Vogel showed that any oriented link diagram $D$ can be deformed to a closed braid by a finite sequence of Reidemeister II moves, each performed on two coherently oriented edges in a face of $D$ such that the edges are contained in distinct Seifert circles. We show that the number of such moves is constant for a given oriented link diagram, and does not depend on the sequence of moves. An easy way of calculating the number is given.

We obtain sufficient conditions for recurrence and sufficient conditions for transience of storage processes. Applying these results, we give a necessary and sufficient condition for transience in case the input process is a stable process and the release rate is a power function. This result is extended using Abelian theorem for Stieltjes transforms. As another application, we refine conditions for recurrence-transience in case the release rate is non-decreasing and bounded. One of these results corresponds to a recurrence-transience criterion for Bessel processes. As a by-product, the necessary and sufficient condition for transience of processes of Ornstein-Uhlenbeck typeis simplified in diagonal drift coefficient case.

In this paper, we study submodules of $L^2(\mathbf{R})^2$. We will give a Lax-type theorem and a result which is analogous to Helson's theory.

We study an eigenvalue problem for the Laplace operator on a planar region with a growing crack. We impose the Neumann boundary condition on the crack and the Dirichlet boundary condition elsewhere. One tip of the crack is fixed at the boundary. We obtain the full asymptotic expansions of the first two eigenvalues of that operator as the other tip of the crack reaches the boundary.

A filtration of the loop group of unitary group by singular complex algebraic varieties defined by S. Mitchell and G. Segal is studied, focusing on determining the singularities of those varieties.

A homogeneous almost Kähler manifold $M$ of negative curvature can be identified with a solvable Lie group $G$ with a left invariant metric $g$ and a left invariant almost complex structure $J$. We prove that if $g$ is an Einstein metric and $G$ is of Iwasawa type, then $J$ is integrable so that $M$ is Kähler, and hence is holomorphically isometric to a complex hyperbolic space of the same dimension.

Exponential decaying states for the wave equations are considered and some examples are given. Moreover, the spectral structure of the operator with coulomb type dissipation is investigated.

We give several examples of towers $\mathcal{F}=(F_{0}, F_{1}, F_{2}, \dotsb)$ of function fields of one variable over a finite field $\mathbb{F}_{q}$, for which the Garcia-Stichtenoth number $$ \lambda(\mathcal{F})=\lim_{m \to \infty}\frac{\text{number of $\mathbb{F}_{q}$-rational places of $F_{m}$}}{\text{genus of $F_{m}$}} $$ is zero. Moreover, we study an upper bound for the limit $\lambda(\mathcal{F})$.

In this paper, we prove the theorems of the Gauss-Bonnet and Chern-Lashof types for low dimensional compact submanifolds in a simply connected symmetric space of compact type. In particular, in the case where the ambient space is a sphere, we need not to give the restriction for the dimension of the submanifold. Those proofs are performed by applying the Morse theory to squared distance functions.

Consider a moduli space ${{\cal M}(\Sigma,d)}$ of reduced curves in ${\textbf{CP}^2}$ with a given degree ${d}$ and having a prescribed configuration of singularities ${\Sigma}$. Let ${C,C'\in {\cal M}(\Sigma,d)}$. The pair of curves ${(C,C')}$ is called a weak Zariski pair if the pairs of spaces ${(\textbf{CP}^2,C)}$ and ${(\textbf{CP}^2,C')}$ are not homeomorphic. There exists two classical ways to detect weak Zariski pairs: (i) showing that the generic Alexander polynomials ${\Delta_C(t)}$ and ${\Delta_{C'}(t)}$ of~${C}$ and ${C'}$ are different; (ii) showing that the fundamental groups ${\pi_1(\textbf{CP}^2-C)}$ and ${\pi_1(\textbf{CP}^2-C')}$ are not isomorphic. In this paper, we give the first example of a weak Zariski pair ${(C,C')}$ such that ${\pi_1(\textbf{CP}^2-C)}$ and ${\pi_1(\textbf{CP}^2-C')}$ are isomorphic (notice that such an isomorphism automatically implies ${\Delta_C(t)} = {\Delta_{C'}(t)}$). We shall call such a pair a ${\pi_1}$-equivalent weak Zariski pair. The members ${C}$ and ${C'}$ of our pair are reducible sextics with the following configuration of singularities ${\{D_{10}+A_5+A_4\}}$. By the way, we determine the fundamental group ${\pi_1(\textbf{CP}^2-D)}$ for any curve~${D}$ in the moduli space ${{\cal M}(\{D_{10}+A_5+A_4\},6)}$. As an application, we give a new weak Zariski ${4}$-ple (we recall that a ${4}$-ple ${(D_1,D_2,D_3,D_4)}$ of curves in ${{\cal M}(\Sigma,d)}$ is called a weak Zariski ${4}$-ple if for any ${1\leq i<j\leq 4}$ the pairs of spaces ${(\textbf{CP}^2,D_i)}$ and ${(\textbf{CP}^2,D_j)}$ are not homeomorphic).

Let $F$ be a compact, orientable surface with negative Euler characteristic, and let $x_1, \cdots, x_n$ be $n$ fixed but arbitrarily chosen points on $\mathrm{int}F$, each of which has a (small) diskal neighborhood $D_i \subset F$. Denote by $\mathcal{S}_n(F)$ a subgroup of $\mathrm{Diff}(F)$ consisting of "sliding" maps $f$ each of which satisfies

(1) $f(\{x_1, \dots , x_n\}) = \{x_1, \dots , x_n\}, f(D_1\cup \cdots \cup D_n) = D_1 \cup \cdots \cup D_n $ and

(2) $f$ is isotopic to the identity map on $F$

Then by restricting such automorphisms to $\hat{F} = F - \mathrm{int}(D_1 \cup \cdots \cup D_n)$, we have automorphisms $\hat{f} : \hat{F} \to \hat{F}$, which form a subgroup $\mathcal{S}_n(\hat{F})$ of $\mathrm{Diff}(\hat{F})$. We give a Nielsen-Thurston classification of elements of $\mathcal{S}_n(\hat{F})$ using braids in $F \times I$ which characterize the elements of $\mathcal{S}_n(\hat{F})$.

By using Hecke type modular forms, we relate the eighth order mock theta functions with theta functions.

The Collatz conjecture is that there exists a positive integer $n$ which satisfies $f^n(m)=1$ for any integer $m \geq 3$, where $f$ is the function on the rational number field defined by $f(m)=m/2$ if the numerator of $m$ is even and $f(m)=(3m+1)/2$ if the numerator of $m$ is odd. Let $m$ be a rational number such that $f^n(m)=m>1$. Then we show that, if $m$ has some simple sequences, then the total number of positive integer $m$ is finite, by estimating $f(m)-m$.