Mathematische Nachrichten最新文献

{"title":"Three-parameter Triebel–Lizorkin spaces associated with a sum of two flag singular integrals","authors":"Yan Chen,&nbsp;Xiangxing Tao,&nbsp;Taotao Zheng","doi":"10.1002/mana.202400208","DOIUrl":"https://doi.org/10.1002/mana.202400208","url":null,"abstract":"<p>In this paper, the authors establish the three-parameter Triebel–Lizorkin spaces and characterize these spaces as the intersection of two flag Triebel–Lizorkin spaces by applying the discrete Littlewood–Paley–Stein analysis. Moreover, they obtain the boundedness of product singular integral operators on the three-parameter Triebel–Lizorkin spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"581-601"},"PeriodicalIF":0.8,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397164","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}

{"title":"Entropy numbers and box dimension of polynomials and holomorphic functions","authors":"Daniel Carando,&nbsp;Carlos D'Andrea,&nbsp;Leodan A. Torres,&nbsp;Pablo Turco","doi":"10.1002/mana.202400042","DOIUrl":"https://doi.org/10.1002/mana.202400042","url":null,"abstract":"<p>We study entropy numbers and box dimension of (the image of) homogeneous polynomials and holomorphic functions between Banach spaces. First, we see that entropy numbers and box dimensions of subsets of Banach spaces are related. We show that the box dimension of the image of a ball under a homogeneous polynomial is finite if and only if it spans a finite-dimensional subspace, but this is not true for holomorphic functions. Furthermore, we relate the entropy numbers of a holomorphic function to those of the polynomials of its Taylor series expansion. As a consequence, if the box dimension of the image of a ball by a holomorphic function <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> is finite, then the entropy numbers of the polynomials in the Taylor series expansion of <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> at any point of the ball belong to <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$ell _p$</annotation>\u0000 </semantics></math> for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$p&gt;1$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"567-580"},"PeriodicalIF":0.8,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397165","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}

{"title":"Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous","authors":"Lorenzo Dello Schiavo,&nbsp;Ronan Herry,&nbsp;Eva Kopfer,&nbsp;Karl-Theodor Sturm","doi":"10.1002/mana.202400169","DOIUrl":"https://doi.org/10.1002/mana.202400169","url":null,"abstract":"<p>For an arbitrary dimension <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>, we study: \u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 1","pages":"244-281"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400169","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120096","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}

{"title":"Twisted conjugacy in soluble arithmetic groups","authors":"Paula M. Lins de Araujo,&nbsp;Yuri Santos Rego","doi":"10.1002/mana.202300448","DOIUrl":"https://doi.org/10.1002/mana.202300448","url":null,"abstract":"<p>Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here, we address a question posed by Gonçalves and Wong in the mid-2000s: we construct an infinite series of compact connected solvmanifolds (that are <i>not</i> <i>nil</i>manifolds) of strictly increasing dimensions and all of whose self-homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so-called property <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <annotation>$R_infty$</annotation>\u0000 </semantics></math>. In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>-arithmetic groups have <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <annotation>$R_infty$</annotation>\u0000 </semantics></math> and suggesting a conjecture in this direction.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"763-793"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300448","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595679","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}

{"title":"On non-Hopf Ricci-pseudosymmetric hypersurfaces in \u0000 \u0000 \u0000 C\u0000 \u0000 P\u0000 2\u0000 \u0000 \u0000 $mathbb {C}P^{2}$\u0000 and \u0000 \u0000 \u0000 C\u0000 \u0000 H\u0000 2\u0000 \u0000 \u0000 $mathbb {C}H^{2}$","authors":"Qianshun Cui,&nbsp;Zejun Hu","doi":"10.1002/mana.202300463","DOIUrl":"https://doi.org/10.1002/mana.202300463","url":null,"abstract":"<p>In this paper, we study an open problem raised by Cecil and Ryan [<i>Geometry of Hypersurfaces</i>, Springer Monographs in Mathematics, p. 531] which asked whether there exist non-Hopf Ricci-pseudosymmetric hypersurfaces in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>. As our main results, we first prove the nonexistence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>. Then, we prove the existence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math>. Finally, applying the preceding results and sharpening Theorem 4.1 of Wang and Zhang [<i>J. Geom. Phys</i>. <b>181</b> (2022), 104648], we prove the nonexistence of non-Hopf weakly Einstein hypersurfaces with constant norm of Riemannian curvature tensor in both <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"527-547"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397235","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}

{"title":"Abstract integro-differential equations with state-dependent integration intervals: Existence, uniqueness, and local well-posedness","authors":"Eduardo Hernandez,&nbsp;Shashank Pandey,&nbsp;Dwijendra N. Pandey","doi":"10.1002/mana.202400126","DOIUrl":"https://doi.org/10.1002/mana.202400126","url":null,"abstract":"<p>In this work, we study a new class of integro-differential equations with delay, where the informations from the past are represented as an average of the state over state-dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro-differential equations arising in the theory of population dynamics.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 1","pages":"356-384"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120066","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}

{"title":"Construction of the log-convex minorant of a sequence \u0000 \u0000 \u0000 \u0000 {\u0000 \u0000 M\u0000 α\u0000 \u0000 }\u0000 \u0000 \u0000 α\u0000 ∈\u0000 \u0000 N\u0000 0\u0000 d\u0000 \u0000 \u0000 \u0000 $lbrace M_alpha rbrace _{alpha in mathbb {N}_0^d}$","authors":"Chiara Boiti,&nbsp;David Jornet,&nbsp;Alessandro Oliaro,&nbsp;Gerhard Schindl","doi":"10.1002/mana.202400135","DOIUrl":"https://doi.org/10.1002/mana.202400135","url":null,"abstract":"&lt;p&gt;We give a simple construction of the log-convex minorant of a sequence &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$lbrace M_alpha rbrace _{alpha in mathbb {N}_0^d}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and consequently extend to the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional case the well-known formula that relates a log-convex sequence &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$lbrace M_prbrace _{pin mathbb {N}_0}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; to its associated function &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$omega _M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, that is, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;sup&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mi&gt;exp&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;m","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"456-477"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400135","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397231","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}

{"title":"Partitions in real quadratic fields","authors":"David Stern,&nbsp;Mikuláš Zindulka","doi":"10.1002/mana.202300480","DOIUrl":"https://doi.org/10.1002/mana.202300480","url":null,"abstract":"<p>We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1–5, 7, and 11 partitions.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"548-566"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300480","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397242","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}

{"title":"Curvature of quaternionic skew-Hermitian manifolds and bundle constructions","authors":"Ioannis Chrysikos,&nbsp;Vicente Cortés,&nbsp;Jan Gregorovič","doi":"10.1002/mana.202400301","DOIUrl":"https://doi.org/10.1002/mana.202400301","url":null,"abstract":"<p>This paper is devoted to a description of the second-order differential geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is, of quaternionic skew-Hermitian manifolds <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>,</mo>\u0000 <mi>Q</mi>\u0000 <mo>,</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(M, Q, omega)$</annotation>\u0000 </semantics></math>. We provide a curvature characterization of such integrable geometric structures, based on the holonomy theory of symplectic connections and we study qualitative properties of the induced Ricci tensor. Then, we proceed with bundle constructions over such a manifold <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>,</mo>\u0000 <mi>Q</mi>\u0000 <mo>,</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(M, Q, omega)$</annotation>\u0000 </semantics></math>. In particular, we prove the existence of almost hypercomplex skew-Hermitian structures on the Swann bundle over <i>M</i> and investigate their integrability.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 1","pages":"87-112"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400301","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120095","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}

{"title":"Laplacian with singular drift in a critical borderline case","authors":"D. Kinzebulatov","doi":"10.1002/mana.202400098","DOIUrl":"https://doi.org/10.1002/mana.202400098","url":null,"abstract":"<p>We establish well-posedness and regularity results for parabolic diffusion equation on the torus in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$L^infty$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"511-526"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397233","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}