We are interested in the question how the spaces of solutions of elliptic equations vary according to the variations of underlying regions and coefficients of the equation. We will discuss this question for the case of equations Δu = Pu considered on noncompact Riemann surfaces R. Typically we ask the properties of mappings τx: (R,P) → dim PX(R) from the space Φ of pairs (R, P) of noncompact Riemann surfaces R and densities P on R, i.e. P(z)dxdy are 2-forms on R such that P(z)dxdy ≢ 0 and P(z) ≥ 0 are Hölder continuous with respect to local parameters z = x + iy, into cardinals, where PX(R) are the linear spaces of solutions of Δu = Pu on R with certain boundedness properties X.

0. Introduction. The purpose of this paper is to consider the following question: if R is a regular Noetherian ring and S ⊃ R is a module-finite R-algebra, is R a direct summand of S as R-modules? An affirmative answer is given if R contains a field, and it is shown that if the completions of the local rings of S possess maximal Cohen-Macaulay modules in the sense of § 1 of [6] then the conclusion is valid in this case too. Hence, if Conjecture E of [6] is true then the question raised here has an affirmative answer without further restriction on the regular Noetherian ring R, and it will be shown here that only a greatly weakened version of Conjecture E is needed.

Soient X un groupe abélien localement compact et dénombrable à l’infini, et ξ sa mesure de Haar. Dans la théorie du potentiel, un noyau de convolution N sur X signifie une mesure de Radon positive dans X. Pour une mesure de Radon réelle μ dans X, N* μ s’appelle le N-potentiel de μ dès que cette convolution a un sens.

Let Ω, I be open intervals in Rx = (— ∞ < x < ∞), Rt = (— ∞ < t < ∞) respectively. For a function a(x, t) ∈ C∞(Ω × I), consider the partial differential operator

(1.1) .

In this note we study the estimate from above and below and the asymptotic behaviour of the class number of positive definite integral quadratic forms.

In a recent paper [2], an operator valued function space integral was defined by the authors as follows.

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

The Laplace-Beltrami operator Δ on a smooth manifold M with a smooth Riemannian metric ds2 = ∑i,jgij(x)dxidxj applied to a smooth function φ takes the form Functions in the class H2(M) = {u∈ C4(M); Δ2u = 0} are called biharmonic.

The main purpose of the present paper is to give the details of the results announced in the P. J. A. note [11], establishing some global theorems on rigidity for a certain class of isometric imbeddings.

Due to Clairin and Goursat, a Bäcklund transformation of the first kind can be associated with Monge-Ampère’s equation. We shall consider Monge-Ampère’s equation of the form s + f(x, y, z, p, q) + g(x, y, z, p, q) t = 0, where p = ∂z/∂x, q = ∂z/∂y, s = ∂2z/∂x∂y, t = ∂2z/∂y2. The following theorems will be obtained:

1. The transformed equation takes on the same form s′ + f′ + g′t′ = 0 if and only if the given equation can be transformed to a Teixeira equation s + L(x, y, z, q)t + M(x, y, z, q)p + N(x, y, z, q) = 0 by a contact transformation.

2. Teixeira equation s + tL + pM + N = 0 is solved by integrable systems of order n if and only if the transformed equation is solved by integrable systems of order n — 1.

Let D be the open unit disk and let K be the unit circle. We say that α is an arc at ζ ∈ K if α is contained in D and is the image of a continuous function z = z(t) (0 ≤ t < 1) such that z(t) → ζ as t → 1. We call α a segment as ζ if the function z = z(t) is linear in t. If P is a property which is meaningful for each point of K, we say that nearly every point of K has property P if the exceptional set is a set of first Baire category in K. We assume that the reader is familiar with the rudiments of cluster set theory, and in particular with the terms ambiguous point, Meier point, and Plessner point of a function (cf. [4] or [7]).

The purpose of this paper is to improve a result of Schnitzer and Seidel [6] and to continue an analogy between entire functions and the class of functions which are holomorphic and unbounded in the unit disk, but bounded on a boundary spiral. The existence of such functions was established by Valiron [7].