Let $M$ be a $\Q$-divisor on a smooth surface over $\C\,$. In this paper we give criteria for very ampleness of the adjoint of $\rup{M}$, the round-up of $M$. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita's Conjecture in dimension 3.) In \S 4 we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.

We give pluriharmonic and harmonic representations of prehomogeneous vector spaces of regular classical parabolic type $(L = \Gamma \times M, V)$ througt the commutative quotients $V \mathbin{{/}\!{/}} M = \mathop{\rm spec} (C[V]^{M})$ of them.

The fourth Painlev\'e equation $P_{IV}$ is known to have symmetry of the affine Weyl group of type $A^{(1)}_2$ with respect to the Bäcklund transformations. We introduce a new representation of $P_{IV}$, called the symmetric form , by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of $P_{IV}$ is given in terms of this representation. Through the symmetric form, it turns out that $P_{IV}$ is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions $P_{IV}$, called Okamoto polynomials , are expressible in terms of the 3-reduced Schur functions.

An extended interpolation problem on a Riemann surface is formulated in terms of local rings and ideals. A sufficient condition for Nevanlinna parametrization is obtained. By means of this, Heins theorem on Pick-Nevanlinna interpolation in doubly connected domains is generalized to extended interpolation.

A martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional "stable-like" processes are presented.

It is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being concerned with circles, we also know that two closed circles in a real space form are congruent if and only if they have the same length. In this paper we study how prime periods of circles on a complex hyperbolic space are distributed on a real line and show that even if two circles have the same length and the same geodesic curvature they are not necessarily congruent each other.

A componentwise linear ideal is a graded ideal $I$ of a polynomial ring such that, for each degree $q$, the ideal generated by all homogeneous polynomials of degree $q$ belonging to $I$ has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal $I_{\Delta}$ arising from a simplicial complex $\Delta$ is componentwise linear if and only if the Alexander dual of $\Delta$ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors' definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors' theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to ${\rm SL}_2$ of the integers of the number field. This gives rise to a theta inversion formula, to which the authors' Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

We study the function $$\zeta(k_1,\dots,k_{n-1};s)=\sum_{0<m_1<m_2<\cdots<m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{s}}$$ and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.