founded by the Swiss National Science Foundation grant nos.200756 and 169542.
Abiotic stresses such as heat and frost limit plant growth and productivity.Image-based field phenotyping methods allow quantifying not only plant growth but also plant senescence.Winter crops show senescence caused b...
The National Natural Science Foundation of China(No.11371088,11571173,11871144);the Natural Science Foundation of Jiangsu Province(No.BK20171348)
The condition of an algebra to be a Hopf algebra or a Hopf(co)quasigroup can be determined by the properties of Galois linear maps.For a bialgebra H,if it is unital and associative as an algebra and counital coassocia...
Supported by the National Nature Science Foundation of China(Grant No.10901098 and No.11271239)
The antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map is proved. It is also proved that the tensor algebra of Yetter-Drinfeld Hopf module is a Yetter-Drinfeld Hopf algebra.
The Leibniz-Hopf algebra is the free associative Z - algebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the 'ring ofnoncommutative symmetric fun...
The Leibniz-Hopf algebra is the free associative algebra with one generator in each positive degree and coproduct given by the Cartan formula. Quasi-symmetric functions are a generalisation of symmetric functions [7],...
the National Natural Science Foundation of China (Nos.10671016; 10771014);the Foundation of Beijing Educational Committee (No.KM200710005013)
In this paper a class of two-parameter weak Hopf algebras wrτ,s(g) corresponding to Borcherds-Cartan matrix is constructed. This class consists of noncommutative and noncocommutative weak Hopf algebras but not Hopf a...
Supported by the National Natural Science Foundation of China;Yangzhou University Natural Science Foundation
Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras (coalgebras), we investigate the basic properties of the graded antipode and the st...
Starting from shikimic acid, the total synthesis of zeylenone was studied. The productwas proved to be the (+)antipode of zeylenone through analysis and comparison of their respectivespectra (including NMR, MS, IR and...
Partially supported by the National Natural Science Foundation of China.
In this paper, we obtain the following main theorem for a free quadratic bialgebra J: (a) For p≠0, J is a pointed cosemisimple coalgebra. For p=0, J is a hyperalgebra. (b) For p≠0 and q≠0, J has antipode S iff p·q...