We find a formula to compute the number of the generators, which generate the n-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight n. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneider’s Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algeb...

A Hopf texture is a vacuum field configuration of isovector fields which is an onto map from the space as a large three sphere to the vacuum manifold S. We construct a Hopf texture with spherically symmetric energy density and discuss the topological charge. A Hopf texture collapses, and we study the collapse process numerically. In our simulations, it is clear that a Hopf texture does not deca...

We give a complete classification of holomorphic foliations on compact complex surfaces which are not uniformisable, i.e., for which universal coverings of the leaves do not glue together in a Hausdorff way. This leads to complex analogs of the Reeb component defined on certain Hopf surfaces and certain Kato surfaces. 2000 Math. Subj. Class. 32J15, 37F75, 57R30.

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally assoc...

By constructing Hopf costructures on closure spaces via homotopy, we give the concepts of cospace (CH-cospace) and cogroup (CH-cogroup). We then prove that retract deformation a CH-cospace are also CH-cospace. construct costructure set with help quotient operator. show space same homotopy type as CH-cogroup is itself CH-cogroup. existence covariant functor between category pointed ($\mathcal{CH...

We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the RotaBaxter double construction, respectively Atkinson’s theorem. We also outline the extension...

In this article, we study the nonlinear dynamics of a quadratic system in the three dimensional space which can be obtained from a scalar third order differential equation. More precisely, we study the stability and bifurcations which occur in a parameter dependent quadratic system in the three dimensional space. We present an analytical study of codimension one, two and three Hopf bifurcations...

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...