Consider the polar coordinate differentials (dr, dθ) on a hyperbolic Riemann surface R with center z0 ∈ R which are given by

where GR(z, ζ) is the Green’s function on R with pole ζ ∈ R.

We are concerned with an algebra S over a commutative ring. Precisely S is a non-commutative ring with identity which is also a finitely generated unital R module such that r(xy) = (rx)y = x(ry) for r in R and x, y ∈ S. In section one, we assume A is a commutative, Artinian ring. Following Goro Azumaya (see (1, p. 273)), we define the canonical module F of A to be the injective hull of A modulo the Jacobson radical of A i.e. F = I(A/J(A)).

The purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

In this paper, we treat the problem of determining the rate of growth of meromorphic functions on the plane, which are solutions of nth order algebraic differential equations whose coefficients are arbitrary entire functions (i.e., equations of the form, Ω(z,y,dy/dz, • • •, dny/dzn) — 0, where Ω is a polynomial in y, dy/dz, • • •, dny/dzn whose coefficients are arbitrary entire functions of z.)

Throughout what follows, let H be a normal subgroup of the group G, be G/H, M a left G module, and HomH(G,M) = a left G module via (τϕ(σ) = (ϕ(στ)

In [1] the present authors compared the 5-term Hochschild Serre sequence to the long exact sequence arising from the short exact

sequence of modules

Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ) = limr→1f(reiθ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ), a < d < b, cover the circumference |w| = 1 infinitely many times.

Soient f = (f0, …, fn) un système transcendant dans le plan |z| < ∞ et X = {F} un ensemble de combinaisons des fonctions f0,…,fn, linéaires, homogènes à coefficients constants et linéairement indépendantes n + 1 à n + 1. Alors, combien de combinaisons exceptionnelles au sens de Nevanlinna y-a-t-il dans X? On sait que 1) il y en a une infinité dénombrable au plus en général ([5]) et 2) si l’ordre inférieur de f est égal à zéro, il y en a n au plus ([4]).

In this paper, I will prove the following theorems;

For given integers n and m such that m ≧ 2n, there exist 1-dimensional local domains which are complete intersections and have embedding dimension n + 1 and multiplicity m.

In [3], M. W. Hirsch obtained some necessary conditions for the existence of an Anosov diffeomorphism on a differentiable manifold. As an application, he constructed many manifolds which do not admit Anosov diffeomorphisms.

W. Sierpisnki [3] demonstrated 82 propositions, called C1-C82, with the aid of the continuum hypothesis. D. A. Martin and R. M. Solovay remarked in [2] that 48 of these propositions followed from Martin’s axiom (MA), 23 were refuted by and three were independent of But the relation of the remaining eight propositions to has been unsettled.

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

Consider a pair (R, Γ) of a Riemann surface R and a period Γ. By a period Γ we mean a real-valued function Γ(γ) on one-dimensional cycles {γ} of the Riemann surface R.

The author discussed in [4] the ultimate boundedness of a system which is governed by a stochastic differential equation

The object of the present paper is to give a method of studying the topological properties of integral manifolds defined by a completely integrable one-form.

Our method is differential-topological. Through the singular points of the variation equation of the given one-form, we investigate the qualitative properties of the integral manifolds.