The purpose of this paper is to provide the reader with an introduction to the theory of generalized Igusa local zeta functions associated to irreducible matrix groups. As an application, the generalized Igusa local zeta function $Z_K\left(s\right)$ associated to the simple Chevalley $K$-groups of type $B_l,$$C_l,$$D_l,$$E_6,$$E_7,$$E_8,$$F_4$ and $G_2$ under the adjoint representation is explicitly exhibited as arational function satisfying a certain functional equation.

We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

In this paper we study the relation between automorphism groups of branched coverings over the complex projective line and automorphism groups of compact Riemann surfaces. We give a criterion for the coincidence of them. We also give examples when the criterion does not hold.

This paper aims to study a variety of general type whose canonical system is composed with a pencil. This kind of variety admits a natural fibration onto a nonsingular curve. A natural problem is whether the geometric genus of the general fiber of this fibration is bounded. A simple classification is given in this paper. When the object is a nonsingular minimal 3-fold of general type, if the canonical system is composed of an irrational pencil, then the geometric genus of the general fibre is bounded. If the canonical system is composed of a rational pencil, it seems that the geometric genus of the general fibre is not bounded though no counter examples have been found.

In this paper, we treat a quantum harmonic oscillator in thermal equilibrium with any systems in certain classes of bosons with infinitely many degrees of freedom. We describe the following results: (i) when a canonical correlation function is given, we so reconstmct a Hamiltonian by the rotating wave approximation from it that the Hamiltonian restores it. Namely, we solve an inverse problem in the quantum field theory at finite temperature in a finite volume. (ii) Taking an infinite volume limit for the result in (i), we consider long-time behavior of the canonical correlation function in the finite volume limit.

In the deformation space of once punctured tori, we investigate the slice determined by moduli equation such that the first and the second moduli are complex conjugate. We show the figure of the slice to some extent.

Let $Z_3$ be the ring of 3-adic integers. For each number field $F$, let $F_{\infty ,3}$ denote the basic $Z_3$-extension over $F;$ let ${\lambda }_3\left(F\right)$ and ${\mu }_3\left(F\right)$ denote respectively the Iwasawa $\lambda \text{-}$ and $\mu$-invariants of $F_{\infty ,3}/F$. Here a number field means a finite extension over the rational field $Q$ contained in the complex field $C;F\subset C,$. Now let $k$ be a number field. Let $L$-denote the infinite set of totally imaginary quadratic extensions in $C$ over $k$(so that $L$-coincides with the set $L^{-}$ in the text when $k$ is totally real); let $L_{+}$ denote the infinite set of quadratic extensions in $C$ over $k$ in which every infinite place of $k$ splits (so that $L_{+}$ coincides with the set $L^{+}$ in the text when $k$ is totally real). After studying the distribution of certain quadratic extensions over $k$, that of certain cubic extensions over $k$, and the relation between the two distributions, this paper proves that, if $k$ is totally real, then a subset of $\{K\in L_{-}|{\lambda }_3\left(K\right)={\lambda }_3\left(k\right),{\mu }_3\left(K\right)={\mu }_3\left(k\right)\}$ has an explicit positive density in $L$-. The paper also proves that a subset of $\{L\in L_{+}|{\lambda }_3\left(L\right)=$${\mu }_3\left(L\right)=0\}$ has an explicit positive density in $L_{+}$ if 3 does not divide the class number of $k$ but is divided by only one prime ideal of $k$. Some consequences of the above results are added in the last part of the paper.

We discuss the problem of a generalization to higher dimensions of an earlier result of ours on a geometrical description of Selmer groups of elliptic curves over global fields of positive characteristic.

In both ordinary and equivariant 3-dimensional topology there are strong uniqueness theorems for connected sum decompositions of manifolds, but in ordinary higher dimensional topology such decompositions need not be unique. This paper constructs families of manifolds with smooth group actions that are equivariantly almost diffeomorphic but have infinitely many inequivalent equivariant connected sum representations for which one summand is fixed. The examples imply the need for restrictions in any attempt to define Atiyah-Singer type invariants for odd dimensional manifolds with nonfree smooth group actions. Applications to other questions are also considered.

A $W$-transform of an operator on white noise functionals is introduced and then characterizations for operators on white noise functionals are given in terms of their $W$-transforms. A simple proof of the analytic characterization theorem for operator symbol and convergence of operators are also discussed.

Based on the free product construction we show that a certain full factor of type $II{I}_{q^2}$ admits a minimal coaction of the compact quantum group $S{U}_q\left(n\right)$. Minimal coactions of compact Kac algebras are also investigated by the same technique.

We consider the asymptotic behavior of the solution of the Cauchy problem for the nonlinear nonlocal Shrödinger equation (NNS) with a source. The source in the NNS equation makes essential alterations to the asymptotic behavior. We study the cases of both small and large initial data.

For the signature of the Milnor fiber of a surface singularity of cyclic type, we prove a certain inequality, which naturally induce an answer of Durfee's conjecture in this case. For the proof, we use a certain perturbation method on the way of Hirzebruch's resolution process.