To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibili...To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.展开更多
A rhodium-catalyzed directing group promoted selective C-H olefination reaction of indolizines at the 8-position is re ported.Di-olefination at 2,8-positions also achieved with silver hexafluoroantimonate as an additi...A rhodium-catalyzed directing group promoted selective C-H olefination reaction of indolizines at the 8-position is re ported.Di-olefination at 2,8-positions also achieved with silver hexafluoroantimonate as an additive under similar reaction conditions.Weakly coordinating groups,such as ketone,alde hyde,amide and ester,were used as directing groups.The ester group can be removed under acid conditions and therefore is used as a traceless directing group.展开更多
Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notion...Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.展开更多
We present the weighted weak group inverse,which is a new generalized inverse of operators between two Hilbert spaces,and we extend the notation of the weighted weak group inverse for rectangular matrices.Some charact...We present the weighted weak group inverse,which is a new generalized inverse of operators between two Hilbert spaces,and we extend the notation of the weighted weak group inverse for rectangular matrices.Some characterizations and representations of the weighted weak group inverse are investigated.We also apply these results to define and study the weak group inverse for a Hilbert space operator.Using the weak group inverse,we define and characterize various binary relations.展开更多
We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for p to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, ...We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for p to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.展开更多
基金The National Natural Science Foundation of China(No.12171083,12071070)Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(No.KYCX22_0231).
文摘To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.
基金Jiangsu Province(No.BK20161307 and“333”Talents Project for H.Hu)Huaiyin Normal University(No.JSKC18014)for their financial support。
文摘A rhodium-catalyzed directing group promoted selective C-H olefination reaction of indolizines at the 8-position is re ported.Di-olefination at 2,8-positions also achieved with silver hexafluoroantimonate as an additive under similar reaction conditions.Weakly coordinating groups,such as ketone,alde hyde,amide and ester,were used as directing groups.The ester group can be removed under acid conditions and therefore is used as a traceless directing group.
基金Supported by the Natural Science Foundation of Shandong Province(ZR2012AL02)
文摘Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.
基金The first author was supported by the Ministry of Education,Science and Technological Development,Republic of Serbia,Grant No.174007(451-03-68/2020-14/200124)The second author was supported by the National Natural Science Foundation of China(Grant Nos.11901079,61672149,11601211)the Scientific and Technological Research Program Foundation of Jilin Province,China(Grant Nos.JJKH20190690KJ,20190201095JC,20200401085GX.)。
文摘We present the weighted weak group inverse,which is a new generalized inverse of operators between two Hilbert spaces,and we extend the notation of the weighted weak group inverse for rectangular matrices.Some characterizations and representations of the weighted weak group inverse are investigated.We also apply these results to define and study the weak group inverse for a Hilbert space operator.Using the weak group inverse,we define and characterize various binary relations.
基金Supported by National Natural Science Foundation of China （Grant No. 10871042）, National Science Foundation of Jiangsu Province （Grant No. BK2009258） and Science Foundation for The Excellent Youth Scholars of Jiaxing University （Grant No. 70609010） and Scientific Research Foundation of Jiaxing University （Grant No. 70509015）
文摘We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for p to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.
