Given a harmonic function u on a Riemann surface R, we define a period function

for every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

Very recently, M. Choda, I. Kasahara and R. Nakamoto [3] extend the concept of free action of automorphisms for C*-algebras and prove several theorems which are hither to known for von Neumann algebras. In the present note, we shall concern with freely acting automorphisms on abelian C*-algebras. In § 2, several equivalent conditions for the free action are obtained. In §3, we shall apply them to an automorphism which has a transversal group.

A function f(z), regular in the unit disk D, is called annular ([1], p. 340) if there is a sequence of closed Jordan curves Jn ⊂ D satisfying

1. (A1) Jn is contained in the interior of Jn+1 for every n,

2. (A2) given ε > 0, there exists a positive number n(ε) such that, for each n > n(ε), Jn lies in the region 1 – ε < | z | < 1 and

3. (A3) lim min {| f(z) |; z ∈ Jn} = + ∞.

Let Ω be a bounded domain (in the complex plane) whose boundary, C, consists of finitely many disjoint, rectifiable, closed Jordan curves.

In this paper we describe a canonical procedure for constructing the extension of a G-foliation on a differentiable manifold X to its tangent bundles of higher order and by applying the Bott-Haefliger’s construction of characteristic classes of G-foliations ([2], [3]) we obtain an infinite sequence of characteristic classes for those foliations (Theorem 4.8).

In this paper we shall prove some congruence relations mod pα between the traces of Hecke operators T(pm) which act on the space of cusp forms of different weights satisfying some congruences mod pα – pα-1. The method of the proof is very simple and is applicable to all the cases where the trace formula for Hecke operators are already known.

In this note, we investigate a relation between the connected sum of manifolds and the distance of manifolds ([2]). Since the smoothing of a piecewise linear equivalence is given by connected sum of exotic spheres ([1]), we have a certain estimate of the smoothing obstruction using the distance of manifolds (Proposition 3). In § 3, an application is given to show the impossibility of the 0.64-pinching of an exotic sphere.

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 the additive group of p-adic integers. We will denote Gal(Q∞/Q) by Γ.

We call a second order differential P(z)dxdy on a Riemann surface R a density if it is not identically zero and P(z) is a nonnegative Hölder continuous function of the local parameter z = x + iy in each parametric disk. To each density P on R we associate the linear space P(R) of C2 solutions of the equation Δu(z) = P(z)u(z) invariantly defined on R. We also consider subspaces PX(R) of P(R) consisting of solutions with certain boundedness properties X.

Let II be a finite group and denote by MII the class of all (finitely generated Z-free) II-modules. In the previous paper [3] we defined an equivalence relation in MII and constructed the abelian semigroup T(II) by giving an addition to the set of all equivalence classes in MII. The investigation of the semigroup T(II) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group II.

A nonnegative locally Hölder continuous function P(z) on 0 < | z | ≤ 1 will be referred to as a density on 0 < | z | ≤ 1. The elliptic dimension of a density P(z) at z = 0, dim P in notation, is defined to be the dimension of the half module of nonnegative solutions of the equation Δu(z) = P(z)u(z) on the punctured unit disk Ω : 0 < | z | < 1 with boundary values zero on | z | = 1. After Bouligand we say that the Picard principle is valid for a density P at z = 0 if dim P = 1.

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

Let X(t) denote Brownian motion on the line 0 ≤ t < ∞, let and let 0 < α < 1. Orey and Taylor [5] have investigated the random set defined by the inequalities

and proved that P{dim Eα = 1 – α2} = 1. Here we prove two theorems on Eα that reflect more subtle properties of Eα than its Hausdorff dimension alone.

Recently G. Shimura [1] constructed modular forms of integral weight from the forms of half integral weight. His construction is rather indirect. Indeed, he proved that the Dirichlet series, obtained from a form of half integral weight, multiplied by a certain L-function, corresponds to a modular form of an integral weight by means of the characterization of modular forms due to Weil.

The purpose of this paper is to study the distribution of Fatou points of annular functions introduced by Bagemihl and Erdös [1]. Recall that a function f(z), regular in the open unit disk D: | z | < 1, is referred to as an annular function if there exists a sequence {Jn} of closed Jordan curves, converging out to the unit circle C: | z | = 1, such that the minimum modulus of f(z) on Jn increases to infinity. If the Jn can be taken as circles concentric with C, f(z) will be called strongly annular.

Let I denote the set of all inner functions in H∞, where H∞ is the Banach algebra of all bounded analytic functions on the open unit disk D. Let I* denote the set of all functions f(z) in H∞ for which the cluster set C(f,α) at any point α on the circumference C = {α| |α| = 1} is either the closed unit disk |w| ≤ 1 or else a single point of modulus one. Clearly, I is a subset of I*. In [3] the author has proved that I is properly contained in I*. Recently, Lohwater and Piranian [7] have shown that there is an outer function in I*. The purpose of this note is to point out some applications of this result. In particular we shall show in Theorem 2.3 that there exist outer functions whose boundary behavior is similar to that of inner functions.

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

Let h: R4 → R3 be a quadratic map defined by

For a natural number t, put