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Ringel has given and studied the N_0I-graded twisted coalgebra and Ringel-Hall algebra in the reference [1]. Subsequently, people found that there is an analogy between Ringel-Hall algebra and Hopf algebra, and then they introduced the concept of twisted Hopf algebras and bitwisted Hopf algebras, and worked on them.

Ringel在[1]中引入并研究了N_0I-分次扭的余代数和Ringel-Hall代数，后来人们发现Ringel-Hall代数有类似Hopf代数的结构，随后，扭Hopf代数、双扭Hopf代数先后被引入和研究。

In chapter 5, local derivations and local automorphisms of nest subalgebras in von Neumann algebras and local higher cohomology -local 2-cocycles are studied. It is proved that every weakly continuous local derivation, respectively, every weakly local automorphism, of nest subalgebra of a factor von Neumann algebra is a derivation, respectively, an automorphism. Every norm continuous local derivation, respectively, every norm local automorphism, of the nest subalgebra associated to a countable nest in a factor von Neumann algebra is a derivation, respectively, an automorphism. Moreover, it is answered Larson's question. Finally, it is shown that every local 2-cocycle of any von Neumann algebra is a 2-cocycle.

第五章研究von Neumann代数中套子代数的局部导子和局部同构以及von Neumann代数的高维局部映射—局部2-上循环，证明了因子von Neumann代数中套子代数的每一个局部强连续导子和局部强连续同构分别是导子和同构；可数套所对应的套子代数的每一个有界局部导子和有界局部同构分别是导子和同构；同时，部分回答了Larson所提的问题；最后，得到von Neumann代数的每一个局部2-上循环是2-上循环。

Put forward a commutative law in the weak fuzzy implication algebra;The properties of weak FI-algebra and FI-algebra are studied.

引入可交换弱FI代数的概念，讨论了弱FI代数和FI代数的几个性质；进一步得到了交换弱FI代数和交换FI代数的一些结果。

We showthat End〓is an algebra in 〓,and then we construct a derived Lie algebra〓 in 〓 and prove existenceness of its enveloping algebra under the condi-tion that the"braiding"in the category 〓 is symmetric on 〓.

Radford的双积定理给出了在Yetter-Drinfel'd范畴〓中构造辫子群[48-50]的一种方法。因为，冲积已被发展为交叉积，许多冲积的相应结果被搬到了交叉积上。

150 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Table 3-2 Distributivity Idempotency Absorption laws 分配律同一律吸收律 a=ab+ac a+= a+a=a aa=a a+ab=a a=a'=a'b''=a'+b' DeMorgan's laws德摩根定理计算机专业英语 1-151 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n.

由于n个元素的有限集有且只有个子集由于个元素的有限集有且只有2n个子集，而且很显然有限布个元素的有限集有且只有个子集，尔代数一定是有限集合代数，所以对某个整数n而言而言，尔代数一定是有限集合代数，所以对某个整数而言，每个有限布尔代数也有且只有2n个元素。例如，上文定义的集合T的限布尔代数也有且只有个元素。

Semilattice graded weak Hopf algebra is a special case of weak Hopf algebra and a generalization of Hopf algebras, in Section 1.3, we will give results and applications to the Maschke's and Dual Maschke's Theorem of semilattice graded weak Hopf algebra.

尽管有限Clifford幺半群的量子偶确实是有限群量子偶的一个推广，但是[L2]中的量子偶一般情况下是不能够看作是Hopf代数量子偶的一个推广，这是因为我们在[L2]中要求所有讨论的弱Hopf代数都是交换的。

The elementary transformation matrix throughout the linear algebra theory has, it makes some of the problems of linear algebra, become more simple, makes it easier to understand, so elementary transformation in the role of linear algebra is very important.

标签：论文摘要，翻译，论文矩阵的初等变换贯穿于整个线性代数理论的始终，它使线性代数里的一些问题变的更加简单，使人更容易理解，所以初等变换在线性代数中作用十分重要。

Based on the Lie algebra that describes the symmetries, it is proved in this dissertation that a sufficiently large class of control Hamiltonians can be provided by the universal enveloping algebra of the Lie algebra, on which the quantum mechanical control systems can be modelled.

基于描述量子力学系统对称性的李代数结构，本文证明该李代数的泛包络代数提供了足够大的一类控制哈密顿量，因而可以在其上建立量子力学控制系统的模型。

It is easy to see that the bialgebra H we find,canbe regarded as the enveloping algebra of a Lie algebra,whose generators are thoseof the Lie algebra of type A and n-1 other elements which commute with genera-tors above.

可以看出，我们找到的H作为代数，恰为在An-1型李代数上增加与之可换的n-1个生成元后得到的包络代数，其自然表示的矩阵元恰好生成代数AR。

We use algebra to describe the semantics. Algebra provides a common base for function, class and algebra specification.

它以代数作为数学基础，这使得函数、类和代数规范等具有统一的合成基础。

We got alerted a couple of times while we were down south that HETs were on the way to bring us back up north because things were going to go hot again, but it was just rumors.

南下的途中我们不只一次得到警告说重型装备运输车将拉着我们重新北上，因为局势正在变得重新紧张起来，但这只是谣传。

It's the one where they find the ghost in the salt mine.

这一集是演他们在盐矿找到鬼