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1. By the method of Dolzhenko’s paper, we studied relations between non-tangetial (angular) boundary behaviors and horocyclic boundary behaviors of arbitrary functions defined in the open unit disc of the complex plane in [8]. Vessey [5], [6] investigated the behavior of arbitrary functions on paths which are “more tangential” than horocycles. The purpose of the present paper is to prove the fact that is sharper than the results in Vessey [5], [6], and generalize the results in [8] to obtain the connection between behaviors on two “more tangential” angles.

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

In their paper [12], Toda and the author have concerned themselves in the following Theorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class O HB (O HD , resp.). Then, for any compact subset K of R such that R−K is connected, R−K as an open Riemann surface belongs to the class O AB (O AD , resp .) (Kuramochi [4]). They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.

In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem—Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the T x of [1] is just the a x of [6] when X is restricted to p 0, see [6], p. 539). In fact after introducing, below, the notion of one affine connection A on a manifold being rigid with respect to another affine connection B on M and making some observations concerning such a relationship, Theorem 1 is seen to be a reformulation of Theorem 2.

In this paper we consider the following quasi-linear parabolic equationsLu = ut — div A(x, t, u, ux)+B(x, t, u, ux= 0,where A is a given vector function of the variables x,t,u,uX, and B is a given scalar function of the some variables. We assume that they are difined in the rectangle

The purpose of this paper is to study the relationship between a certain isomorphism of some rings of functions on Riemann surfaces and a quasi-conformal mapping.It is well known that two compact Hausdorff spaces are topologically equivalent if and only if their rings of continuous functions are isomorphic. We shall establish an analougous result concerning a function ring on a Riemann surface and the quasi-conformal equivalence.

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

The object of the present paper is to prove some theorems concerning the existence and the uniqueness of the solution of the initial value problem for the evolution equation(E).

In the former paper, I have developped a theory of meromorphic functions in a neighbourhood of a bounded closed set E of logarithmic capacity zero, by means of Evans’ potential fnnction u(z), which tends to ∞, when z tends to any point of E. It is not known, whether such a potential function exists on an open Riemann surface with null boundary, but by a substitute of Evans’ function. we shall develop the similar theory of meromorphic functions on an open Riemann surface with null boundary.

The purpose of this paper is to prove the following result.Theorem 1. Let K be a real algebraic number field of degree m = n + 1. Let 1, β1, …, βn be a basis of K.

Introduction. The purpose of this paper is to discuss implications of Hadamard's condition for elastic stability (2, §269) with respect to uniqueness of solutions of boundary value problems in the theory of small deformations superimposed on large. We show that a slightly refined form of his condition implies a uniqueness theorem for displacement boundary value problems. We construct a counter-example showing that his condition does not imply uniqueness of solutions for one type of stress boundary value problem.

The main purpose of the present paper is to show that the local homogeneous holonomy group of the conformally flat Riemannian manifold is the full rotation group with some exceptions.