Unlike graphs, for an integer r > 3 and many classical families of r -uniform hypergraphs M , there are perhaps more than one M -free r -uniform hypergraphs with n vertices and the maximum possible number of edges (such hypergraphs are called extremal con(cid:12)gurations). Moreover, those extremal con(cid:12)gurations are far from each other in edit-distance. Such a phenomenon is called not stable and is a fundamental barrier to determining the Tur(cid:19)an number of M . Liu and Mubayi (2022) gave the (cid:12)rst example for 3-uniform hypergraphs to be not stable. A simple argument shows that for r > 4, one can get a family of r -uniform hypergraphs which is not stable through a not stable family of 3-uniform hypergraphs. In this paper, we construct a (cid:12)nite family of 4-uniform hypergraphs M directly such that two near-extremal M -free con(cid:12)gurations are far from each other in edit-distance. This is the (cid:12)rst unstable example that does not depend on 3-uniform hypergraphs. We also prove its Andr(cid:19)asfai-Erd}os-S(cid:19)os type stability theorem: Every M -free 4-uniform hypergraph whose minimum degree is close to the average degree of extremal con(cid:12)gurations is a subgraph of one of these two near-extremal con(cid:12)gurations. As a corollary, our main result shows that the boundary of the feasible region of M has exactly two global maxima.
与图不同，对于整数 r > 3 和许多经典的 r -Uniform 超图族 M，可能存在不止一个具有 n 个顶点和最大可能边数的无 M 的 r -Uniform 超图（这种超图称为极值 con(cid:12)gurations）。此外，这些极值 con(cid:12)gurations 在编辑距离上彼此相距甚远。这种现象被称为不稳定，是确定 M 的 Tur(cid:19)an 数的基本障碍。Liu 和 Mubayi（2022 年）给出了 (cid:12)3-uniform 超图不稳定的第一个例子。一个简单的论证表明，当 r > 4 时，可以通过一个不稳定的 3-Uniform 超图族得到一个不稳定的 r-Uniform 超图族。在本文中，我们直接构造了一个 (cid:12)nite 的 4-Uniform 超图 M 族，使得两个近极值的无 M con(cid:12)gurations 在编辑距离上相距很远。这是第一个不依赖于 3-Uniform 超图的不稳定例子。我们还证明了它的 Andr(cid:19)asfai-Erd}os-S(cid:19)os 型稳定性定理：每一个无 M 的 4-Uniform 超图，其最小度数都接近极值图的平均度数，都是这两个近极值图之一的子图。作为推论，我们的主要结果表明 M 的可行区域边界恰好有两个全局最大值。
{"title":"A family of 4-uniform hypergraphs with no stability","authors":"Yixiao Zhang, Li Heng, Jianfeng Hou","doi":"10.1360/ssm-2023-0059","DOIUrl":"https://doi.org/10.1360/ssm-2023-0059","url":null,"abstract":"Unlike graphs, for an integer r > 3 and many classical families of r -uniform hypergraphs M , there are perhaps more than one M -free r -uniform hypergraphs with n vertices and the maximum possible number of edges (such hypergraphs are called extremal con(cid:12)gurations). Moreover, those extremal con(cid:12)gurations are far from each other in edit-distance. Such a phenomenon is called not stable and is a fundamental barrier to determining the Tur(cid:19)an number of M . Liu and Mubayi (2022) gave the (cid:12)rst example for 3-uniform hypergraphs to be not stable. A simple argument shows that for r > 4, one can get a family of r -uniform hypergraphs which is not stable through a not stable family of 3-uniform hypergraphs. In this paper, we construct a (cid:12)nite family of 4-uniform hypergraphs M directly such that two near-extremal M -free con(cid:12)gurations are far from each other in edit-distance. This is the (cid:12)rst unstable example that does not depend on 3-uniform hypergraphs. We also prove its Andr(cid:19)asfai-Erd}os-S(cid:19)os type stability theorem: Every M -free 4-uniform hypergraph whose minimum degree is close to the average degree of extremal con(cid:12)gurations is a subgraph of one of these two near-extremal con(cid:12)gurations. As a corollary, our main result shows that the boundary of the feasible region of M has exactly two global maxima.","PeriodicalId":513480,"journal":{"name":"SCIENTIA SINICA Mathematica","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140272105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuhuan Li, Jianyu Jing, Qianqian Liu, Wang Cheng, Wenbin Chen
{"title":"A third-order positivity-preserving and energy stable numerical scheme for the Cahn-Hilliard equation with logarithmic potential","authors":"Yuhuan Li, Jianyu Jing, Qianqian Liu, Wang Cheng, Wenbin Chen","doi":"10.1360/ssm-20223-0014","DOIUrl":"https://doi.org/10.1360/ssm-20223-0014","url":null,"abstract":"","PeriodicalId":513480,"journal":{"name":"SCIENTIA SINICA Mathematica","volume":"21 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139889590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuhuan Li, Jianyu Jing, Qianqian Liu, Wang Cheng, Wenbin Chen
{"title":"A third-order positivity-preserving and energy stable numerical scheme for the Cahn-Hilliard equation with logarithmic potential","authors":"Yuhuan Li, Jianyu Jing, Qianqian Liu, Wang Cheng, Wenbin Chen","doi":"10.1360/ssm-20223-0014","DOIUrl":"https://doi.org/10.1360/ssm-20223-0014","url":null,"abstract":"","PeriodicalId":513480,"journal":{"name":"SCIENTIA SINICA Mathematica","volume":"43 43","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139829907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity when the initial velocity is sufficiently small in the critical Besov space $dot{B}^{frac 12}$. Compared with the previous result of Abidi and Zhang (Science China Mathematics 58 (6) (2015) 1129-1150), we remove the smallness assumption of the viscosity $mu(rho_0)-1$ in $L^{infty}$-norm.
{"title":"On the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity","authors":"Dongjuan Niu, Lu Wang","doi":"10.1360/SCM-2023-0384","DOIUrl":"https://doi.org/10.1360/SCM-2023-0384","url":null,"abstract":"In this paper, we are concerned with the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity when the initial velocity is sufficiently small in the critical Besov space $dot{B}^{frac 12}$. Compared with the previous result of Abidi and Zhang (Science China Mathematics 58 (6) (2015) 1129-1150), we remove the smallness assumption of the viscosity $mu(rho_0)-1$ in $L^{infty}$-norm.","PeriodicalId":513480,"journal":{"name":"SCIENTIA SINICA Mathematica","volume":"46 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139634839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A multi-level correction scheme for biharmonic eigenvalue problem by nonconforming finite element method","authors":"Yingxia Xi, Xie Hehu, Ji Xia","doi":"10.1360/scm-2022-0215","DOIUrl":"https://doi.org/10.1360/scm-2022-0215","url":null,"abstract":"","PeriodicalId":513480,"journal":{"name":"SCIENTIA SINICA Mathematica","volume":"233 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139636383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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