Let $R$ be a compact bordered Riemann surface which is non-planar. We solve a conjecture posed by Gouma concerning the distribution of degrees of Ahlfors functions on $R$ whose double is hyperelliptic. Besides we consider the problem when a linear transformation of an Ahlfors function on $R$ is again an Ahlfors function. We give a necessary and sufficient condition for this problem when the degree of the Ahlfors function is maximal.

In the first paper of the same title, we introduced the concept of singular inner functions of $L^1$-type and obtained results for singular inner functions which are reminiscent of the results for weak infinite powers of Blaschke products. In this Paper, we investigate singular inner functions for discrete measures. We give equivalent conditions on a measure for which it is a Blaschke type. And we prove that two discrete measures are mutually singular if and only if the associated common zero sets of singular inner functions of $t_{+}^{\infty }$-type do not meet.

We study the first positive eigenvalue ${\lambda }_1^{\left(p\right)}$ of the Laplacian on $p$-forms for oriented closed Riemannian manifolds. It is known that the inequality ${\lambda }_1^{\left(1\right)}\le {\lambda }_1^{\left(0\right)}$ holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality holds. We show that any oriented closed manifold $M$ with the first Betti number $b_1\left(M\right)=0$ whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.

In knot theory, it is well-known that two links in the Euclidean 3-space are ambient isotopic if and only if they are related by a finite number of combinatorial moves along 2-simplices. This fact is generalized for submanifolds in a manifold whose codimensions are positive.

In 1985, P. C. Roberts [14] proved the vanishing theorem of intersection multiplicities for a local ring that satisfies ${\tau }_{A/S}\left(\right[A\left]\right)=[SpecA{]}_{\dim A}$, where ${\tau }_{A/S}$ is the Riemann-Roch map for $SpecA$ with regular base scheme $SpecS$. We refer such rings as Roberts rings. For rings of positive characteristic, we can characterize Roberts rings by the Frobenius maps. For rings with field of fractions of characteristic 0, we can characterize Roberts rings by some Galois extensions. We shall give basic properties and examples of Roberts rings in the paper.

Let $f:N\to P$ be a smooth map between $n$-dimensional oriented manifolds which has only folding singularities. Such a map is called a folding map. We prove that a folding map $f$ : $N\to P$ canonically determines the homotopy class of a bundle map of $TN\oplus {\theta }_N$ to $TP\oplus {\theta }_P$, where ${\theta }_N$ and ${\theta }_P$ are the trivial line bundles over $N$ and $P$ respectively. When $P$ is a closed manifold in addition, we define the set ${\Omega }_{\mathrm{fold}}\left(P\right)$ of all cobordism classes of folding maps of closed manifolds into $P$ of degree 1 under a certain cobordism equivalence. Let $SG$ denote the space ${\lim }_{k\to \infty }S{G}_k$, where $S{G}_k$ denotes the space of all homotopy equivalences of $S^{k-1}$ of degree 1. We prove that there exists an important map of ${\Omega }_{\mathrm{fold}}\left(P\right)$ to the set of homotopy classes $[P,SG]$. We relate ${\Omega }_{\mathrm{fold}}\left(P\right)$ with the set of smooth structures on $P$ by applying the surgery theory.

We determine the module structure of Morava $K$-theory for the special orthogonal groups and the spinor groups at the prime 2, using the Atiyah-Hirzebruch spectral sequence.

The problem discussed in that paper is to find a characterization on the Arthur's conjectural parameters for the representations occuring in the residual spectrum for a classical group. Only conjectures are given but it is proved that the global conjectures can be reduced to very natural local conjectures, in the case of cohomological square integrable automorphic forms.

Suppose a group $G$ acts on a tensor category $\mathcal{C}$ over a field $k$. Then we have the tensor category ${\mathcal{C}}^G$ of $G$-invariant objects in $\mathcal{C}$, and the semi-direct product tensor category $\mathcal{C}\left[G\right]$. We show that if $G$ is finite and $k\left[G\right]$ is semi-simple, there exists a one-to-one correspondence between categories with action of ${\mathcal{C}}^G$ and categories with action of $\mathcal{C}\left[G\right]$.

We study a problem of constructing an algebraic torus (an abelian variety) over a $p$-adic field whose Néron model would have a given connected commutative unipotent group as the identity component of its special fibre.

In this paper, the blow-up of solutions of ordinary differential equations, which are deduced from the equation of equivariant harmonic maps, is studied. Its direct consequence is the non-existence or existence result of equivariant harmonic maps between warped product manifolds. As another application we prove the non-existence of a harmonic map from an Euclidean space to a Hadamard manifold with a certain nondegeneracy condition at infinity, provided sectional curvatures of the Hadamard manifold are bounded from above by a slowly decaying negative function of the distance from a fixed point.