The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n≤7. For dimension n 〉 7, the same result is established under the spin assumption.
Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension result...Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.展开更多
With the great advancement of experimental tools,a tremendous amount of biomolecular data has been generated and accumulated in various databases.The high dimensionality,structural complexity,the nonlinearity,and enta...With the great advancement of experimental tools,a tremendous amount of biomolecular data has been generated and accumulated in various databases.The high dimensionality,structural complexity,the nonlinearity,and entanglements of biomolecular data,ranging from DNA knots,RNA secondary structures,protein folding configurations,chromosomes,DNA origami,molecular assembly,to others at the macromolecular level,pose a severe challenge in their analysis and characterization.In the past few decades,mathematical concepts,models,algorithms,and tools from algebraic topology,combinatorial topology,computational topology,and topological data analysis,have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge.In this work,we introduce biomolecular topology,which concerns the topological problems and models originated from the biomolecular systems.More specifically,the biomolecular topology encompasses topological structures,properties and relations that are emerged from biomolecular structures,dynamics,interactions,and functions.We discuss the various types of biomolecular topology from structures(of proteins,DNAs,and RNAs),protein folding,and protein assembly.A brief discussion of databanks(and databases),theoretical models,and computational algorithms,is presented.Further,we systematically review related topological models,including graphs,simplicial complexes,persistent homology,persistent Laplacians,de Rham-Hodge theory,Yau-Hausdorff distance,and the topology-based machine learning models.展开更多
文摘The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n≤7. For dimension n 〉 7, the same result is established under the spin assumption.
基金Research partially supported by NSF Grant DMS-0404729
文摘Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.
基金supported by Nanyang Technological University Startup Grant M4081842Singapore Ministry of Education Academic Research fund Tier 1 RG109/19,MOE-T2EP20120-0013 and MOE-T2EP20220-0010+10 种基金supported by NIH grant GM126189NSF grants DMS-2052983,DMS-1761320,and IIS-1900473supported by Natural Science Foundation of China(NSFC)grant(11971144)Highlevel Scientific Research Foundation of Hebei Provincethe Start-up Research Fund from Yanqi Lake Beijing Institute of Mathematical Sciences and Applicationssupported by Tianjin Natural Science Foundation(Grant No.19JCYBJC30200)supported by National Natural Science Foundation of China(NSFC)grant(12171275)Tsinghua University Spring Breeze Fund(2020Z99CFY044)Tsinghua University Start-up FundTsinghua University Education Foundation fund(042202008)National Center for Theoretical Sciences(NCTS)for providing an excellent research environment while part of this research was done。
文摘With the great advancement of experimental tools,a tremendous amount of biomolecular data has been generated and accumulated in various databases.The high dimensionality,structural complexity,the nonlinearity,and entanglements of biomolecular data,ranging from DNA knots,RNA secondary structures,protein folding configurations,chromosomes,DNA origami,molecular assembly,to others at the macromolecular level,pose a severe challenge in their analysis and characterization.In the past few decades,mathematical concepts,models,algorithms,and tools from algebraic topology,combinatorial topology,computational topology,and topological data analysis,have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge.In this work,we introduce biomolecular topology,which concerns the topological problems and models originated from the biomolecular systems.More specifically,the biomolecular topology encompasses topological structures,properties and relations that are emerged from biomolecular structures,dynamics,interactions,and functions.We discuss the various types of biomolecular topology from structures(of proteins,DNAs,and RNAs),protein folding,and protein assembly.A brief discussion of databanks(and databases),theoretical models,and computational algorithms,is presented.Further,we systematically review related topological models,including graphs,simplicial complexes,persistent homology,persistent Laplacians,de Rham-Hodge theory,Yau-Hausdorff distance,and the topology-based machine learning models.
