This paper is devoted totwo aspects: one is the smash product of a semisimple Hopf algebra and its transitivemodule algebra which has a 1-dimensional ideal; the other is faithfully flat Hopf-bi-Galois extensions.

The resonant double Hopf bifurcations and the nonresonant ones exist due to the technological delay τ_1.

The dynamic behavior of a resonant double Hopf bifurcation is examined for a van der Pol-Duffing oscillator with delayed feedback,and the influence on double Hopf bifurcation is investigated with time delay and amplitude variation.

It is found that the excitatory self-connections can lead to non-resonant double Hopf bifurcations since the double Hopf bifurcation disappears without self delayed connecti.

Study on the simplest normal forms of nonresonant double Hopf bifurcation;

In this paper, an analog of the neural network of the fourth order aiming to store the oscillatory memory patterns is designed using the normal form equations for double Hopf bifurcation with two remaining dimensions.

It is well known that the kG-Hopf bimodule category■s equivalent to the direct product categoryΠ_(C∈K(G))Mkz_(u(C)).

Let D2 be a dihedral group and r=sum from C∈K(D_2) to (r_CC) be a ramification of D2, the kD2-actions on the Hopf bimodule kQ1 and the structure of Hopf algebra kD2[kQ1] are presented for ra,rb and rba being nonzero natural numbers.

It is well known that the kG-Hopf bimodule category ~(kG)_(kG)M~(kG)_(kG) is equivalent to the direct product category ∏(C∈K(G))M_(kZ_(u(C))), where(K(G)) is the set of conjugate classes in G,u:K(G)→G is a map such that u(C)∈C for any C∈K(G),Z_(u(C))={g∈G|gu(C)=u(C)g} and M_(kZ_(u(C))) denoted the category of right kZ_(u(C)) modules.