The ergodicity of classical dynamical systems which appear really in the statistical mechanics was discussed by Ya. G. Sinai [9]. He announced that the dynamical system of particles with central potential of special type in a rectangular box is ergodic. However no proofs have been given yet. Sinai [11] has given a proof of the ergodicity of a simple one-particle model which is called a Sinai billiard system.

The deflection, under a point load, of a thin elastic plate clamped at the edges is the biharmonic Green’s function β with the boundary data β = ∂β/∂n = 0. If the boundary of the region is reasonably smooth, the construction of β offers no difficulty. In contrast, nothing is known about the existence of β in the general case. The purpose of our study is to give a sufficient condition for the existence of β on a given Riemannian manifold of arbitrary dimension and to construct β. Our results will have, apart from their physical meaning in elasticity, consequences in the biharmonic classification theory of Riemannian manifolds.

In recent papers with S. S. Chern [3] and T.Ishihara [4], the author studied both the volume—and distance—decreasing properties of harmonic mappings thereby obtaining real analogues and generalizations of the classical Schwarz-Ahlfors lemma, as well as Liouville’s theorem and the little Picard theorem. The domain M in the first case was the open ball with the hyperbolic metric of constant negative curvature, and the target was a negatively curved Riemannian manifold with sectional curvature bounded away from zero. In this paper, it is shown that M may be taken to be any complete Riemannian manifold of non-positive curvature.

Let Ω be a bounded domain of n-dimensional Euclidean space Rn (n ≥ 2). On Ω we consider the biharmonic equation

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

The Airy integral is a formula concerning the Fourier transform of a function like sin x3 or cos x3, and is written, for instance in [2], as

for x ≧ 0.

In this paper we investigate the connection between the u-invariant, u(F), of a formally real field F as defined by Elman and Lam [2] and certain Hasse Principles studied by Elman, Lam and Prestel in [3].

In [2], chap. IV, 2me partie, (7.4.8), Grothendieck considered the following problem: is any m-adic completion of an excellent ring A also excellent?

In [8] I proved that, if A is an algebra of finite type over an arbitrary field k, the answer is positive.

We are concerned with the behaviour of analytic functions near the boundary. Let T and D be the unit circle |z| = 1 and the unit disk |z| < 1, respectively. The element of T is denoted by θ (0 ≤ θ < 2π). Let be analytic in D. The area integral A(f, θ) of f at θ is defined by

Let V be a complex vector space of dimension n, L a lattice of V and E = V/L a complex torus. Let H be a Hermitian form on V. We introduce a multiplication in L × C* by

This note is to be a supplement of the preceeding paper in the journal by Matsushima, settling a question raised by him. In his paper he associates a holomorphic vector bundle over a complex torus to a holomorphic representation of what he calls Heisenberg group. We shall show that a simple holomorphic vector bundle is determined in this manner if and only if the associated projective bundle admits an integrable holomorphic connection. A theorem by Morikawa ([3], Theorem 1) is the motivation of this problem and is somewhat strengthened by our result.

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote the Riemann sphere by Ω. If f(z) is a function defined on D with values belonging to Ω, if ζ ∈Γ, and if Λ is an arc at ζ then C∈(f, ζ) denotes the cluster set of f at ζ along Λ. If there exist three mutually exclusive arcs Λ1, Λ2, Λ3 at ζ such that

In the following we shall fix an algebraically closed field K of arbitrary characteristic. The term variety will mean an algebraic scheme over K which is integral. In general we shall use the notations and the terminology of Éléments de Géométrie Algébrique of A. Grothendieck and J. Dieudonné. For instance, if Z is a variety and x ∈ Z is a point, then OZ, x means the local ring of Z at x and mx – the maximal ideal of 0Z, x.