In this paper, a trace theorem and several lemmas for a class of anisotropic Sobolev spaces are given. These results are used to study the initial boundary value problem for symmetric hyperbolic systems with characteristic boundary of constant multiplicity. Especially, the trace theorem is needed for showing the necessity of the compatibility condition for the existence of solution to the characteristic mixed problem mentioned above.

We study the initial boundary value problem for symmetric hyperbolic systems with characteristic boundary. The regularity of solutions is shown in a refined form, taking into account of the break down of the full regularity.

In the case of elliptic root system $A_l^{(1,1)}$ ($l \geq 2$), a $q$-deformation algebra of the hyperbolic extension of the elliptic Weyl group is constructed by using a representation according to Kazhdan-Lusztig.

We show that a closed complex-tangential $C^2$-curve $\gamma$ of constannt curvature on the unit sphere $\partial B_2$ of $\mathbf{C}^2$ is unitarily equivalent to \[ \gamma_{l,m}(t) = \left( \sqrt{l/d} e^{it\sqrt{m/l}}, \sqrt{m/d} e^{-it\sqrt{l/m}} \right) \] where $d = l + m$, $l, m \geq 1$ integers. As an application, we propose a conjecture that if a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ admits a complex-tangential analytic curve on $\partial B_2$ with $\pi(\gamma(t)) = 1$ then $\pi$ is unitarily equivalent to a monomial \[ \pi_{l,m}(z,w) = \sqrt{\frac{d^d}{l^l m^m}} z^l w^m \] where $l, m \geq 1$ integers and show that the conjecture is true for homogeneous polynomial of degree $\leq 5$. A relevant conjecture and partial answer on the maximum modulus set of a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ is also given.

Let $D$ be a Euclidian domain, $G$ be a $GL_n(D)$ and $f$ be an endomorphism of $G$ which preserves conjugacy classes. Then we shall prove $f$ must be an inner automorphism, namely $G$ enjoys the Hasse principle.