Journal of Functional Analysis最新文献

{"title":"Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise","authors":"Fabrice Baudoin ,&nbsp;Maria Gordina ,&nbsp;David P. Herzog ,&nbsp;Jina Kim ,&nbsp;Tai Melcher","doi":"10.1016/j.jfa.2024.110814","DOIUrl":"10.1016/j.jfa.2024.110814","url":null,"abstract":"<div><div>For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified Γ calculus on finite-dimensional projections of the equation in order to produce explicit functional inequalities that can be scaled to infinite dimensions. The choice of our Γ operator appears canonical in our context, as the estimates depend only on the induced control distance. We apply the general analysis to a number of examples, exploring implications for quasi-invariance and uniqueness of stationary distributions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110814"},"PeriodicalIF":1.7,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174943","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":"A Banach space whose set of norm-attaining functionals is algebraically trivial","authors":"Miguel Martín","doi":"10.1016/j.jfa.2024.110815","DOIUrl":"10.1016/j.jfa.2024.110815","url":null,"abstract":"<div><div>We construct a Banach space <span><math><mi>X</mi></math></span> for which the set of norm-attaining functionals <span><math><mi>NA</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on <span><math><mi>X</mi></math></span>, no other element of the segment between them attains its norm. Equivalently, the intersection of <span><math><mi>NA</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> with a two-dimensional subspace of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is contained in the union of two lines. In terms of proximinality, we show that for every closed subspace <em>M</em> of <span><math><mi>X</mi></math></span> of codimension two, at most four elements of the unit sphere of <span><math><mi>X</mi><mo>/</mo><mi>M</mi></math></span> have a representative of norm-one. We further relate this example with an open problem on norm-attaining operators.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110815"},"PeriodicalIF":1.7,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174942","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":"Propagation of regularity for a class of systems of real vector fields on torus","authors":"I.A. Ferra,&nbsp;L.A. Carvalho dos Santos","doi":"10.1016/j.jfa.2024.110807","DOIUrl":"10.1016/j.jfa.2024.110807","url":null,"abstract":"<div><div>We characterize the global hypoellipticity, almost hypoellipticity and solvability for a class of systems of real vector fields on the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional torus as well as the same properties about the sum of squares associated to the system. The key result is a theorem about propagation of regularity for solutions of the system.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110807"},"PeriodicalIF":1.7,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174929","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":"One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations","authors":"Bruno Premoselli ,&nbsp;Frédéric Robert","doi":"10.1016/j.jfa.2024.110808","DOIUrl":"10.1016/j.jfa.2024.110808","url":null,"abstract":"<div><div>We investigate in this work families <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>ε</mi><mo>&gt;</mo><mn>0</mn></mrow></msub></math></span> of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type:<span><span><span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>g</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>ε</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mo>|</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>−</mo><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub></math></span></span></span> in a closed Riemannian manifold <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>ε</mi></mrow></msub></math></span> converges to <em>h</em> in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>→</mo><msubsup><mrow><mn>2</mn></mrow><mrow><mo>−</mo></mrow><mrow><mo>⋆</mo></mrow></msubsup><mo>:</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>. Assuming that <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>ε</mi><mo>&gt;</mo><mn>0</mn></mrow></msub></math></span> blows-up as <em>a single sign-changing bubble</em>, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between <em>h</em>, the geometry of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110808"},"PeriodicalIF":1.7,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174945","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":"Quantitative spectral stability for the Neumann Laplacian in domains with small holes","authors":"Veronica Felli ,&nbsp;Lorenzo Liverani ,&nbsp;Roberto Ognibene","doi":"10.1016/j.jfa.2024.110817","DOIUrl":"10.1016/j.jfa.2024.110817","url":null,"abstract":"<div><div>The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are “small” in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110817"},"PeriodicalIF":1.7,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174947","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":"Sobolev differentiability properties of logarithmic modulus of real analytic functions","authors":"Ziming Shi ,&nbsp;Ruixiang Zhang","doi":"10.1016/j.jfa.2024.110804","DOIUrl":"10.1016/j.jfa.2024.110804","url":null,"abstract":"<div><div>Let <em>f</em> be the germ of a real analytic function at the origin in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, and suppose the codimension of the zero set of <em>f</em> at <strong>0</strong> is at least 2. We show that <span><math><mi>log</mi><mo>⁡</mo><mo>|</mo><mi>f</mi><mo>|</mo></math></span> is <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msubsup></math></span> near <strong>0</strong>. In particular, this implies that the differential inequality <span><math><mo>|</mo><mi>∇</mi><mi>f</mi><mo>|</mo><mo>≤</mo><mi>V</mi><mo>|</mo><mi>f</mi><mo>|</mo></math></span> holds 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></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110804"},"PeriodicalIF":1.7,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128188","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":"Strongly convergent unitary representations of limit groups","authors":"Larsen Louder ,&nbsp;Michael Magee ,&nbsp;Will Hide","doi":"10.1016/j.jfa.2024.110803","DOIUrl":"10.1016/j.jfa.2024.110803","url":null,"abstract":"<div><div>We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that ‘strongly converge’ to the regular representation of the group. The corresponding statement for finitely generated free groups was proved by Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary representations to arise from representations of the group by permutation matrices, as was proved for free groups by Bordenave and Collins.</div><div>As for Haagerup and Thorbjørnsen, the existence of such representations implies that for any non-abelian limit group, the Ext-invariant of the reduced <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra is not a group (has non-invertible elements).</div><div>An important special case of our main theorem is in application to the fundamental groups of closed orientable surfaces of genus at least two. In this case, our results can be used as an input to the methods previously developed by the authors of the appendix. The output is a variation of our previous proof of Buser's 1984 conjecture that there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first eigenvalue of the Laplacian tending to <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>. In this variation of the proof, the systoles of the surfaces are bounded away from zero and the surfaces can be taken to be arithmetic.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 6","pages":"Article 110803"},"PeriodicalIF":1.7,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174940","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":"Deformed single ring theorems","authors":"Ching-Wei Ho ,&nbsp;Ping Zhong","doi":"10.1016/j.jfa.2024.110797","DOIUrl":"10.1016/j.jfa.2024.110797","url":null,"abstract":"<div><div>Given a sequence of deterministic matrices <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and a sequence of deterministic nonnegative matrices <span><math><mi>Σ</mi><mo>=</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> such that <span><math><mi>A</mi><mo>→</mo><mi>a</mi></math></span> and <span><math><mi>Σ</mi><mo>→</mo><mi>σ</mi></math></span> in ⁎-distribution for some operators <em>a</em> and <em>σ</em> in a finite von Neumann algebra <span><math><mi>A</mi></math></span>. Let <span><math><mi>U</mi><mo>=</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and <span><math><mi>V</mi><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of <span><math><mi>U</mi><mi>Σ</mi><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>+</mo><mi>A</mi></math></span> converges to the Brown measure of <span><math><mi>T</mi><mo>+</mo><mi>a</mi></math></span>, where <span><math><mi>T</mi><mo>∈</mo><mi>A</mi></math></span> is an <em>R</em>-diagonal operator freely independent from <em>a</em> and <span><math><mo>|</mo><mi>T</mi><mo>|</mo></math></span> has the same distribution as <em>σ</em>. The assumptions can be removed if <em>A</em> is Hermitian or unitary. By putting <span><math><mi>A</mi><mo>=</mo><mn>0</mn></math></span>, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110797"},"PeriodicalIF":1.7,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128208","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":"Stability threshold of Couette flow for the 3D MHD equations","authors":"Yulin Rao ,&nbsp;Zhifei Zhang ,&nbsp;Ruizhao Zi","doi":"10.1016/j.jfa.2024.110796","DOIUrl":"10.1016/j.jfa.2024.110796","url":null,"abstract":"<div><div>In this paper, we consider the stability of 3D Couette flow <span><math><msup><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mrow><mo>⊤</mo></mrow></msup></math></span> in a uniform background magnetic field <span><math><mi>α</mi><msup><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>⊤</mo></mrow></msup></math></span>. In particular, the MHD equations on <span><math><mi>T</mi><mo>×</mo><mi>R</mi><mo>×</mo><mi>T</mi></math></span> that we are concerned with are of different viscosity coefficient <em>ν</em> and magnetic diffusion coefficient <em>μ</em>. It is shown that if the background magnetic field <span><math><mi>α</mi><msup><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>⊤</mo></mrow></msup></math></span> with <span><math><mi>σ</mi><mo>∈</mo><mi>R</mi><mo>﹨</mo><mi>Q</mi></math></span> satisfying a generic Diophantine condition is so strong that <span><math><mo>|</mo><mi>α</mi><mo>|</mo><mo>≫</mo><mfrac><mrow><mi>ν</mi><mo>+</mo><mi>μ</mi></mrow><mrow><msqrt><mrow><mi>ν</mi><mi>μ</mi></mrow></msqrt></mrow></mfrac></math></span>, and the initial perturbations <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>in</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>in</mi></mrow></msub></math></span> satisfy <span><math><msub><mrow><mo>‖</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>in</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>in</mi></mrow></msub><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn>2</mn></mrow></msup></mrow></msub><mo>≪</mo><mi>min</mi><mo>⁡</mo><mrow><mo>{</mo><mi>ν</mi><mo>,</mo><mi>μ</mi><mo>}</mo></mrow></math></span> for sufficiently large <em>N</em>, then the resulting solution remains close to the steady state in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> at the same order for all time. Compared with the result of Liss [Comm. Math. Phys., 377(2020), 859–908], we use a more general energy method to address the physically relevant case <span><math><mi>ν</mi><mo>≠</mo><mi>μ</mi></math></span> based on some new observations.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110796"},"PeriodicalIF":1.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128170","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":"Second order Sobolev regularity results for the generalized p-parabolic equation","authors":"Yawen Feng ,&nbsp;Mikko Parviainen ,&nbsp;Saara Sarsa","doi":"10.1016/j.jfa.2024.110799","DOIUrl":"10.1016/j.jfa.2024.110799","url":null,"abstract":"<div><div>We study a general class of parabolic equations<span><span><span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msup><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup><mo>(</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>2</mn><mo>)</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math></span></span></span> which can be highly degenerate or singular. This class contains as special cases the standard parabolic <em>p</em>-Laplace equation and the normalized version that arises from stochastic game theory. Utilizing the systematic approach developed in our previous work we establish second order Sobolev regularity together with a priori estimates and improved range of parameters. In addition we derive second order Sobolev estimate for a nonlinear quantity. This quantity contains many useful special cases. As a corollary we also obtain that a viscosity solution has locally <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-integrable Sobolev time derivative.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110799"},"PeriodicalIF":1.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128171","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}