G. Mokobodzki proved [5] that on any harmonic space with countable basis satisfying the axioms 1, 2, T+, K0 [2] [1] any equally bounded set of harmonic functions is equicontinuous. P. Loeb and B. Walsh showed [4] that the same property holds on a harmonic space without countable basis, if Brelot’s axiom 3 is fulfilled. The aim of this paper is to generalize these results to a harmonic space X satisfying only the axioms 1, 20, K1, [2] [1] where 20 is a weakened form of axiom 2. As a corollary we get: if any point of X possesses two open neighbourhoods U, V such that the set of harmonic functions on U separates the points of U∩V, then X has locally a countable basis.

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).

A space ring R is defined as a domain whose complement in the Moebius space consists of two components. The modulus of R can be defined in variously different but essentially equivalent ways (see e.g. Gehring [3] and Krivov [5]), which is denoted by mod R. Following Gehring [2], we refer to a homeomorphism y(x) of a space domain D as a k-quasiconformal mapping, if the modulus condition

is satisfied for all bounded rings R with their closure , where y(R) denotes the image of R by y = y(x). Then, it is evident that the inverse of a k-quasi-conformal mapping is itself k-quasiconformal and that a k1-quasiconformal mapping followed by a k2-quasiconformal one is k1k2-quasiconformal. It is also well known that the restriction of a Moebius transformation to a space domain is equivalent to a 1-quasiconformal mapping of its domain.

Let l be a rational prime. For each n≧0, denote by ζln a primitive ln-th root of unity and by Q(ζl,n) the cyclotomic field obtained by adjoining ζl,n to the rational field Q. Then a theorem which was proved by H. Weber is well known:

Aronson proved, in his paper [1], the existence and the uniqueness property of weak solutions of the initial boundary value problem for parabolic equations of second order with measurable coefficients. On the uniqueness of solutions of the Cauchy problem for such equations he also gave some interesting results in [2].

In our previous paper [1], we proved that any Cantor set E with successive ratios ξn satisfying the condition

is a Picard set, that is, each single-valued meromorphic function with E as the set of essential singularities takes on every value infinitely often in any neighborhood of each singularity with the possible exception of two values.

Recently Aronson [1] proved the uniqueness property of weak solutions of the initial boundary value problem for second order parabolic equations with discontinuous coefficients. An analogous result to Aronson’s was proved by Kuroda M in the case of some parabolic equations of higher order, where the method due to Aronson [2] plays an essential role. In this paper, under the same idea we shall be concerned with the asymptotic behavior of weak solutions for parabolic equations of higher order of the divergence form, when the data are prescribed on a portion of a time-like surface.

Dans ce mémoire, on continue d’étudier sur les fonctions à des limites fines que J. L. Doob a trouvées [5]. On a trouvé quelques propriétés des fonctions dans [10]. Dans le paragraphe 2 on donne une condition suffisante afin qu’une fonction admette une limite fine et on étudie sur l’indice harmonique d’un spot asymptotique d’une fonction à une limite fine. Le paragraphe 3 est consacré aux études d’autres propriétés.

This paper is devoted to the problem of the realisation of homology classes of a PL-manifold by PL-submanifolds.

The present study is founded on the consideration of Thorn complexes M(PLk), M(SPLk) for PL-microbundles which is defined by R. Williamson [5]. We shall apply Thom’s method [4] to PL-manifolds.

The author is grateful to Professors R. Shizuma and K. Shiraiwa for their kind criticisms.

As Fuks [3] stated, every domain of holomorphy or meromorphy over Cn is analytically convex in the sense of Hartogs. Oka [6] proved that every domain over Cn analytically convex in the sense of Hartogs is a domain of holomorphy. Therefore a domain of meromorphy over Cn coincides with a domain of holomorphy over Cn.

In the present paper, we first prove the following

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

The theorem of identity for analytic subsets of a reduced complex space is stated as follows;

In the present note we shall be concerned with the improvement of fundamental definitions in higher order enumerative geometry which has been recently given by W. F. Pohl. Pohl’s definition of q-th derivative of vector bundle is very complicated. We shall give a simpler and more reasonable definition of the q-th derivative of vector bundle in terms of sheaf theory and simplify the proofs in [P]. We shall also give a definition of higher order singularity of map.

In the previous paper [6], we have considered flows on the measure space (Ω, ℬ, P), which was the projective limit of a certain subspace (Ωn, ℬn, Pn) of the measure space (Sn, ℬ(Sn), Pn), where Sn is the (n − 1) · sphere with radius and Pn is the uniform probability distribution over Sn.

Given an arbitrary Riemannian n-space V let σ be a harmonic field in the complement V-V0 of a regular region V0. The problem of constructing in v a harmonic field ρ with the property was given a complete solution in [2]. The corresponding problem for harmonic forms σ, ρ remains open iri the general case. In the special case of locally flat spaces the construction can be carried out by replacing by the point norm [3].

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

The observation by Poincaré that Möbius transformations in the complex plane can be lifted to a half-space raises the need to be able to handle motions in hyperbolic space of more than two dimensions by means of an analytic apparatus of not too forbidding complexity. In my experience the best way to do so is to be guided by analogies with the familiar twodimensional case. The purpose of this little paper is to collect a few formulas that the writer has found useful when working with certain hyperbolically invariant operators.

Let

be a power series with Hadamard gaps,

convergent in |z|<1.

This report is a survey of some of the many different ways of characterizing a class of plane quasiconformal mappings. This class was considered by Ahlfors [4] in his treatment of the Teichmiiller problem, and it has been studied rather extensively in the last ten years.