":"10.1016/j.jmaa.2024.128972","url":null,"abstract":"<div><div>We are interested in the validity of effective boundary conditions for a heat equation on a coated body as the thickness of the coating shrinks to zero. The coating is optimally aligned in the sense that the normal vector in the coating is an eigenvector of the thermal tensor. If the heat equation satisfies Neumann boundary condition on the outer boundary of the coating, Chen et al. (Arch. Ration. Mech. Anal. 206 (2012) 911-951) derived the complete list of effective boundary conditions satisfied by the limiting model. In this paper we provide explicit error estimates between the full model and the effective model. Moreover, our error estimates are independent of time, which shows that the maximal time interval in which the effective boundary conditions remain valid are infinite. The proof is based on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> estimates for solutions of the full model, characterization of large time behaviors for solutions of the effective model, and interaction estimates between the two models.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 1","pages":"Article 128972"},"PeriodicalIF":1.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445607","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
Octahedrality and Gâteaux smoothness 八面体性和伽陀平滑性
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2024-10-16 DOI: 10.1016/j.jmaa.2024.128968
Ch. Cobollo , P. Hájek
{"title":"Octahedrality and Gâteaux smoothness","authors":"Ch. Cobollo ,&nbsp;P. Hájek","doi":"10.1016/j.jmaa.2024.128968","DOIUrl":"10.1016/j.jmaa.2024.128968","url":null,"abstract":"<div><div>We prove that every Banach space admitting a Gâteaux smooth norm and containing a complemented copy of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> has an equivalent renorming which is simultaneously Gâteaux smooth and octahedral. This is a partial solution to an open problem from the early nineties.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128968"},"PeriodicalIF":1.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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