A discrete subset S of a topological gyrogroup G with the identity 0 is said to be a suitable set for G if it generates a dense subgyrogroup of G and S∪{0}is closed in G.In this paper,it is proved that each countable...A discrete subset S of a topological gyrogroup G with the identity 0 is said to be a suitable set for G if it generates a dense subgyrogroup of G and S∪{0}is closed in G.In this paper,it is proved that each countable Hausdorff topological gyrogroup has a suitable set;moreover,it is shown that each separable metrizable strongly topological gyrogroup has a suitable set.展开更多
In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannia...In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannian manifolds.The authors rigorously establish statistical properties of the estimators,providing formal proofs of their consistency and asymptotic behaviors.The effectiveness of our method is demonstrated through extensive simulations and applications to real-world datasets which highlight its practical utility for complex data with non-Euclidean structures.展开更多
Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algeb...Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algebraic subsets of G forms a family of closed sets for another T_(1)topology Z_(G) on G called the Zarski topology.A subgroup H of G is said to be Markov(resp.Zarski)embedded if the equality M_(G|H)=M_(H)(resp.Z_(G|H)=Z_(H))holds.I's proved that an abirary subgroup of a free group is both Zariski and Markov embedded in it.展开更多
基金supported by Fujian Provincial Natural Science Foundation of China(2024J02022)the NSFC(11571158)+1 种基金supported by the NSFC(12071199)supported by the Young and middle-aged project in Fujian Province(JAT190397)。
文摘A discrete subset S of a topological gyrogroup G with the identity 0 is said to be a suitable set for G if it generates a dense subgyrogroup of G and S∪{0}is closed in G.In this paper,it is proved that each countable Hausdorff topological gyrogroup has a suitable set;moreover,it is shown that each separable metrizable strongly topological gyrogroup has a suitable set.
文摘In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannian manifolds.The authors rigorously establish statistical properties of the estimators,providing formal proofs of their consistency and asymptotic behaviors.The effectiveness of our method is demonstrated through extensive simulations and applications to real-world datasets which highlight its practical utility for complex data with non-Euclidean structures.
基金Supported by the Grant-in-Aid for Scientific Research(C)by the Japan Society for the Promotion of Science(20K03615)。
文摘Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algebraic subsets of G forms a family of closed sets for another T_(1)topology Z_(G) on G called the Zarski topology.A subgroup H of G is said to be Markov(resp.Zarski)embedded if the equality M_(G|H)=M_(H)(resp.Z_(G|H)=Z_(H))holds.I's proved that an abirary subgroup of a free group is both Zariski and Markov embedded in it.
