This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the de...

In this paper, we review a number of basic results about so-called Hopf algebras. We begin by giving a historical account of the results obtained in the 1930's and 1940's about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations betwee...

We investigate the relation between instantons and monopoles in the Laplacian Abelian Gauge using analytical methods in the continuum. Our starting point is the fact that the ’t Hooft instanton with its high symmetry leads to a pointlike defect with Hopf invariant one. In order to generalise this result we partly break the symmetry by a local perturbation. We find that for generic configuration...

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t-Lying Over (t-LO), and Going Up (GU), analogous to the classical Krull relations for p...

A commutative but not cocommutative graded Hopf algebra HN , based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees HC , developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to de...

Let τ be an invertible skew pairing on (B,H), where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category HYD and there exists a Hopf algebra isomorphism w : B∞H → B τH in C, where B∞H is a Hopf algebra with (co)algebra structure the smash (c...

In recent studies, global Hopf branches were investigated for delayed model of HTLV-I infection with delay-independent parameters. It is shown in [8, 9] that when stability switches occur, global Hopf branches tend to be bounded, and different branches can overlap to produce coexistence of stable periodic solutions. In this paper, we investigate global Hopf branches for delayed systems with del...

This exposition concerns two different notions of Frobenius-Schur indicators for finite-dimensional Hopf algebras. These two versions of indicators coincide when the underlying Hopf algebra is semisimple. We are particularly interested in the family of pivotal finite-dimensional Hopf algebras with unique pivotal element; both indicators are gauge invariants of this family of Hopf algebras. We o...

In this paper, we propose and analyze a mathematical model describing the effect of awareness programs by public media on the prevalence of Typhoid fever. A threshold quantity $R_{0}$, similar to the basic reproduction number is derived. We investigate the biologically meaningful equilibrium points and their local stability analysis. The global stability analysis has been performed with respect...

The analysis of dynamic of chemical reaction networks by computing Hopf bifurcation is a method to understand the qualitative behavior of the network due to its relation to the existence of oscillations. For low dimensional reaction systems without additional constraints Hopf bifurcation can be computed by reducing the question of its occurrence to quantifier elimination problems on real closed...