We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grA. Then grA is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π : grA → A0; let R be the algebra ...

In this paper, we study Hopf-zero bifurcation in a generalized Gopalsamy neural network model. By using multiple time scales and center manifold reduction methods, we obtain the normal forms near a Hopf-zero critical point. A comparison between these two methods shows that the two normal forms are equivalent. Moreover, bifurcations are classified in two-dimensional parameter space near the crit...

A kind of delay neural network with n elements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. 1988 and delay differential equations due to Wu 1998 , and combining the normal form theory of functional...

We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use ...

Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of [12]. We first show that on the k-vector space A⊗H∗ we can define an algebra structure, denoted by A # H∗, in the monoidal category of left H-modules (i.e. A # H∗ is an Hmodule algebra in the sense of [2]). Then we will prove that the category of two-sided (A,H)bimodules HM H A is is...

We consider the regularity of Leray-Hopf solutions to impressible Navier-Stokes equations on critical case u ∈ L 2 w (0, T ; L ∞ (R 3)). By a new embedding inequality in Lorentz space we prove that if u L 2 w (0,T ;L ∞ (R 3)) is small then as a Leray-Hopf solution u is regular. Particularly, an open problem proposed in [8] is solved.

We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case u ∈ L 2 w (0, T ; L ∞ (R 3)) was obtained. By a new embedding inequality in Lorentz space we proved that if u L 2 w (0,T ;L ∞ (R 3)) is small then the Leray-Hopf solutions are regular. Particularly, an open problem proposed in [KK] was solved.

Tensor operators in graded representations of Z2−graded Hopf algebras are defined and their elementary properties are derived. WignerEckart theorem for irreducible tensor operators for Uq[osp(1 | 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra Uq[osp(1 | 2)] are considered. The reduced matrix elements for the irreducible tensor operators are c...

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two...