Let K be a field of characteristic zero. We characterize coordinates and tame coordinates in K[z][x,y], i.e. the images of x respectively under all automorphisms and under the tame automorphisms of K[z][x,y]. We also construct a new large class of wild automorphisms of K[z][x,y] which maps x to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, ...

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K〈x, y, z〉 over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordin...

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmüller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic in its homotopy class. We study this pairing function using the Fenchel–Nielse...

When a strictly dissipative term arises in the energy equation, Cauchy problem for Euler–Poisson systems admits global smooth solutions small initial data. This was proved previous studies. In this paper, we study stability full system without dissipation equation. We prove existence of near constant equilibrium states one space dimension. The results are obtained both one-fluid and two-fluid s...

Lagrangian coordinates describe a local frame moving and deforming with a flow, where by flow we mean either a physical flow or the solution of a smooth dynamical system. The rôle of chaos in these coordinates is reflected by a mean exponential stretching of fluid elements along characteristic directions. Many partial differential equations, such as the kinematic advection–diffusion equation, a...

We construct an explicit diffeomorphism between the Sasaki-Einstein spaces Y p,q and the product space S × S in the cases q 6 2. When q = 1 we express the Kähler quotient coordinates as an SU(2) bundle over S which we trivialize. When q = 2 the quotient coordinates yield a non-trivial SO(3) bundle over S with characteristic class p, which is rotated to a bundle with characteristic class 1 and r...