In this paper we shall study the holomorphic sectional curvature of the Bergman metric on a domain

The present paper is a continuation of [6] and [7] in which the principle of limiting absorption has been verified for symmetric systems of first order with long-range perturbations, but the operators considered there are of constant rank. On the other hand, operators with non-constant rank are also important in application as well as from a purely theoretical point of view. In this paper, we consider the linearized equation of magnetogasdynamics with long-range perturbation as an important example of such operators.

Let (A, m, k) be a Noetherian local ring. We define

and call it the Rees algebra of A. Let X be an indeterminate over A, then R(A) can be identified with the A-subalgebra .

A convergent power series f(z) in the open unit disk D is called Hadamard lacunary if it is expressed as follows:

In this paper we will discuss p-adic Artin L-functions. The existence of these functions is a simple consequence of a theorem of Deligne and Ribet [4]. One can formulate a “p-adic Artin conjecture” for these functions. Our primary purpose here is to relate this conjecture to the “main conjecture” discussed by Coates in [3]. We will describe the precise formulations of these conjectures that we will use later. Our main result will be that in fact the main conjecture implies the p-adic Artin conjecture.

In this paper we shall study relations among the spaces of modular forms mod p attached to Sk(Np, ψχ) and Sk,(N, ψ) by using certain identities between dimensions of these spaces.

Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Let D be a division algebra of finite dimension n2 over it’s center F. Suppose D has an involution, τ, of the first kind, of symplectic type (e.g. [1], p. 169). By the theory of the pfaffian, τ symmetric elements have degree less than n/2 over F. On the other hand, Tamagawa has shown (unpublished) that involutions like τ are closely related to minimal symmetric idempotents in D ⊗ F D. This author began by examining and trying to generalize these relationships. But before any theory seemed possible for division algebras, a theory relating subfields and symmetric idempotents was required. This investigation gave rise to the results presented here, especially the main theorem in Section Two.

In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.