Satellite constructions and geometric classification of Brunnian links

We study satellite operations on Brunnian links. First, we find two special satellite operations, both of which can construct infinitely many distinct Brunnian links from almost every Brunnian link. Second, we give a geometric classification theorem for Brunnian links, characterize the companionship graph defined by Budney in [JSJ-decompositions of knot and link complements in [Formula: see text], Enseign. Math. 3 (2005) 319–359], and develop a canonical geometric decomposition, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links then turn out to be Hopf [Formula: see text]-links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements. Third, we define an operation to reduce a Brunnian link in an unlink-complement into a new Brunnian link in [Formula: see text] and point out some phenomena concerning this operation.

Drinfel’d construction for Hom–Hopf T-coalgebras

Let [Formula: see text] be a Hom–Hopf T-coalgebra over a group [Formula: see text] (i.e. a crossed Hom–Hopf [Formula: see text]-coalgebra). First, we introduce and study the left–right [Formula: see text]-Yetter–Drinfel’d category [Formula: see text] over [Formula: see text], with [Formula: see text], and construct a class of new braided T-categories. Then, we prove that a Yetter–Drinfel’d module category [Formula: see text] is a full subcategory of the center [Formula: see text] of the category of representations of [Formula: see text]. Next, we define the quasi-triangular structure of [Formula: see text] and show that the representation crossed category [Formula: see text] is quasi-braided. Finally, the Drinfel’d construction [Formula: see text] of [Formula: see text] is constructed, and an equivalent relation between [Formula: see text] and the representation of [Formula: see text] is given.

The structure theorem and duality theorem for endomorphism algebras of weak Hopf group coalgebras

In this paper, we investigate the HOM-functor and state the structure theorem for endomorphism algebras of weak two-sided [Formula: see text]-Hopf [Formula: see text]-modules in order to explore homological algebras for weak Hopf [Formula: see text]-modules, and present the duality theorem for weak group “big” Smash products which extends the result of Menini and Raianu [Morphisms of relative Hopf modules, Smash products and duality, J. Algebra 219 (1999) 547–570] in the setting of weak Hopf group coalgebras.

Topological tools for the prescribed scalar curvature problem on S n

In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

The representation formula for solutions of some class Hamilton–Jacobi equations

The lower semicontinious solutions of Hamilton–Jacobi equation are contructed by Hopf formula, when hamiltonian is maximum of linear functions.