Let $(S,L)$ be a polarized surface and let $\pi$ : $\tilde{S}\to S$ be the blow-up at $r$ points $p_1,$ . . . , $p_r$ on $S$. Set $\tilde{L}={\pi }^{*}L-{\displaystyle \sum }{a}_i{E}_i$, where $a_i\text{'}s$ are positive integers and $E_i\text{'}s$ are the (-1)-curves over $p_i$. We consider whether $\tilde{L}$ is ample or not if $p_i\text{'}s$ are in a general position. The cases of sectional genus two are studied especially precisely.

We show that the existence of a precipitous ideal over the successor of some limit cardinals implies the existence of some large cardinals, in the sense of consistency. Moreover we use the same technique to evaluate the consistency strength of precipitousness of Woodin's stationary tower.

In this paper, we shall obtain a reducible theorem for a linear almost periodic system with an almost zero coefficient matrix. This reducible theorem states that the system can be transforms into two systems with size smaller than the original system. Of course, the transformation is linear and almost periodic.

First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space $X$ of nondegenerate unramified hermitian forms on a $\mathfrak{p}$-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on $X$, and parametrize of all spherical functions on $X$. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.

We define a Rohlin property for $Z^2$-actions on UHF algebras and show a non-commutative Rohlin type theorem. Among those actions with the Rohlin property, we classify product type actions up to outer conjugacy. We consider two classes of UHF algebras. For UHF algebras in one class including the CAR algebra, there is one and only one outer conjugacy class of product type actions and for UHF algebras in the other class, contrary to the case of $Z$-actions, there are infinitely many outer conjugacy classes of product type actions.

The Wick product of operators on Fock space is introduced on the basis of the analytic characterization theorem for operator symbols established within the framework of white noise distribution theory. Existence and uniqueness of solutions are proved for a certain class of ordinary differential equations for Fock space operators. Quantum stochastic differential equations of Itô type and their generalizations involving higher powers of quantum white noises enter into our consideration.

These varieties are conjectured to be abelian varieties up to finite étale coverings. This conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory. In particular, this is true for $n=3$.

G. Raby proved in [4] that the Godbillon-Vey invariant for codimension-one foliations is invariant by $C^1$-diffeomorphisms. In this paper we generalize the result for foliations of arbitrary codimension.

We determine the conjugacy classes of the Mathieu simple groups by using the combinatorial properties of the Steiner system and the binary Golay code.

We describe the notion of the dimension group for subshifts in terms of symbolic dynamical system and show that the dimension group is a conjugacy invariant. We will show that the the hereditary subsets invariant under the dimension group automorphism exactly corresponds to the gauge invariant ideals of the associated $C^{*}$-algebra ${\mathcal{O}}_{\Lambda }$ for the subshift $(\Lambda ,\sigma )$. As a result, we give the conditions that the $C^{*}$-algebra ${\mathcal{O}}_{\Lambda }$ becomes simple and purely infinite in terms of symbolic dynamics.

We study entire functions of exponential type which are eigenfunctions of the Laplacian. We represent them by an integral on the complex light cone. The integral formula is closely related to the Fourier-Borel transformation for analytic functionals on the complex sphere.

The purpose of this paper is to derive that the square of Fourier coefficients $a\left(n\right)$ at a square free positive integer $n$ of modular forms $f$ of half integral weight belonging to Kohnen's spaces of arbitrary odd level and of arbitrary primitive character is essentially equal to the critical value of the zeta function attached to the modular form $F$ of integral weight which is the image of $f$ under the Shimura correspondence. Previously, KohnenZagier had obtained an analogous result in the case of Kohnen's spaces of square free level and of trivial character. Our results give some generalizations of them of KohnenZagier. Our method of the proof is similar to that of Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight over totally real fields.

We call a formal morphism between completions of complex spaces convergent if it comes from a holomorphic mapping between the complex spaces. We assume always that the source space is compact. Then a formal morphism is either convergent everywhere or nowhere, under very general conditions.

We study the initial-boundary value problems and the corresponding stationary problems of the one-dimensional discrete Boltzmann equation in a bounded region. The boundary conditions considered are of mixed type and involve both the reflection and diffusion parts. It is shown that a unique solution to the initial-boundary value problem exists globally in time under the general situation that the reflection parts of both the boundary conditions do not increase the number of gas particles. Furthermore, it is proved that stationary solutions exist under the restriction that the reflection part of the boundary condition on one side really decreases the number of gas particles. This restriction plays an essential role in proving the existence result.