Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

Soit f(z) une fonction algébroïde dans | z | < ∞ définie par

où cette équation est irréducible dans | z | < ∞. Rf est la surface de Riemann définie par f(z) comme surface de recouvrement de | z | < ∞.

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.

The main purpose of the present paper is to introduce a new understanding of formal theories.

It has been a traditional pattern of formal theories to presuppose a logic and an axiom system for each formal theory. The axiom system of any formal theory consists of a finite number of axiom schemata in general, but occasionally it can be regarded as consisting of a finite number of axioms. I will call any formal theory of this kind an axiomatic theory or an axiom-schematic theory according as its axiom system is regarded as consisting of a finite number of axioms or as consisting of a finite number of axiom schemata.

Let G be a connected Lie group and H a closed subgroup. The homogeneous space M = G/H is called reductive if in the Lie algebra g of G there exists a subspace m such that (subspace direct sum) and where is the Lie algebra of H, see [8].

It has been pointed out by K. ONO that there is a pair of mutually contradictory abstractions, each of which is self-consistent. Afterwards, Y. INOUE pointed out that there is a vast class of such pairs which is as vast as the class of all the Russell-type paradoxes. It must be a natural course of matter to ask the following question: For every number n, is there a system of mutually contradictory abstractions whose proper subsystems are all mutually consistent?