In this paper, we give explicit formulae of certain higher annihilators of the ideal class groups defined by V. Kolyvagin and K. Rubin, which come from Euler systems of Stickelberger elements and cyclotomic units. Further, using these explicit formulae, we reformulate Kolyvagin-Rubin's structure theorem of the ideal class groups of abelian number fields.

We will study a new special solution of the fourth Painlevé equation, for which we can calculate the linear monodromy exactly. We will show the relation between Umemura's classical solutions and our solutions.

The Kleiman-Mori cone plays important roles in the birational geometry. In this paper, we construct complete varieties whose Kleiman-Mori cones have interesting properties. First, we construct a simple and explicit example of complete non-projective singular varieties for which Kleiman's ampleness criterion does not hold. More precisely, we construct a complete non-projective toric variety $X$ and a line bundle $L$ on $X$ such that $L$ is positive on $\overline{\mathit{NE}}(X)\setminus \{0\}$. Next, we construct complete singular varieties $X$ with $\mathit{NE}(X)=N_1(X)\simeq \mathbf{R}^k$ for any $k$. These explicit examples seem to be missing in the literature.

In this note, we will completely classify all of rational solutions of the $A_4$ Painlevé equation, which is a generalization of the fourth Painlevé equation. The rational solutions are classified to three types by the Bäcklund transformation group.

Using the splitting-up method, we establish a new existence result for an initial boundary value problem for the doubly degenerate stochastic quasilinear parabolic equation \begin{equation*} d \bigl( \left|y\right|^{\alpha-2}y\bigr) -\left[ \sum_{i=1}^{n} \frac{\partial}{\partial x_{i}} \left( \left| \frac{\partial y}{\partial x}\right|^{p-2} \frac{\partial y}{\partial x_{i}}\right) - g(t,y) \right] dt = \sum_{l=0}^{d}h_{l} (t,y) dW_{t}^{l}, \end{equation*} where $W_{t}^{l}$ are one-dimensional Wiener process defined on a complete probability space, $p$, $\alpha$ and the functions $g$ and $h_{l}$ satisfy appropriate restrictions.