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Clustering and Spectral Analysis of the Infinite Cucker–Smale Model 无限cucker - small模型的聚类与谱分析
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5431-z
Seung-Yeal Ha, Xinyu Wang, Xiaoping Xue
{"title":"Clustering and Spectral Analysis of the Infinite Cucker–Smale Model","authors":"Seung-Yeal Ha,&nbsp;Xinyu Wang,&nbsp;Xiaoping Xue","doi":"10.1007/s10114-026-5431-z","DOIUrl":"10.1007/s10114-026-5431-z","url":null,"abstract":"<div><p>We study clustering dynamics for the infinite Cucker–Smale (ICS) model and its connection to the spectrum of the graph Laplacian. For the ICS model, we overcome the challenge of estimating the velocities of particles and derive a system of dissipative differential inequalities (SDDI) in terms of infinite norms. As in the finite ensemble, we show that mono-cluster flocking emerges exponentially fast, and additionally establish a sufficient framework for algebraic multi-cluster flocking. Moreover, for the CS model with a finite system size, we offer a complete spectral characterization of multi-cluster flocking. Specifically, the emergence of <i>n</i>-cluster behavior corresponds to the limit of the <i>n</i>-th eigenvalue of the time-varying Laplacian approaching zero. In contrast, the lower bound of the (<i>n</i>+1)-th eigenvalue remains strictly positive. Furthermore, we extend this framework to the ICS model by characterizing <i>weak n</i>-cluster flocking via spectral asymptotics, where the <i>n</i>-th eigenvalue tends to zero, while both the (<i>n</i>+1)-th eigenvalue and the infimum of the essential spectrum remain strictly positive. Our results bridge spectral analysis and clustering dynamics, providing indirect evidence for the fast emergence of mono-cluster flocking and the slow relaxation of multi-cluster patterns in both finite and infinite particle systems.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"821 - 880"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752464","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}
引用次数: 0
Mixing for Generic Passive Scalars by Incompressible Flows 不可压缩流动中一般被动标量的混合
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5290-7
Zeyu Jin, Ruo Li
{"title":"Mixing for Generic Passive Scalars by Incompressible Flows","authors":"Zeyu Jin,&nbsp;Ruo Li","doi":"10.1007/s10114-026-5290-7","DOIUrl":"10.1007/s10114-026-5290-7","url":null,"abstract":"<div><p>Mixing by incompressible flows is a ubiquitous yet incompletely understood phenomenon in fluid dynamics. While previous studies have focused on optimal mixing rates, the question of its genericity, i.e., whether mixing occurs for typical incompressible flows and typical initial data, remains mathematically unclear. In this paper, it is shown that classical mixing criteria, e.g. topological mixing or non-precompactness in <i>L</i><sup>2</sup> for all nontrivial densities, fail to persist under arbitrarily small perturbations of velocity fields. A Young-measure theory adapted to <i>L</i><sup>∞</sup> data is then developed to characterize exactly which passive scalars mix. As a consequence, the existence of a single mixed density is equivalent to mixing for generic bounded data, and this equivalence is further tied to the non-precompactness of the associated measure-preserving flow maps in <i>L</i><sup><i>p</i></sup>. These results provide a foundation for a general theory of generic mixing in non-autonomous incompressible flows.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"723 - 739"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10114-026-5290-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752471","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}
引用次数: 0
Closing Lemma and KAM Normal Form 闭引理与KAM范式
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-4598-7
Jinxin Xue
{"title":"Closing Lemma and KAM Normal Form","authors":"Jinxin Xue","doi":"10.1007/s10114-026-4598-7","DOIUrl":"10.1007/s10114-026-4598-7","url":null,"abstract":"<div><p>In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing <i>C</i><sup>1</sup> perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré’s original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh–Robinson’s <i>C</i><sup>1</sup>-closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish–Hofer in this setting and a problem of Arnold. We also discuss the relation of our models to the recent researches on many–body localization in physics.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"541 - 567"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752470","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}
引用次数: 0
Global Regularity of Axisymmetric Euler Equations Without Swirl in Higher Dimensions 高维无旋流轴对称欧拉方程的全局正则性
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5110-0
Feng Shao, Dongyi Wei, Zhifei Zhang
{"title":"Global Regularity of Axisymmetric Euler Equations Without Swirl in Higher Dimensions","authors":"Feng Shao,&nbsp;Dongyi Wei,&nbsp;Zhifei Zhang","doi":"10.1007/s10114-026-5110-0","DOIUrl":"10.1007/s10114-026-5110-0","url":null,"abstract":"<div><p>We consider the axisymmetric incompressible Euler equations without swirl in ℝ<sup><i>d</i></sup> or in a cylinder domain for <i>d</i> ≥ 3. For 3 ≤ <i>d</i> ≤ 6, we prove the global regularity under the following conditions: <span>(u_{0}in L^{2}(mathbb{R}^{d}), {omega_{0}over r^{d-2}}in L^{{dover d-2},infty}(mathbb{R}^{d}))</span> and <span>(min{1,r^{3-d}}{omega_{0}over r^{alpha}}in L^{infty}(mathbb{R}^{d}))</span> for some α ∈ (0, 1). Moreover, if the domain is a cylinder or if <i>ω</i><sub>0</sub> is single-signed, we prove the same global regularity result for all <i>d</i> ≥ 3. Additionally, for 3 ≤ <i>d</i> ≤ 6, we obtain the same growth bounds as in [Lim and Jeong, <i>Arch. Ration. Mech. Anal.</i>, <b>249</b>, Paper No. 32, 31 pp. (2025)], without assuming compact support on the initial data.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"663 - 679"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752462","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}
引用次数: 0
Monotonicity of Harmonic Functions on 3-Manifolds with an Asymptotically Flat End 具有渐近平坦端点的3-流形上调和函数的单调性
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5343-y
Pengzi Miao
{"title":"Monotonicity of Harmonic Functions on 3-Manifolds with an Asymptotically Flat End","authors":"Pengzi Miao","doi":"10.1007/s10114-026-5343-y","DOIUrl":"10.1007/s10114-026-5343-y","url":null,"abstract":"<div><p>We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is also given.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"806 - 820"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752463","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}
引用次数: 0
Total Positivity in the Space of Maximal Tori 最大环面空间中的总正性
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5208-4
George Lusztig
{"title":"Total Positivity in the Space of Maximal Tori","authors":"George Lusztig","doi":"10.1007/s10114-026-5208-4","DOIUrl":"10.1007/s10114-026-5208-4","url":null,"abstract":"<div><p>Let <i>G</i> be a connected reductive group over the complex numbers with a fixed pinning. We define and study the totally positive part of the set of maximal tori of <i>G</i>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"680 - 686"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752472","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}
引用次数: 0
Signal Recovery on Algebraic Varieties Using Linear Samples 线性样本在代数变量上的信号恢复
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-03-15 DOI: 10.1007/s10114-026-5299-y
Zhiqiang Xu
{"title":"Signal Recovery on Algebraic Varieties Using Linear Samples","authors":"Zhiqiang Xu","doi":"10.1007/s10114-026-5299-y","DOIUrl":"10.1007/s10114-026-5299-y","url":null,"abstract":"<div><p>The recovery of an unknown signal from its linear measurements is a fundamental problem spanning numerous scientific and engineering disciplines. Commonly, prior knowledge suggests that the underlying signal resides within a known algebraic variety. This context naturally leads to a question: what is the minimum number of measurements required to uniquely recover any signal belonging to such an algebraic variety? In this survey paper, we introduce a method that leverages tools from algebraic geometry to address this question. We then demonstrate the utility of this approach by applying it to two problems: phase retrieval and low-rank matrix recovery. We also highlight several open problems, which could serve as a basis for future investigations in this field.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 3","pages":"740 - 754"},"PeriodicalIF":0.9,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752469","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}
引用次数: 0
On the Boundedness of Degenerate Hypergraphs 关于退化超图的有界性
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4419-z
Jianfeng Hou, Caiyun Hu, Heng Li, Xizhi Liu, Caihong Yang, Yixiao Zhang
{"title":"On the Boundedness of Degenerate Hypergraphs","authors":"Jianfeng Hou,&nbsp;Caiyun Hu,&nbsp;Heng Li,&nbsp;Xizhi Liu,&nbsp;Caihong Yang,&nbsp;Yixiao Zhang","doi":"10.1007/s10114-026-4419-z","DOIUrl":"10.1007/s10114-026-4419-z","url":null,"abstract":"<div><p>We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs (including graphs). We say an <i>r</i>-graph <i>F</i> is bounded if there exist constants <i>α, β</i> &gt; 0 such that for large <i>n</i>, every <i>n</i>-vertex <i>F</i>-free <i>r</i>-graph with a vertex of degree at least <span>(alpha left({matrix{{n - 1} cr {r - 1}}}right))</span> has fewer than (1 − <i>β</i>) · ex(<i>n, F</i>) edges. The boundedness property is crucial for recent works that aim to extend the classical Hajnal–Szemerédi Theorem (Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs. <i>J. Combin. Theory Ser. B</i>, <b>172</b>, 221–262 (2025)) and the anti-Ramsey theorems of Erdős–Simonovits–Sós (Tight bounds for rainbow partial <i>F</i>-tiling in edge-colored complete hypergraphs. <i>J. Graph Theory</i>, <b>110</b>(4), 457–467 (2025)). We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for 3-graphs, strengthening a theorem by Kostochka–Mubayi–Verstraëte (Turán problems and shadows III: expansions of graphs. <i>SIAM J. Discrete Math.</i>, <b>29</b>(2), 868–876 (2015)).</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"464 - 480"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339045","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}
引用次数: 0
Twist Monomials of Binary Delta-matroids 二元三角拟阵的扭转单项式
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4460-y
Qi Yan, Xian’an Jin
{"title":"Twist Monomials of Binary Delta-matroids","authors":"Qi Yan,&nbsp;Xian’an Jin","doi":"10.1007/s10114-026-4460-y","DOIUrl":"10.1007/s10114-026-4460-y","url":null,"abstract":"<div><p>Recently, we introduced the twist polynomials of delta-matroids and gave a characterization of even normal binary delta-matroids whose twist polynomials have only one term and posed a problem: what would happen for odd binary delta-matroids? In this paper, we show that a normal binary delta-matroid whose twist polynomials have only one term if and only if each connected component of the intersection graph of the delta-matroid is either a complete graph of odd order or a single vertex with a loop.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"377 - 383"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339044","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}
引用次数: 0
Asymptotic Lower and Upper Bounds for Linear Elasticity Eigenvalues 线性弹性特征值的渐近下界和上界
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4391-7
Yifan Yue, Hongtao Chen, Shuo Zhang
{"title":"Asymptotic Lower and Upper Bounds for Linear Elasticity Eigenvalues","authors":"Yifan Yue,&nbsp;Hongtao Chen,&nbsp;Shuo Zhang","doi":"10.1007/s10114-026-4391-7","DOIUrl":"10.1007/s10114-026-4391-7","url":null,"abstract":"<div><p>Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete <b>displacement</b> may lead to lower bound and that of <b>pressure</b> to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"357 - 376"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339039","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}
引用次数: 0
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