In this paper, we shall show some representations of Nevanlinna-type spaces $N^p$, $1\leqq p<\infty$, as unions of weighted $H^q$-spaces, $0<q<\infty$. Moreover, we shall prove that the usual metric topology on $N^p$ is equivalent to an inductive limit topology on $N^p$.

In three dimensional Minkowski space, we show that a generalized helicoid is isometric to a rotation surface so that helices on the helicoid correspond to parallel circles on the rotation surface. Moreover, if these surfaces have the same Gauss map, we can determine them.

An algorithm to decide if an orientable atoroidal 3-manifold contains closed incompressible acylindrical surfaces, and construct closed incompressible acylindrical surfaces is given. Mainly, the normal surface theory is used. To assure that the algorithm stops after finite steps, we show that each acylindrical surface is isotopic to some ``edge surface'' which is constructible.

In this paper, for a compact symmetric space $M$ we consider orbits of the linear isotropy action of $M$ and the canonical action on $M$ of its isotropy group. We reduce some geometric conditions of these orbits to those of their starting points. Consequently we get some relations among the geometric conditions of orbits.

We consider a differentiable map $f$ from an open interval to a real Banach space of all bounded continuous real-valued functions on a topological space. We show that $f$ can be approximated by the solution to the differential equation $x'(t)=\lambda x(t)$, if $||f'(t)-\lambda f(t)||_\infty\leq\varepsilon$ holds.

In this paper, we will consider the equation $\mathcal{P}u=f$, where $\mathcal{P}$ is the linear Fuchsian partial differential operator \[ \mathcal{P}=(tD_t)^m+\sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}a_{j,\alpha}(t, z)(\mu(t)D_z)^\alpha(tD_t)^j . \] We will give a sharp form of unique solvability in the following sense: we can find a domain $\Omega$ such that if $f$ is defined on $\Omega$, then we can find a unique solution $u$ also defined on $\Omega$.

We consider the weakly coupled system of reaction-diffusion equations \[ u_t=\Delta u+|x|^{\sigma_1}v^p, \quad v_t=\Delta v+|x|^{\sigma_2}u^q , \] \[ u(x,0)=\lambda^\mu\varphi(x), \quad v(x,0)=\lambda^\nu\psi(x) \] where $x\in\mathbf{R}^N$, $t>0$, $p$, $q>1$ and $0\leq\sigma_1<N(p-1)$, $0\leq\sigma_2<N(q-1)$. The existence of solutions, blow-up conditions, and global solutions of the above equations are studied by Mochizuki-Huang. In this paper, we consider the estimate of maximal existence time of blow-up solutions in $I^{\delta_1}\times I^{\delta_2}$ as $\lambda$ goes to 0 or $\infty$.

Let $U$ be a finite set of points in general position in the plane. We consider the following graph $\mathcal{G}$ determined by $U$. A vertex of $\mathcal{G}$ is a spanning tree of $U$ whose edges are straight line segments and do not cross. Two such trees $\mathbf{t}$ and $\mathbf{t}'$ are adjacent if for some vertex $u\in U$, $\mathbf{t}-u$ is connected and coincides with $\mathbf{t}'-u$. We show that $\mathcal{G}$ is 2-connected, which is the best possible result.

In this paper we shall give $L^p(\mathbf{R}^2)$-boundedness of the Bochner-Riesz operator $S_\delta$ for $2<p<\infty$ and $\delta>0$, restricting it to functions of product type. In this range, $2<p<\infty$ and $\delta>0$, the strong $L^p$-estimate is valid for functions of product type but not for general functions.

We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.