Journal of Geometric Analysis最新文献

{"title":"On the Cheeger Inequality in Carnot-Carathéodory Spaces.","authors":"Martijn Kluitenberg","doi":"10.1007/s12220-025-01912-w","DOIUrl":"https://doi.org/10.1007/s12220-025-01912-w","url":null,"abstract":"<p><p>We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carathéodory spaces and we describe a concrete method to lower bound the Cheeger constant. The proof is geometric, and works for Dirichlet, Neumann and mixed boundary conditions. One of the main technical tools in the proof is a generalization of Courant's nodal domain theorem, which is proven from scratch for Neumann and mixed boundary conditions. Carnot groups and the Baouendi-Grushin cylinder are treated as examples.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 3","pages":"82"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11880090/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143574622","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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Free Boundary Hamiltonian Stationary Lagrangian Discs in <ns0:math> <ns0:msup><ns0:mrow><ns0:mi>C</ns0:mi></ns0:mrow> <ns0:mn>2</ns0:mn></ns0:msup></ns0:math>.","authors":"Filippo Gaia","doi":"10.1007/s12220-025-01962-0","DOIUrl":"https://doi.org/10.1007/s12220-025-01962-0","url":null,"abstract":"<p><p>Let <math><mrow><mi>Ω</mi> <mo>⊂</mo> <msup><mrow><mi>C</mi></mrow> <mn>2</mn></msup> </mrow> </math> be a smooth domain. We establish conditions under which a weakly conformal, branched <math><mi>Ω</mi></math> -free boundary Hamiltonian stationary Lagrangian immersion <i>u</i> of a disc in <math> <msup><mrow><mi>C</mi></mrow> <mn>2</mn></msup> </math> is a <math><mi>Ω</mi></math> -free boundary minimal immersion. We deduce that if <math><mi>u</mi></math> is a weakly conformal, branched <math> <mrow><msub><mi>B</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mn>0</mn> <mo>)</mo></mrow> </mrow> </math> -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then <math><mrow><mi>u</mi> <mo>(</mo> <msup><mi>D</mi> <mn>2</mn></msup> <mo>)</mo></mrow> </math> is a Lagrangian equatorial plane disc. Furthermore, we present examples of <math><mi>Ω</mi></math> -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 5","pages":"160"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971164/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797164","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":"Geometric Bounds for Low Steklov Eigenvalues of Finite Volume Hyperbolic Surfaces.","authors":"Asma Hassannezhad, Antoine Métras, Hélène Perrin","doi":"10.1007/s12220-025-01990-w","DOIUrl":"https://doi.org/10.1007/s12220-025-01990-w","url":null,"abstract":"<p><p>We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the corresponding Steklov eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 5","pages":"158"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971064/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797170","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":"On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian.","authors":"Sam Farrington","doi":"10.1007/s12220-024-01887-0","DOIUrl":"https://doi.org/10.1007/s12220-024-01887-0","url":null,"abstract":"<p><p>We consider the problem of minimising the <i>k</i>-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension <math><mrow><mi>d</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as <math><mrow><mi>k</mi> <mo>→</mo> <mo>+</mo> <mi>∞</mi></mrow> </math> . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension <math><mrow><mi>d</mi> <mo>=</mo> <mn>2</mn></mrow> </math> . We also consider these problems for Robin eigenvalues and mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 2","pages":"62"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11811466/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143410705","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":"More Weakly Biharmonic Maps from the Ball to the Sphere.","authors":"Volker Branding","doi":"10.1007/s12220-024-01852-x","DOIUrl":"10.1007/s12220-024-01852-x","url":null,"abstract":"<p><p>In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable critical points of the bienergy.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 1","pages":"23"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11584471/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142711340","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":"On Isolated Singularities and Generic Regularity of Min-Max CMC Hypersurfaces.","authors":"Costante Bellettini, Kobe Marshall-Stevens","doi":"10.1007/s12220-025-01956-y","DOIUrl":"https://doi.org/10.1007/s12220-025-01956-y","url":null,"abstract":"<p><p>In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure in Bellettini and Wickramasekera (arXiv:2010.05847, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real <math><mi>λ</mi></math> we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature <math><mi>λ</mi></math> ; the minimal case ( <math><mrow><mi>λ</mi> <mo>=</mo> <mn>0</mn></mrow> </math> ) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 4","pages":"126"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11920008/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143671740","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":"On the Regularity Problem for Parabolic Operators and the Role of Half-Time Derivative.","authors":"Martin Dindoš","doi":"10.1007/s12220-025-01991-9","DOIUrl":"https://doi.org/10.1007/s12220-025-01991-9","url":null,"abstract":"<p><p>In this paper we present the following result on regularity of solutions of the second order parabolic equation <math> <mrow><msub><mi>∂</mi> <mi>t</mi></msub> <mi>u</mi> <mo>-</mo> <mrow><mspace></mspace> <mtext>div</mtext> <mspace></mspace></mrow> <mrow><mo>(</mo> <mi>A</mi> <mi>∇</mi> <mi>u</mi> <mo>)</mo></mrow> <mo>+</mo> <mi>B</mi> <mo>·</mo> <mi>∇</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn></mrow> </math> on cylindrical domains of the form <math><mrow><mi>Ω</mi> <mo>=</mo> <mi>O</mi> <mo>×</mo> <mi>R</mi></mrow> </math> where <math><mrow><mi>O</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> </mrow> </math> is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is <math><mrow><mi>n</mi> <mo>-</mo> <mn>1</mn></mrow> </math> -Ahlfors regular. Let <i>u</i> be a solution of such PDE in <math><mi>Ω</mi></math> and the non-tangential maximal function of its gradient in spatial directions <math> <mrow><mover><mi>N</mi> <mo>~</mo></mover> <mrow><mo>(</mo> <mi>∇</mi> <mi>u</mi> <mo>)</mo></mrow> </mrow> </math> belongs to <math> <mrow><msup><mi>L</mi> <mi>p</mi></msup> <mrow><mo>(</mo> <mi>∂</mi> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> for some <math><mrow><mi>p</mi> <mo>></mo> <mn>1</mn></mrow> </math> . Furthermore, assume that for <math> <mrow> <msub><mrow><mi>u</mi> <mo>|</mo></mrow> <mrow><mi>∂</mi> <mi>Ω</mi></mrow> </msub> <mo>=</mo> <mi>f</mi></mrow> </math> we have that <math> <mrow><msubsup><mi>D</mi> <mi>t</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msubsup> <mi>f</mi> <mo>∈</mo> <msup><mi>L</mi> <mi>p</mi></msup> <mrow><mo>(</mo> <mi>∂</mi> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> . Then both <math> <mrow><mover><mi>N</mi> <mo>~</mo></mover> <mrow><mo>(</mo> <msubsup><mi>D</mi> <mi>t</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msubsup> <mi>u</mi> <mo>)</mo></mrow> </mrow> </math> and <math> <mrow><mover><mi>N</mi> <mo>~</mo></mover> <mrow><mo>(</mo> <msubsup><mi>D</mi> <mi>t</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msubsup> <msub><mi>H</mi> <mi>t</mi></msub> <mi>u</mi> <mo>)</mo></mrow> </mrow> </math> also belong to <math> <mrow><msup><mi>L</mi> <mi>p</mi></msup> <mrow><mo>(</mo> <mi>∂</mi> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> , where <math><msubsup><mi>D</mi> <mi>t</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msubsup> </math> and <math><msub><mi>H</mi> <mi>t</mi></msub> </math> are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the <math><msup><mi>L</mi> <mi>p</mi></msup> </math> parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 5","pages":"154"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11965225/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797176","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":"Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves.","authors":"Martin Bauer, Patrick Heslin, Cy Maor","doi":"10.1007/s12220-024-01652-3","DOIUrl":"https://doi.org/10.1007/s12220-024-01652-3","url":null,"abstract":"<p><p>We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order <math><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math>. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if <math><mrow><mi>q</mi><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if <math><mrow><mi>q</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math>, whereas if <math><mrow><mi>q</mi><mo><</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math> then finite-time blowup may occur. The geodesic completeness for <math><mrow><mi>q</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math> is obtained by proving metric completeness of the space of <math><msup><mi>H</mi><mi>q</mi></msup></math>-immersed curves with the distance induced by the Riemannian metric.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 7","pages":"214"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11068588/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140857102","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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The <ns0:math><ns0:msub><ns0:mi>A</ns0:mi><ns0:mi>∞</ns0:mi></ns0:msub></ns0:math> Condition, <ns0:math><ns0:mi>ε</ns0:mi></ns0:math>-Approximators, and Varopoulos Extensions in Uniform Domains.","authors":"S Bortz, B Poggi, O Tapiola, X Tolsa","doi":"10.1007/s12220-024-01666-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01666-x","url":null,"abstract":"<p><p>Suppose that <math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math>, <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>, is a uniform domain with <i>n</i>-Ahlfors regular boundary and <i>L</i> is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in <math><mi>Ω</mi></math>. We show that the corresponding elliptic measure <math><msub><mi>ω</mi><mi>L</mi></msub></math> is quantitatively absolutely continuous with respect to surface measure of <math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math> in the sense that <math><mrow><msub><mi>ω</mi><mi>L</mi></msub><mo>∈</mo><msub><mi>A</mi><mi>∞</mi></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math> if and only if any bounded solution <i>u</i> to <math><mrow><mi>L</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math> in <math><mi>Ω</mi></math> is <math><mi>ε</mi></math>-approximable for any <math><mrow><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math>. By <math><mi>ε</mi></math>-approximability of <i>u</i> we mean that there exists a function <math><mrow><mi>Φ</mi><mo>=</mo><msup><mi>Φ</mi><mi>ε</mi></msup></mrow></math> such that <math><mrow><msub><mrow><mo>‖</mo><mi>u</mi><mo>-</mo><mi>Φ</mi><mo>‖</mo></mrow><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><mi>ε</mi><msub><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub></mrow></math> and the measure <math><msub><mover><mi>μ</mi><mo>~</mo></mover><mi>Φ</mi></msub></math> with <math><mrow><mi>d</mi><mover><mi>μ</mi><mo>~</mo></mover><mo>=</mo><mrow><mo>|</mo><mi>∇</mi><mi>Φ</mi><mrow><mo>(</mo><mi>Y</mi><mo>)</mo></mrow><mo>|</mo></mrow><mspace></mspace><mi>d</mi><mi>Y</mi></mrow></math> is a Carleson measure with <math><msup><mi>L</mi><mi>∞</mi></msup></math> control over the Carleson norm. As a consequence of this approximability result, we show that boundary <math><mrow><mspace></mspace><mtext>BMO</mtext><mspace></mspace></mrow></math> functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy <math><msup><mi>L</mi><mn>1</mn></msup></math>-type Carleson measure estimates with <math><mrow><mspace></mspace><mtext>BMO</mtext><mspace></mspace></mrow></math> control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 7","pages":"218"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11087277/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140913353","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":"On the Dimension of the Singular Set of Perimeter Minimizers in Spaces with a Two-Sided Bound on the Ricci Curvature.","authors":"Alessandro Cucinotta, Francesco Fiorani","doi":"10.1007/s12220-024-01784-6","DOIUrl":"https://doi.org/10.1007/s12220-024-01784-6","url":null,"abstract":"<p><p>We show that the Hausdorff dimension of the singular set of perimeter minimizers in noncollapsed limits of manifolds with two-sided bounds on the Ricci curvature is at most <math><mrow><mi>n</mi> <mo>-</mo> <mn>5</mn></mrow> </math> , where <i>n</i> is the dimension of the ambient space. The estimate is sharp.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 12","pages":"381"},"PeriodicalIF":1.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11845540/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143484951","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}