Let D = {z : |z| < 1} and C = {z : |z| = 1}. If W denotes the Riemann sphere equipped the chordal metric X, let f: D → W be meromorphic. A chord T lying in D except for an endpoint γ ∈ C is called a Julia segment for f if for each Stolz angle Δ in D at γ which contains T, f assumes infinitely often in Δ all values of W with at most two exceptions. We call γ ∈ C a Julia point for f if every chord in D ending at γ is a Julia segment for f, and we denote by J(f) the set of Julia points of f.

Consider an open Riemann surface R and a density P(z)dxdy (z = x + iy), well defined on R. As was shown by Myrberg in [3], if P ≢ 0 is a nonnegative α-Hölder continuous density on R (0 < α ≤ 1) then there exists the Green’s functions of the differential equation

p>on R, where Δ means the Laplace operator. As a consequence, there always exists a nontrivial solution on R.

The well-known first limit formula of Kronecker asserts that

where z = x + iy is contained in the complex upper halfplane H, C = the Euler-Mascheroni constant, and η(z) is the Dedekind eta-function defined by

In this paper, we study a certain difference between meromorphic mappings and holomorphic mappings into taut complex analytic spaces. We prove in §2 that, for any complex analytic space X, there exists a unique proper modification of X with center Sg (X) which is minimal with respect to the property that M(X) is normal and, for any T-meromorphic mapping f: X → Y (see Definition 1.3) into a complex analytic space Y, there exists a unique holomorphic mapping such that except some nowhere dense complex analytic set, where Sg(X) denotes the set of all singular points of X.

Consider a nonnegative Hölder continuous 2-form P(z)dxdy (z = x + iy) on a connected Riemann surface R. We denote by P(R) the linear space of solutions u of the equation Δu = Pu on R and by PX(R) the subspace of P(R) consisting of those u with a certain boundedness property X. We also use the standard notations H(R) and HX(R) for P(R) and PX(R) with P ≡ 0.

This paper continues the study of stochastic integrals in abstract Wiener space previously given in [14]. We will present, among other things, the detailed discussion and proofs of the results announced in [16]. Let H ⊂ B be an abstract Wiener space.

Dans l’article précédent [6], nous définissions deux classes des noyaux de convolution symétriques sur un groupe abélien localement compact et dénombrable à l’infini; l’une est régulière et l’autre est singulière.

Dans toute la suite X désigne un groupe abélien localement compact et dénombrable à l’infini; ξ est sa mesure de Haar. Nous rappelons qu’un noyau de convolution N sur X signifie une mesure de Radon positive dans X. Il est connu que le principe de domination pour N joue un grand rôle dans la théorie du potentiel. Nous remarquons ici qu’il existe deux sortes des principes de domination pour N; l’un est défini par G. Choquet et J. Deny (cf. [3]), qui est très utile pour discuter le principe du balayage, et l’autre est introduit de celui dans le cardre plus large (cf. [8]). Nous montrerons d’abord que ceux sont équivalents. Par conséquent, on pourra discuter, en même temps, sur les mesures balayées relativement au noyau N et sur la résolvante associée au noyau N.

Let X be a locally compact and σ-compact Abelian group and ξ be the Haar measure of X. A positive Radon measure N on X is called a convolution kernel when we regard it as a kernel of potentials of convolution type. M. Itô [4], [6] characterized the convolution kernel which satisfies the domination principle. The purpose of this paper is to characterize the relative domination principle for the convolution kernels.

Let A and B be densely defined closed linear operators in complex Banach spaces X, Y, respectively, with nonempty resolvent sets.

In this note, we shall examine some results of Bloch [2] and Cartan [3] concerning complex projective space minus hyperplanes in general position. The purpose is to restate their results in a more general setting by using the intrinsic pseudo-distance defined on a complex space [16] and the concept of tautness introduced by Wu in [18]. In the process we shall generalize some results of Dufresnoy [4] and Fuj imoto [7].

Let S be a ring and d be a derivation of S. The Oreextension S(X,d) is the ring generated by S and an indeterminate X satisfying the ralation Xa — aX = da for all a in S.

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.