Categorification and Quasi-Hopf Algebras

We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the copentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.

一类Laplace方程的Neumann问题的边界梯度估计

在二阶椭圆偏微分方程中,边值问题仍然是十分重要的问题之一,其中Neumann问题是大家极力想解决的问题。本文主要借助梯度内估计、Hopf引理、极大值原理给出一类Laplace方程的Neumann问题的边界梯度估计的一个证明。主要根据所在领域分为三种情况:1)若φ(x)在■Ωμ0∩Ω上达到极大值则归结为梯度内估计;2)若φ(x)在■Ω上达到极大值则可由Hopf引理可得|Du(x0)|有界;3)对固定小的正常数μ0>0,若φ(x)在Ωμ0上达到极大值则有极大值原理证明|Du(x0)|有界。三种情况采用不同方式证明,综合得到最终的结果:supΩμ0|Du|≤max{M1,M2}。

Bistability in Coupled Oscillators Exhibiting Synchronized Dynamics

We report some new results associated with the synchronization behavior of two coupled double-well Duffing oscillators （DDOs）. Some sufficient algebraic criteria for global chaos synchronization of the drive and response DDOs via linear state error feedback control are obtained by means of Lyapunov stability theory. The synchronization is achieved through a bistable state in which a periodic attractor co-exists with a chaotic attractor. Using the linear perturbation analysis, the prevalence of attractors in parameter space and the associated bifurcations are examined. Subcritical and supercritical Hopf bifurcations and abundance of Arnold tongues -- a signature of mode locking phenomenon are found.

Cohen-Fischman-Westreich's Double Centralizer Theorem for Almost-Triangular Hopf Algebras

In this paper, the authors study the Cohen-Fischman-Westreich's double centralizer theorem for triangular Hopf algebras in the setting of almost-triangular Hopf algebras.

A HOPF INDEX THEOREM FOR A REAL VECTOR BUNDLE

The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.

HOPF BIFURCATION ANALYSIS IN A 4D-HYPERCHAOTIC SYSTEM

This paper investigates the Hopf bifurcation of a 4-dimensional hyperchaotic system withonly one equilibrium.A detailed set of conditions are derived,which guarantee the existence of theHopf bifurcation.Furthermore,the standard normal form theory is applied to determine the directionand type of the Hopf bifurcation,and the approximate expressions of bifurcating periodic solutions andtheir periods.In addition,numerical simulations are used to justify theoretical results.

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants AB = {a ∈ A ｜ b · a = ε（b）a, Ab ε B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AH as in the classical case. The results are applied to the case H = （kG）＊ for a finite group G to get a Galois 1-1 correspondence.

Hopf Superquivers

In this paper,a super version of the Hopf quiver theory is developed.The notion of Hopf superquivers is introduced.It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures.A complete classification of such graded Hopf superalgebras is given.A superquiver setting for general pointed Hopf superalgebras is also built up.In particular,a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given.

Tensor products of ideal codes over Hopf algebras

We study indecomposable codes over a family of Hopf algebras introduced by Radford.We use properties of Hopf algebras to show that tensors of ideal codes are ideal codes,extending the corresponding result that was previously given in the case of Taft Hopf algebras and showing the differences with that case.

Hopf bifurcation analysis in a turbidostat model with Beddington.DeAngelis functional response and discrete delay

In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington-DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.

HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS

In this paper, two sunflower equations are considered. Using delay T as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem＇s increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

A Note on Residue Formulas for the Euler Class of Sphere Fibrations

This paper presents a definition of residue formulas for the Euler class ot eohomology-oriented sphere fibrations ε. If the base of ε is a topological manifold, a Hopf index theorem can be obtained and, for the smooth category, a generalization of a residue formula is derived for real vector bundles given in [2].

On the Same n-Types for the Wedges of the Eilenberg-Maclane Spaces

Given a connected CW-space X, SNT（X） denotes the set of all homotopy types [X＇] such that the Postnikov approximations X（n） and X＇^（n） are homotopy equivalent for all n. The main purpose of this paper is to show that the set of all the same homotopy n- types of the suspension of the wedges of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type of itself, i.e., SNT（Σ（K（Z, 2a1） ∨ K（Z, 2a2）∨… ∨ K（Z,2ak））） = ＊ for a1 〈 a2 〈 … 〈 ak, as a far more general conjecture than the original one of the same n-type posed by McGibbon and Moller （in [McGibbon, C. A. and Moller, J. M., On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres, Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Lecture Notes in Math., 1509, 1992, 285-293].）

THE SERRE RELATIONS IN RINGEL-HALL ALGEBRAS

The author constructs the Casimir element of Hall algebras. By the method of Gabber-Kac theorem (see [4]), it is proved that the Serre relations are the defining relations in composition algebra.

On Smash Products of Transitive Module Algebras

Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a l-dimensional ideal. It is proved that the smash product A#H is isomorphic to a full matrix algebra over some right coideal subalgebra N of H. The correspondence between A and such N, and the special case A = k（X） of function algebra on a finite set X are considered.

Mathematical analysis of a model for thymus infection with discrete and distributed delays

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

Bifurcation behaviors analysis of a plankton model with multiple delays

A mathematical model describing the dynamics of toxin producing phytoplankton- zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.

STABILITY ANALYSIS OF A PREDATOR-PREY MODEL

In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.