In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

In this paper we are concerned with transversely orientable codimension one foliations. Let be a Cr-foliation as above in a smooth manifold M, r ≧ 1, and let F0 be a closed leaf of . A neighborhood U of F0 is called a bicollar of F0 in this paper if there is a normal line bundle : U → F0 with respect to a fixed Riemannian metric on M such that each fibre of is transverse to .

Consider a parabolic end Ω of a Riemann surface in the sence of Heins [2] (cf. Nakai [3]). A density P = P(z)dxdy (z = x + iy) is a 2-form on with nonnegative locally Hölder continuous coefficients P(z). A density P is said to be finite if the integral

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

In [10] and [11], the second author proved the following theorem by using Synge’s formula:

THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.

Let M be a compact, oriented Riemannian manifold of dimension dr and let Γ be the fundamental group of M. For a finite dimensional representation ρ of Γ on a vector space F, Ray and Singer [10] have defined the analytic torsion T(M, ρ) as follows: We denote by E the vector bundle over M with typical fibre F defined by the representation ρ.

The multi-parameter Brownian motion introduced by P. Lévy is not only a basic random field but also gives us interesting fine probabilistic structures as well as important properties from the view point of analysis. We shall be interested in investigation of such structures and properties by expressing the Brownian motion in terms of the multiparameter white noise. The expression naturally requires basic tools from analysis, in particular the Radon transform. While there arises the special linear group SL(n + 1, R), to which the Radon transform is adapted, and the group plays an important role in observing probabilistic structures of the Brownian motion. To be more interested, we can give some deep insight to unitary representations of SL(n + 1, R) through our discussion.

In his study of non-linear differential equations of the second order, N. Levinson [3] defined the dissipative systems (D-systems) which arise in many important cases in practice. To a dissipative system a transformation T: R2 → R2 called the Poincaré transformation is associated. Levinson used the Poincaré transformation in the qualitative study of dissipative systems, and he [3] and Massera [5] obtained certain equalities between the number of subharmonic solutions of a dissipative systems under suitable conditions. We call these the Levinson-Massera’s equalities.

Let p be a prime. If one adjoins to Q all pn-th roots of unity for n = 1,2,3, …, then the resulting field will contain a unique subfield Q∞ such that Q∞ is a Galois extension of Q with Gal (Q∞/Q) Zp, the additive group of p-adic integers. We will denote Gal (Q∞/Q) by Γ. In a previous paper [6], we discussed a conjecture relating p-adic L-functions to certain arithmetically defined representation spaces for Γ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces for Γ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

Let k be a totally real algebraic number field, the maximal order of k, and let L (resp. M) be a Z-lattice of a positive definite quadratic space U (resp. V) over the field Q of rational numbers. Suppose that there is an isometry σ from L onto M. We have shown that the assumption implies σ(L) = M in some cases in [2]. Our aim in this paper is to improve the results of [2]. In § 1 we introduce the notion of E-type: Let L be a positive definite quadratic lattice over Z.

Let f(z) be a non-degenerate meromorphic mapping of the n-dimensional complex Euclidean space Cn into the N-dimensional complex projective space PNC. A generalization of results of Edrei-Fuchs [2] for meromorphic mappings of C into PNC was given by Toda [5], and an estimate of K(λ) for meromorphic mappings of Cn into PNC was done by Noguchi [4]. In this note we generalize several results of Edrei-Fuchs [2] in the case of meromorphic mappings of Cn into PNC.

In this note we exhibit a condition under which a sequence of elements a1, …, an in a commutative noetherian local ring A form an A-sequence, and derive a number of corollaries.