We shall give a new elementary proof for the conjugacy theorem of Cartan subalgebras of a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero.

We show that all extremal elliptic surfaces in characteristic 2 and 3 are obtained from rational extremal elliptic surfaces as purely inseparable base extensions. As a corollary, we can show that the automorphism group of every supersingular elliptic $K3$ surface has an element of infinite order which acts trivially on the global sections of the sheaf of differential forms of degree 2. We also determine the structures of Mordell- Weil groups for extremal rational elliptic surfaces in these characteristics.

We study linearly normal projective curves with degenerate general hyperplane section, in terms of the ‘‘amount of degeneracy’’ of it, giving a characterization and/or a description of such curves.

We consider the asymptotic behavior of solutions to the initial value problem for a certain class of hyperbolic-elliptic coupled systems. It will be proved that the solution is time asymptotically approximated by the superposition of diffusion waves constructed in terms of the self-similar solutions of generalized Burgers equations. We will give space-time decay estimates for the residual term through a pointwise estimate for the Green’s function of the linearized system.