One of the authors has shown the ergodicity of the perturbed billiard system which can describe the motion of a particle in a potential field of a special type [5], [6]. Since then, some development has been made, and we are now able to show the Bernoulli property of the system in this article. We hope, the result gives a new progress in statistical mechanics. Our method of the proof is inspired by the idea of D. S. Ornstein and B. Weiss [9], which has been used by G. Gallavotti and D. S. Ornstein [3] for a Sinai billiard system.

For a given Riemannian manifold M and its submanifold N, one can find various types of geodesies on M starting from any point of N and ending in any point of N. For example, geodesies which start perpendicularly from N and end perpendicularly in N are treated by many mathematicians. K. Grove has stated a condition in a general case for the existence of such a geodesic ([4]), where he has used the method of the infinite dimensional critical point theory. This method is very useful for the study of geodesies and many geometricians have used it successfully. It has two aspects: one is an existence theory and the other is a quantitative theory, which one can find, for instance, in the excellent theory for closed geodesies of W. Klingenberg ([1], [7]) and so on.

Il est bien connu que le groupe des classes relatives d’une extension abélienne du corps des rationnels est annulé par tout êlément entier de l’idéal de Stickelberger. Soit l un nombre premier impair. Nous montrons dans les pages qui suivent qu’un certain groupe de l-classes réelles associé à un caractère x d’ordre g et considéré comme -module est annulé par la valeur en 1 de la fonction L l-adique: ou bien par un multiple bien défini de cette valeur.

Most of the problems in differential geometry can be reduced to problems in differential equations and differential inequalities on Riemannian manifolds. Our main purpose of this paper is to study such differential inequalities on complete Riemannian manifolds. In [5], H. Omori proved a very important theorem. S. Y. Cheng and S. T. Yau gave a simplification and a generalization of it which was called the generalized maximum principle in [2] and [7], and many interesting applications in differential geometry in [2], [3], [7], and [8].

Let B be a B-ring with a nonarchimedean valuation | |, i.e., B is an integral domain satisfying the following conditions: (i) B is bounded (| a | ≤ 1 for every a ∊ B), (ii) the boundary forms a multiplicative group.

In this paper, we shall give some notes on the order functions of holomorphic curves and, applying these facts, we may formulate the theory of holomorphic curves more precisely than those in Ahlfors [2], Cowen and Griffiths [4], Weyl [7] or Wu [8] in several cases.

Let X be a complex analytic space of positive dimension and A a complex analytic subvariety of X. We call A a direct factor of X if there exist a complex analytic space B and a biholomorphic mapping f: A × B → X such that, for some b ∊ B, f(a, b) = a on A, and a complex analytic space X to be primary if X has no direct factor, not equal to X itself, of positive dimension.

Let k be a field, and X = [xij] be an n × (n + m) matrix whose elements are algebraically independent over k.

We shall study the canonical module of the graded ring R, which is a quotient ring of the polynomial ring A = k[X] by the ideal αn(X) generated by all the n × n minors of X.

Let us start with a short survey concerning complex white noise (for details see Hida [3]): Let be the space of real valued rapidly decreasing testing functions and its complexification.

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integer t ≥ 1 and a cusp form of weight k/2 (k odd, ≥ 3), level N (divisible by 4) and character ϰ, he associated a certain function f(t)(z) (Ft(z) in Shimura’s notation).

As is well known, for a real vector space V, the Fourier transformation

gives an isometry between L2(V) and L2(Vv), where Vv is the dual vector space of V and < , >: V×Vv → R is the canonical pairing.

Stochastic processes on a Hilbert space have been discussed in connection with quantum field theory, theory of partial differential equations involving random terms, filtering theory in electrical engineering and so forth, and the theory of those processes has greatly developed recently by many authors (A. B. Balakrishnan [1, 2], Yu. L. Daletskii [7], D. A. Dawson [8, 9], Z. Haba [12], R. Marcus [18], M. Yor [26]).