The vertical bundle of the twistor fibration over a 4-dimensional self-dual manifold is a holomorphic line bundle and plays an important role in a study of the twistor space. On the other hand, the vertical bundle of the twistor space over a quaternionic manifold is not a holomorphic line bundle, in general. We shall give the condition for a vertical bundle to be a holomorphic line bundle.

Based upon an intuition from electrostatics one might suspect that there is no topological ball in Euclidean space of dimension $d\ge 2$ which carries a nonconstant Dirichlet finite harmonic measure. This guess is certainly true for $d=2$. However, contrary to the above intuition, it is shown in this paper that there does exist a topological ball in Euclidean space of every dimension $d\ge 3$ on which there exists a nonconstant Dirichlet finite harmonic measure.

We try to investigate whether an element of order $2$ in homotopy groups of spheres has a lift to an element of homotopy groups of mod $2$ Moore spaces or not. A typical element of the affirmative property is Thomeier's element of ${\pi }_{8n+7}\left(S^{4n+3}\right)$. And the Whitehead square $[l_{2n+1},l_{2n+1}]$ of the identity class $l_{2n+1}$ of $S^{2n+1}$ is a negative example except for $n=0,1$ and $3$.

We study holomorphic solutions for convolution equations in tube domains. Let ${\mathcal{O}}^{\tau }$ be the sheaf of holomorphic functions in tube domains on the purely imaginary space $\sqrt{-1}{R}^n$ and $\mathcal{S}$ the complex $0\rightarrow \mathscr{O}^{\tau}\rightarrow \mathscr{O}^{\tau}\mu*\rightarrow 0$ generated by the convolution operator $\mu *$ with hyperfunction kernel $\mu$. In this paper, we give a new definition of "the characteristic set" Char$(\mu *)$ using terms of zeros of the total symbol of $\mu *$, and show, under the abstract condition $\left(S\right)$, the equivalence between two notions of characteristics outside of the zero section $T^{*}_{\sqrt{-1}^R^{n}}(\sqrt{-1}R^{n})$. Moreover we conclude that the micro-support SS$\left(\mathcal{S}\right)$ of $\mathcal{S}$ coincides with the characteristics Char$(\mu *)$.

Passing from regular variation of a function $f$ to regular variation of its integral transform $k*f$ of Mellin-convolution form with kernel $k$ is an Abelian problem; its converse, under suitable Tauberian conditions, is a Tauberian one. In either case, one has a comparison statement that the ratio of $f$ and $k*f$ tends to a constant at infinity. Passing from a comparison statement to a regular-variation statement is a Mercerian problem. The prototype results here are the Drasin-Shea theorem (for non-negative $k$) and Jordan's theorem (for $k$ which may change sign). We free Jordan's theorem from its non-essential technical conditions which reduce its applicability. Our proof is simpler than the counter-parts of the previous results and does not even use the Pólya Peak Theorem which has been so essential before. The usefulness of the extension is highlighted by an application to Hankel transforms.

We consider a quasi-linear second order elliptic differential equation on a euclidean domain. After developing necessary potential theory for the equation which extends some part of the theories in the book by Heinonen-Kilpeläinen-Martio, we show that the ideal boundary of the Royden type compactification of the domain is resolutive with respect to the Dirichlet problem for the equation.

In this paper we determine isomorphism classes of connected solvable Lie groups with some conditions such that their group $C*$-algebras have stable rank one, and give its applications. Also, we show that stable rank of group $C*$-algebras of connected Lie groups of type I is estimated in terms of their closed normal subgroups and quotient groups.

In this paper we study certain kinds of links; proper links, algebraically split links and $Z_2$-algebraically split links. These links have 'algebraic' definitions. In fact these are defined in terms of the linking number. We shall give these links certain 'geometric' definitions. By using the geometric definitions, we study the Arf invariants of these links.

We discuss the global existence of small solutions to the Cauchy problem for systems of quasilinear wave equations in three space dimensions, when their nonlinear terms have quadratic nonlinearity. A global existence theorem is established on the null condition which is extended to the condition for systems of wave equations with different propagation speeds.

We prove that the image of the relative dualizing sheaf of a fibration from a smooth projective surface onto a smooth projective curve is ample under some extra conditions.

We consider the global properties of nonnegative solutions of the semilinear elliptic equations in the entire space. By employing Pohozaev identity in the entire space and the results concerning the asymptotic behavior of nonnegative solutions, we establish some theorems of Liouville type.

To Fuchsian partial differential equations in the sense of M. S. Baouendi and C. Goulaouic, which is a natural extension of ordinary differential equations with regular singularity at a point, all the solutions in a complex domain are constructed along the same line as the method of Frobenius to ordinary differential equations, without any assumptions on the characteristic exponents. The same idea can be applied to Fuchsian hyperbolic equations considered by H. Tahara.

We consider a semilinear elliptic equation in a varying thin domain of $R^n$. This thin domain degenerates into a geometric graph when a certain parameter tends to zero. We determine a limit equation on the graph and we prove that a solution of the PDE converges to a solution of the limit equation. Conversely, when a solution of the limit equation is given, we construct a solution of the PDE approaching a solution of the limit equation.