(f,ω)-Compatible Pair (B,H) for ω-Smash Coproduct Hopf Algebras

In this paper we introduce the notion of (f,ω)-compatible pair (B,H), by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

Banach空间中含$(H,\eta)$单调算子变分包含组解的迭代法

In this paper, we introduce and study a new system of variational inclusions involving （H, η）-monotone operators in Banach space. Using the resolveut operator associated with （H, η）- monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the iterative sequence generated by the algorithm.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

Wely定理与$(\omega)$性质的等价性

We call T C B（H） consistent in Fredholm and index （briefly a CFI operator） if for each B ∈ B（H）, TB and BT are Fredholm together and the same index of B, or not Fredholm together. Using a new spectrum defined in view of the CFI operator, we give the equivalence of Weyl＇s theorem and property （ω） for T and its conjugate operator T^＊. In addition, the property （ω） for operator matrices is considered.     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

A Note on Generalized Long Mo dules

Let HLB be the category of generalized Long modules, that is, H-modules and B-comodules over Hopf algebras B and H. We describe a new Turaev braided group category over generalized Long module HLB （S（π）） where the opposite group S（π） of the semidirect product of the opposite group πopof a group π by π. As an application, we show that this is a Turaev braided group-category HLBfor a quasitriangular Turaev group-coalgebra H and a coquasitriangular Turaev group-algebra B.     

Generalized braided Hopf algebras

The concept of （f, σ）-pair （B, H）is introduced, where B and H are Hopf algebras. A braided tensor category which is a tensor subcategory of the category ^HM of left H-comodules through an （f, σ）-pair is constructed. In particularly, a Yang-Baxter equation is got. A Hopf algebra is constructed as well in the Yetter-Drinfel＇d category H^HYD by twisting the multiplication of B.     

$(\omega')$性质及其摄动

In this note we define the property （ω′）, a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property （ω′） holds by means of the variant of the essential approximate point spectrum σ1（·） and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property （ω′） is discussed.     

The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.     

群Yetter-Drinfel'd范畴中的Hopf代数

In this note we first show that if H is a finite-dimensional Hopf algebra in a group Yetter-Drinfel＇d category L^LyD（π） over a crossed Hopf group-coalgebra L, then its dual H^＊ is also a Hopf algebra in the category L^LyD（π）. Then we establish the fundamental theorem of Hopf modules for H in the category L^LyD（π）.     

$(\omega)$性质及其摄动

In the note,we establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for the stability of property(ω) by means of the variant of the essential approximate point spectrum and the induced spectrum of consistency in Fredholm and index.In addition,the stability of property(ω) for H(P) operators is considered.     

On Rees Factor $S$-Posets Satisfying Condition $(P_w)$

In this paper, we present some homological classifications of pomonoids by using Rees factor $S$-posets satisfying condition $(P_w)$.     

Universal approach to the Takesaki-Takai $\gamma $-duality for crossed products

In this article, we generalize and simplify the proof of the Takesaki-Takai $\gamma $-duality theorem. Assume a morphism \textbf{\textit{$\omega \; :\; G\to Aut\left({\rm A}\right)$}} is a projective representation of the locally compact Abel group \textbf{\textit{$G$}} in \textbf{\textit{$Aut\left({\rm A}\right)$}}, mapping $\gamma \; :\; G\to G$ is continuous, and $\left({\rm A},\; G,\; \omega \right)$ is a dynamic system then there exists isomorphism \[\Upsilon \; :\; Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\to {\rm A}\otimes LK\left(L^{2} \left(G\right)\right) \] which is the equivariant for the double dual action \[\hat{\hat{\omega }}\; :\; G\to Aut\left(Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\right).\] These results deepen our understanding of the representation theory and are especially interesting given their possible applications to problems of the quantum theory.     

$(m,d)$-内射盖与Gorenstein $(m,d)$-平坦模

研究了$(m,d)$-内射$R$-模作成的类是(预)盖类的条件,证明了$(m,d)$-凝聚环上的每一个左$R$-模都具有$(m,d)$-内射盖.在此基础上,又引入研究了Gorenstein $(m,d)$-平坦模和Gorenstein $(m,d)$-内射模,证明了$(m,d)$-凝聚环上的左$R$-模$M$是Gorenstein$(m,d)$-平坦模的充分必要条件是它的特征模$M^{+}$是Gorenstein $(m,d)$-内射模.推广了Goresntein平坦模和Goresntein $n$-平坦模上的一些结果.     

广义(ω)性质及亚循环算子

研究了Weyl定理的一种变化形式:广义$(\omega)$性质; 给出了广义$(\omega)$性质成立的充要条件.同时, 广义$(\omega)$性质及算子的亚(超)循环性之间的关系得到了研究.     

Characterization of $(c)$-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and $(c)$-Bell Polynomials

Here presented are the definitions of(c)-Riordan arrays and(c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials.The characterization of(c)-Riordan arrays by means of the A-and Z-sequences is given,which corresponds to a horizontal construction of a(c)-Riordan array rather than its definition approach through column generating functions.There exists a one-to-one correspondence between GegenbauerHumbert-type polynomial sequences and the set of(c)-Riordan arrays,which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences.The sequence characterization is applied to construct readily a(c)-Riordan array.In addition,subgrouping of(c)-Riordan arrays by using the characterizations is discussed.The(c)-Bell polynomials and its identities by means of convolution families are also studied.Finally,the characterization of(c)-Riordan arrays in terms of the convolution families and(c)-Bell polynomials is presented.     

$(p,q)$-Analogue of Mittag-Leffler Function with $(p,q)$-Laplace Transform

The aim of this paper is to define $(p, q)$-analogue of Mittag-Leffler Function, by using $(p, q)$-Gamma function. Some transformation formulae are also derived by using the $(p, q)$-derivative. The $(p, q)$-analogue for this function provides elegant generalization of $q$-analogue of Mittag-Leffler function in connection with $q$-calculus. Moreover, the $(p, q)$-Laplace Transform of the Mittag-Leffler function has been obtained. Some special cases have also been discussed.