Journal of Functional Analysis最新文献

{"title":"Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces","authors":"Anton Tselishchev","doi":"10.1016/j.jfa.2025.110895","DOIUrl":"10.1016/j.jfa.2025.110895","url":null,"abstract":"<div><div>For any sequence of positive numbers <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> such that <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>∞</mo></math></span> we provide an explicit simple construction of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> such that <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msubsup><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><mo>∞</mo></math></span> there exists a <span><math><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110895"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The ideal separation property for reduced group C⁎-algebras","authors":"Are Austad ,&nbsp;Hannes Thiel","doi":"10.1016/j.jfa.2025.110904","DOIUrl":"10.1016/j.jfa.2025.110904","url":null,"abstract":"<div><div>We say that an inclusion of an algebra <em>A</em> into a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra <em>B</em> has the ideal separation property if closed ideals in <em>B</em> can be recovered by their intersection with <em>A</em>. Such inclusions have attractive properties from the point of view of harmonic analysis and noncommutative geometry. We establish several permanence properties of locally compact groups for which <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>⊆</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>red</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has the ideal separation property.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110904"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strictly outer actions of locally compact groups: Beyond the full factor case","authors":"Basile Morando","doi":"10.1016/j.jfa.2025.110897","DOIUrl":"10.1016/j.jfa.2025.110897","url":null,"abstract":"<div><div>We show that, given a continuous action <em>α</em> of a locally compact group <em>G</em> on a factor <em>M</em>, the relative commutant <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∩</mo><mo>(</mo><mi>M</mi><msub><mrow><mo>⋊</mo></mrow><mrow><mi>α</mi></mrow></msub><mi>G</mi><mo>)</mo></math></span> is contained in <span><math><mi>M</mi><msub><mrow><mo>⋊</mo></mrow><mrow><mi>α</mi></mrow></msub><mi>H</mi></math></span> where <em>H</em> is the subgroup of elements acting without spectral gap. As a corollary, we answer a question of Marrakchi and Vaes by showing that if <em>M</em> is semifinite and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> is not approximately inner for all <span><math><mi>g</mi><mo>≠</mo><mn>1</mn></math></span>, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∩</mo><mo>(</mo><mi>M</mi><msub><mrow><mo>⋊</mo></mrow><mrow><mi>α</mi></mrow></msub><mi>G</mi><mo>)</mo><mo>=</mo><mi>C</mi></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 2","pages":"Article 110897"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions","authors":"Yohei Fujishima ,&nbsp;Norisuke Ioku ,&nbsp;Bernhard Ruf ,&nbsp;Elide Terraneo","doi":"10.1016/j.jfa.2025.110922","DOIUrl":"10.1016/j.jfa.2025.110922","url":null,"abstract":"<div><div>By introducing a new classification of the growth rate of exponential functions, singular solutions for <span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity “close” to <em>f</em>, which admits an explicit singular solution. Then, using a transformation as in <span><span>[8]</span></span>, one obtains an approximate singular solution, and then one concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way, e.g., <span><math><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>r</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></msup><mspace></mspace><mo>(</mo><mi>q</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msup></math></span> (<span><math><mi>q</mi><mo>&gt;</mo><mn>1</mn></math></span>, <span><math><mi>q</mi><mo>/</mo><mn>2</mn><mo>&gt;</mo><mi>r</mi><mo>&gt;</mo><mn>0</mn></math></span> or <span><math><mn>1</mn><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><mn>4</mn></math></span>, <span><math><mi>r</mi><mo>=</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span>), <span><math><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup></mrow></msup></math></span>. As a special case, our result contains a pioneering contribution by Ibrahim–Kikuchi–Nakanishi–Wei <span><span>[15]</span></span> for <span><math><mi>u</mi><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110922"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143576836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Campanato regularity theory for multi-valued functions with applications to minimal surface regularity theory","authors":"Paul Minter","doi":"10.1016/j.jfa.2025.110908","DOIUrl":"10.1016/j.jfa.2025.110908","url":null,"abstract":"<div><div>The regularity theory of the Campanato space <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> has found many applications within the regularity theory of solutions to various geometric variational problems. Here we extend this theory from single-valued functions to multi-valued functions, adapting for the most part Campanato's original ideas (<span><span>[4]</span></span>). We also give an application of this theory within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain <em>blow-up classes</em> of multi-valued functions, which typically arise when studying blow-ups of sequences of stationary integral varifolds converging to higher multiplicity planes or unions of half-planes. In such a setting, based in part on ideas in <span><span>[16]</span></span>, <span><span>[11]</span></span>, and <span><span>[3]</span></span>, we are able to deduce a boundary regularity theory for multi-valued harmonic functions; such a boundary regularity result would appear to be the first of its kind for the multi-valued setting. In conjunction with <span><span>[9]</span></span>, the results presented here establish a regularity theorem for stable codimension one stationary integral varifolds near classical cones of density <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110908"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hyperinvariant subspaces for trace class perturbations of normal operators and decomposability","authors":"Eva A. Gallardo-Gutiérrez ,&nbsp;F. Javier González-Doña","doi":"10.1016/j.jfa.2025.110903","DOIUrl":"10.1016/j.jfa.2025.110903","url":null,"abstract":"<div><div>We prove that a large class of trace-class perturbations of diagonalizable normal operators on a separable, infinite dimensional complex Hilbert space have non-trivial closed hyperinvariant subspaces. Moreover, a large subclass consists of decomposable operators in the sense of Colojoară and Foiaş <span><span>[3]</span></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 2","pages":"Article 110903"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143578529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimal surface equation and Bernstein property on RCD spaces","authors":"Alessandro Cucinotta","doi":"10.1016/j.jfa.2025.110907","DOIUrl":"10.1016/j.jfa.2025.110907","url":null,"abstract":"<div><div>We show that if <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> is an <span><math><mrow><mi>RCD</mi></mrow><mo>(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> space and <span><math><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a solution of the minimal surface equation, then <em>u</em> is harmonic on its graph (which has a natural metric measure space structure). If <span><math><mi>K</mi><mo>=</mo><mn>0</mn></math></span> this allows to obtain an Harnack inequality for <em>u</em>, which in turn implies the Bernstein property, meaning that any positive solution to the minimal surface equation must be constant. As an application, we obtain oscillation estimates and a Bernstein Theorem for minimal graphs in products <span><math><mi>M</mi><mo>×</mo><mi>R</mi></math></span>, where <span><math><mi>M</mi></math></span> is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 2","pages":"Article 110907"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143578527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions","authors":"Rubén Medina ,&nbsp;Abraham Rueda Zoca","doi":"10.1016/j.jfa.2025.110896","DOIUrl":"10.1016/j.jfa.2025.110896","url":null,"abstract":"<div><div>Let <em>M</em> be a metric space and <em>X</em> be a Banach space. In this paper we address several questions about the structure of <span><math><mi>F</mi><mo>(</mo><mi>M</mi><mo>)</mo><msub><mrow><mover><mrow><mo>⊗</mo></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>π</mi></mrow></msub><mi>X</mi></math></span> and <span><math><msub><mrow><mi>Lip</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>. Our results are the following:<ul><li><span>(1)</span><span><div>We prove that if <em>M</em> is a length metric space then <span><math><msub><mrow><mi>Lip</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> has the Daugavet property. As a consequence, if <em>M</em> is length we obtain that <span><math><mi>F</mi><mo>(</mo><mi>M</mi><mo>)</mo><msub><mrow><mover><mrow><mo>⊗</mo></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>π</mi></mrow></msub><mi>X</mi></math></span> has the Daugavet property. This gives an affirmative answer to <span><span>[9, Question 1]</span></span> (also asked in <span><span>[16, Remark 3.8]</span></span>).</div></span></li><li><span>(2)</span><span><div>We prove that if <em>M</em> is a non-uniformly discrete metric space or an unbounded metric space then the norm of <span><math><mi>F</mi><mo>(</mo><mi>M</mi><mo>)</mo><msub><mrow><mover><mrow><mo>⊗</mo></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>π</mi></mrow></msub><mi>X</mi></math></span> is octahedral, which solves <span><span>[4, Question 3.2 (1)]</span></span>.</div></span></li><li><span>(3)</span><span><div>We characterise all the Banach spaces <em>X</em> such that <span><math><mi>L</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> is octahedral for every Banach space <em>Y</em>, which solves a question by Johann Langemets.</div></span></li></ul></div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110896"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Schauder-type estimates for fully nonlinear degenerate elliptic equations","authors":"Thialita M. Nascimento","doi":"10.1016/j.jfa.2025.110900","DOIUrl":"10.1016/j.jfa.2025.110900","url":null,"abstract":"<div><div>In this paper, we examine regularity estimates for solutions to fully nonlinear, degenerated elliptic equations, at interior vanishing source points. By means of a geometric approach, we obtain, at such points, Schauder-type regularity estimates, which depend on the Hölder-like source-ellipticity vanishing rate.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110900"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a kinetic Poincaré inequality and beyond","authors":"Lukas Niebel ,&nbsp;Rico Zacher","doi":"10.1016/j.jfa.2025.110899","DOIUrl":"10.1016/j.jfa.2025.110899","url":null,"abstract":"<div><div>In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot <span><span>[12]</span></span> in several directions. We use kinetic trajectories along the vector fields <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>v</mi><mo>⋅</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>d</mi></math></span> and do not rely on higher-order commutators such as <span><math><mo>[</mo><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>v</mi><mo>⋅</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>]</mo><mo>=</mo><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span> or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by investigating a Kolmogorov equation with <em>k</em> steps.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 1","pages":"Article 110899"},"PeriodicalIF":1.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}