Consider a single-valued meromorphic function w = f(p) defined on an open Riemann surface R with an ideal boundary β. In [1], Collingwood and Cartwright introduced the global cluster set for a function meromorphic on the unit disk. Generalizing the definition of global cluster sets to our present setting, we define the global cluster set for w = f(p) as follows;

A value w in the extended complex plane is called a cluster value at β if there exists a sequence in R converging to β such that

A nonnegative locally Holder continuous function P(z) on 0<|z |≤1 will be referred to as a density on Ω 0 < |z | < 1 with singularity at δ : z = 0, removable or genuine. A density P on Ω is said to be finite if it is integrable over Ω :

(1) ∫ Ω P(z)dxdy < ∞.

A nonnegative locally Hölder continuous second order differential P = P(z)dxdy (z = x + iy) on a Riemann surface R is referred to as a density on R. A density P is said to be finite if P is integrable over R, i.e.

(1) ∫ R P(z)dxdy < ∞.

Consider an open Riemann surface R and a single-valued meromorphic function w = f(p) defined on R. A value w0 in the extended complex plane is said to be a cluster value for w = f(p) if there exists a sequence {pn } in R accumulating at the ideal boundary of R such that

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows that

where χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

As a continuation of the previous paper [7], we shall consider in this paper the problem of prediction given bounded intervals and obtain integral representations of predictors and prediction errors. For that purpose we shall introduce innovation processes well matched with bounded intervals. We follow the notation and terminology in [6].

In this paper we use a general technique, due to Larfeldt-Lech [9], to show that there exist local noetherian rings which are “bad” with respect to some properties which are obviously verified in the algebro-geometric case. In particular we show that there exist local Gorenstein rings which are not homomorphic images of regular (local) rings and that there exist local rings which do have canonical module but do not have canonical algebra.

In this paper we consider the periodic problems for certain nonlinear parabolic differential equations in domains with periodically moving boundaries. The typical problem, which is going to be discussed in the present paper, is to solve the following:

Let us consider the following two problems.

(A) Does either

or

hold for the not identically vanishing solution of the equation

for x ∈ Ω ⊂ Rn (n ≥ 3), where λ is a constant satisfying λ > E0 and V(x) is a 2-body or many-body potential?

The theory of ordinal diagrams has been a most powerful means for consistency proofs of some systems of second order arithmetic. The last existing result in this line is the consistency proofs of the systems with the provably -comprehension axiom and the -comprehension axiom respectively (cf. [6]). In order to pursue the consistency problem further, one needs investigate the theory of ordinal diagrams in two directions—refinement and strengthening of the theory.

For this purpose we have begun to search for some properties concerning ordinal diagrams and some variations of the theory of ordinal diagrams. The reader is requested to refer to §26 of [7] for the basic knowledge of ordinal diagrams.

The existence of condenser potentials was established in the framework of Dirichlet spaces by Beurling and Deny, cf. Deny [5] or Landkof [10], simply by choosing the potential of minimal energy within a certain convex set. This same idea works for non-symmetric Dirichlet spaces, cf. Bliedtner [3].

An ordered ring with the least positive element 1 is a “Z-ring” if for each natural number n,

In the previous papers [2], [3] we treated the following two questions. Let L,M,N be positive definite quadratic lattices over Z:

• (i) If L, M are indecomposable, then is L⊗M indecomposable?

• (ii) Does L ⊗ M ⋍ L ⊗ N imply M ⋍ N?

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively :

1. (1) the class number of the cyclotomic Zℓ-extension of k,

2. (2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,

3. (3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and

4. (4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).