By Dieudonne J.

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**Extra resources for A Generalization of Rolle's Theorem with Application to Entire Functions**

**Example text**

Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules, Applications, Adv. Math. 145 (1999), 239-290. [11] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules II, in "Hopf algebras and quantum groups" , S. Caenepeel and F. ), Lect. Notes Puere Appl. Math. 209 Marcel Dekker, New York, 2000. [12] S. Caenepeel, G. Militaru, and S. Zhu, A Maschke type theorem for Doi-Hopf modules, J. Algebra 187 (1997), 388-412.

5 A semi-commutative graded algebra is said to be a graded quantum affine space if the monomials ya are k-linearly independent (and hence, form a k-basis for A). Of course, every graded quantum affine space is a PBW algebra with respect to any admissible order. Our aim is to show that the PBW algebras are, precisely, those filtered algebras which have a quantum affine space as associated graded algebra. 6. First, let us recall that an algebra R over k is (positively) filtered, if it is endowed with an ascending chain FR = {FnR \ n ^> 0} of vector subspaces, the filtration of R, satisfying for all n,m ^ 0 f.

2. We invite the reader to write down explicit results. Our final aim is to link the results in this Section to the ones in Section 4, at least in the case of finitely generated, projective B. Let (A,C,if)) be a right-right entwining structure, with C finitely generated and projective, and put B = (C*)°p. Let {ci,c* i = 1, . . , n} be a dual basis for C. There is a bijective correspondence between right-right entwining structures (A, C, tj}} and smash product structures (C*op, A,R). (a®c*) = (c*,cf)cJ(gia V ) , tp(c® a) = Copyright © 2001 by Marcel Dekker, Inc.