Topology and its Applications最新文献

{"title":"G-movability and large subgroups","authors":"Hugo Juárez-Anguiano ,&nbsp;Raúl Juárez-Flores","doi":"10.1016/j.topol.2026.109731","DOIUrl":"10.1016/j.topol.2026.109731","url":null,"abstract":"<div><div>In this paper, we prove the following result: Let <em>H</em> be a closed subgroup of a compact metrizable group <em>G</em>. Then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is <em>G</em>-movable if and only if <em>H</em> is a large subgroup of <em>G</em>. It provides a new characterization of large subgroups and generalizes a result of Gevorgyan <span><span>[12]</span></span> about compact Lie groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109731"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038150","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}

{"title":"On a variation of selective separability using ideals","authors":"Debraj Chandra ,&nbsp;Nur Alam ,&nbsp;Dipika Roy","doi":"10.1016/j.topol.2026.109725","DOIUrl":"10.1016/j.topol.2026.109725","url":null,"abstract":"<div><div>A space <em>X</em> is H-separable (Bella et al. (2009) <span><span>[6]</span></span>) if for every sequence <span><math><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> of dense subspaces of <em>X</em> there exists a sequence <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> such that for each <em>n</em> <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a finite subset of <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and every nonempty open set of <em>X</em> intersects <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all but finitely many <em>n</em>. In this paper, we introduce and study an ideal variant of H-separability, called <span><math><mi>I</mi></math></span>-H-separability.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109725"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978202","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}

{"title":"Round twin groups on few strands","authors":"Jacob Mostovoy","doi":"10.1016/j.topol.2026.109727","DOIUrl":"10.1016/j.topol.2026.109727","url":null,"abstract":"<div><div>We study the space <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all configurations of <em>n</em> ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for <span><math><mi>n</mi><mo>&lt;</mo><mn>6</mn></math></span> and describe its homology for <span><math><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>7</mn></math></span>. For arbitrary <em>n</em>, we compute its first homology and its Euler characteristic.</div><div>We use three geometric approaches. On one hand, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is naturally defined as the complement of an arrangement of codimension-2 subtori in a real torus. On the other hand, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> can also be assembled from no-3-equal manifolds of the real line.</div><div>We also observe that, up to homotopy, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> may be identified with a subspace of the oriented double cover of the moduli space <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of stable real rational curves with <em>n</em> marked points. This gives an embedding of <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> into the pure cactus group. As a corollary, we see that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is residually nilpotent.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109727"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038152","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}

{"title":"Bridging graph-theoretical and topological approaches: Connectivity and Jordan curves in the digital plane","authors":"Yazmin Cote,&nbsp;Carlos Uzcátegui-Aylwin","doi":"10.1016/j.topol.2026.109730","DOIUrl":"10.1016/j.topol.2026.109730","url":null,"abstract":"<div><div>This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and the subsequent introduction of the topological digital plane <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> transforming subsets of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> into subsets of <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> correspond to topological properties in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Moreover, we address whether the topological Jordan curve theorem for <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> can be derived from the graph-theoretical version on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109730"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038155","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}

{"title":"Extending quasi-alternating links III","authors":"Kirandeep Kaur ,&nbsp;Nafaa Chbili","doi":"10.1016/j.topol.2026.109736","DOIUrl":"10.1016/j.topol.2026.109736","url":null,"abstract":"<div><div>Champanerkar and Kofman <span><span>[1]</span></span> introduced a method for constructing quasi-alternating links by replacing a quasi-alternating crossing in a link diagram with a rational tangle of the same type. This approach, however, does not generally extend to alternating tangles of the opposite type or to non-alternating tangles.</div><div>In this paper, we identify sufficient conditions under which the construction remains valid when the crossing is replaced by an alternating rational tangle of opposite type. We also prove that this method applies to certain non-alternating pretzel tangles. As an application, we provide a table of non-alternating quasi-alternating knots with 13 crossings obtained using this construction. Additionally, we describe an infinite family of quasi-alternating links featuring a non-twisted quasi-alternating crossing that satisfies these sufficient conditions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109736"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038151","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}

{"title":"On limit sets and equicontinuity in the hyperspace of continua in dimension one","authors":"Domagoj Jelić ,&nbsp;Piotr Oprocha","doi":"10.1016/j.topol.2025.109690","DOIUrl":"10.1016/j.topol.2025.109690","url":null,"abstract":"<div><div>The paper studies the structure of <em>ω</em>-limit sets of map <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> induced on the hyperspace <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of all connected compact sets, by dynamical system <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> acting on a topological graph <em>G</em>. In the case of the base space being a topological tree we additionally show that <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> is always almost equicontinuous and characterize its Birkhoff center.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109690"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760745","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}

{"title":"Diagonals of separately pointwise Lipschitz functions of n variables","authors":"Olena Karlova ,&nbsp;Volodymyr Mykhaylyuk","doi":"10.1016/j.topol.2026.109726","DOIUrl":"10.1016/j.topol.2026.109726","url":null,"abstract":"<div><div>We study the diagonals <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>…</mo><mo>,</mo><mi>x</mi><mo>)</mo></math></span> of (strongly) separately Lipschitz mappings <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span>. It is shown that for any metric space <em>X</em> and any normed space <em>Y</em> the diagonals of strongly separately pointwise Lipschitz mappings <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span> are exactly stable limits of sequences of pointwise Lipschitz mappings (a mapping on the product of <em>n</em> metric spaces is <em>strongly separately pointwise Lipschitz</em> if it is jointly pointwise Lipschitz mapping with respect to any <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> variables). We introduce classes <span><math><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> of mappings between metric spaces <em>X</em> and <em>Y</em> which are recursively defined from pointwise Lipschitz mappings, analogously as mappings of stable Baire classes are recursively defined from continuous mappings. It was shown that <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> for a metric space <em>X</em> and a Banach space <em>Y</em> if and only if there exists a sequence <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> of ambiguous sets <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊆</mo><mi>X</mi></math></span> of the class <em>n</em> such that every restriction <span><math><mi>f</mi><msub><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span> is Lipschitz. Moreover, for any metric space <em>X</em>, any normed space <em>Y</em> and every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> we construct a separately pointwise Lipschitz mapping <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span> with given diagonal <span><math><mi>g</mi><mo>∈</mo><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109726"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978201","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}

{"title":"Meshed continua have unique n-fold symmetric product suspension","authors":"Felipe de J. Aguilar-Romero,&nbsp;David Herrera-Carrasco,&nbsp;Fernando Macías-Romero","doi":"10.1016/j.topol.2025.109706","DOIUrl":"10.1016/j.topol.2025.109706","url":null,"abstract":"<div><div>Let <em>X</em> be a metric continuum, and let <em>n</em> be a positive integer. We denote by <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the hyperspace consisting of all nonempty closed subsets of <em>X</em> with at most <em>n</em> points. For <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span>, the <em>n-fold symmetric product suspension</em> of <em>X</em> is the quotient space <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In this paper, we prove that if <em>X</em> is a meshed continuum, <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, and <em>Y</em> is a continuum such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is homeomorphic to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo></math></span>, then <em>X</em> is homeomorphic to <em>Y</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109706"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884994","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}

{"title":"Nonlocal loss of first homotopy in polyhedral approximations of Peano continua","authors":"Jeremy Brazas ,&nbsp;Hanspeter Fischer","doi":"10.1016/j.topol.2025.109710","DOIUrl":"10.1016/j.topol.2025.109710","url":null,"abstract":"<div><div>If a Peano continuum <em>X</em> is semilocally simply connected, then it has a finite polyhedral approximation whose fundamental group is isomorphic to that of <em>X</em>. In general, this fails to be true. It is known that the fundamental group of a locally complicated Peano continuum may contain nontrivial elements that are persistently undetectable by polyhedral approximations, at all scales. However, we show that such failure is not inherently local.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109710"},"PeriodicalIF":0.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145972948","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}

{"title":"Cardinal invariants of a meager ideal","authors":"Will Brian","doi":"10.1016/j.topol.2025.109516","DOIUrl":"10.1016/j.topol.2025.109516","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> denote the ideal of meager subsets of a topological space <em>X</em>. We prove that if <em>X</em> is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of <em>X</em>, denoted <span><math><mrow><mi>non</mi></mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, is exactly <span><math><mrow><mi>non</mi></mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo><mo>=</mo><mrow><mi>cf</mi></mrow><msup><mrow><mo>[</mo><mi>κ</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>⋅</mo><mrow><mi>non</mi></mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>, where <em>κ</em> is the minimum weight of a nonempty open subset of <em>X</em>. We also characterize the additivity and covering numbers for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> in terms of simple topological properties of <em>X</em>. Some bounds are proved and some questions raised concerning the cofinality of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and the cofinality of the related ideal of nowhere dense subsets of <em>X</em>.</div><div>We also show that if <em>X</em> is a compact Hausdorff space with <em>π</em>-weight <em>κ</em>, then <span><math><mrow><mi>non</mi></mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>cf</mi></mrow><msup><mrow><mo>[</mo><mi>κ</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>⋅</mo><mrow><mi>non</mi></mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>. This bound for compact Hausdorff spaces is not sharp, in the sense that it is consistent for such a space to have non-meager subsets of even smaller cardinality.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109516"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948065","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}