The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left...The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.展开更多
Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which ...Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.展开更多
基金The National Natural Science Foundation of China(No.10871042)the Natural Science Foundation of Jiangsu Province(No.BK2009258)
文摘The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.
基金the National Natural Science Foundation of China(NSFC)under Grant No.12171321.
文摘Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.
