This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin h...This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.展开更多
Let G be a locally compact Lie group and its Lie algebra. We consider a fuzzy analogue of G, denoted by called a fuzzy Lie group. Spherical functions on are constructed and a version of the existence result of the Hel...Let G be a locally compact Lie group and its Lie algebra. We consider a fuzzy analogue of G, denoted by called a fuzzy Lie group. Spherical functions on are constructed and a version of the existence result of the Helgason-spherical function on G is then established on .展开更多
Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent...Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.展开更多
Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent m...Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).展开更多
In this work,we report long-term trends in the abundance and breeding performance of Adélie penguins(Pygoscelis adeliae)nesting in three Antarctic colonies(i.e.,at Martin Point,South Orkneys Islands;Stranger Poin...In this work,we report long-term trends in the abundance and breeding performance of Adélie penguins(Pygoscelis adeliae)nesting in three Antarctic colonies(i.e.,at Martin Point,South Orkneys Islands;Stranger Point/Cabo Funes,South Shetland Islands;and Esperanza/Hope Bay in the Antarctic Peninsula)from 1995/96 to 2022/23.Using yearly count data of breeding groups selected,we observed a decline in the number of breeding pairs and chicks in crèche at all colonies studied.However,the magnitude of change was higher at Stranger Point than that in the remaining colonies.Moreover,the index of breeding success,which was calculated as the ratio of chicks in crèche to breeding pairs,exhibited no apparent trend throughout the study period.However,it displayed greater variability at Martin Point compared to the other two colonies under investigation.Although the number of chicks in crèche of Adélie penguins showed a declining pattern,the average breeding performance was similar to that reported in gentoo penguin colonies,specifically,those undergoing a population increase(even in sympatric colonies facing similar local conditions).Consequently,it is plausible to assume a reduction of the over-winter survival as a likely cause of the declining trend observed,at least in the Stranger Point and Esperanza colonies.However,we cannot rule out local effects during the breeding season affecting the Adélie population of Martin Point.展开更多
The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vect...The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.展开更多
文摘This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.
文摘Let G be a locally compact Lie group and its Lie algebra. We consider a fuzzy analogue of G, denoted by called a fuzzy Lie group. Spherical functions on are constructed and a version of the existence result of the Helgason-spherical function on G is then established on .
文摘Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.
文摘Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).
基金Nacional de Promoción Científica y Tecnológica(Grant:PICTO 2010-0111)the Instituto Antártico Argentino-Dirección Nacional del Antártico(PINST-05)provided financial and logistical support.
文摘In this work,we report long-term trends in the abundance and breeding performance of Adélie penguins(Pygoscelis adeliae)nesting in three Antarctic colonies(i.e.,at Martin Point,South Orkneys Islands;Stranger Point/Cabo Funes,South Shetland Islands;and Esperanza/Hope Bay in the Antarctic Peninsula)from 1995/96 to 2022/23.Using yearly count data of breeding groups selected,we observed a decline in the number of breeding pairs and chicks in crèche at all colonies studied.However,the magnitude of change was higher at Stranger Point than that in the remaining colonies.Moreover,the index of breeding success,which was calculated as the ratio of chicks in crèche to breeding pairs,exhibited no apparent trend throughout the study period.However,it displayed greater variability at Martin Point compared to the other two colonies under investigation.Although the number of chicks in crèche of Adélie penguins showed a declining pattern,the average breeding performance was similar to that reported in gentoo penguin colonies,specifically,those undergoing a population increase(even in sympatric colonies facing similar local conditions).Consequently,it is plausible to assume a reduction of the over-winter survival as a likely cause of the declining trend observed,at least in the Stranger Point and Esperanza colonies.However,we cannot rule out local effects during the breeding season affecting the Adélie population of Martin Point.
基金Fundamental Research Funds for the Central Universities,China(No.2232021G13)。
文摘The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.
