Soit V une variété complexe satisfaisant aux conditions suivantes:

• 1) un groupe de Lie semi-simple connexe G opère sur V de manière transitive et holomorphe;

• 2) la variété V admet une mesure invariante par G.

On dira par abus de langage que la variété V est semi-simple. On se propose d’étudier dans ce travail des variétés complexes compactes semi-simples. Les variétés de ce genre font l’objet de deux travaux de Wang [5], [6]. Dans le mémoire [5] il a étudié le cas simplement connexe et dans [6] le cas parallélisable. Le résultat obtenu dans ce travail montre que toute variété complexe compacte semi-simple est en un certain sens une combinaison de deux types de variétés étudiées par Wang. On établit en effet le théorème suivant.

Let R be a ring with unit, and e be an idem potent in R such that (1 - e)Re = 0. In this note we shall explore the relationships between homological properties of R and those of its subring eRe.

Examples of such rings are abundant, the most common being perhaps the ring R of all two-by-two upper triangular matrices over a field, where—

The definition of the determinant of an endomorphism of a free module depends on the following fact: If F is a free R-module of rank n, then the homogeneous component ∧nF, of degree n, of the exterior algebra ∧ F of F is a free R-module of rank one. If a is an endomorphism of F, then a extends to an endomorphism of ∧ F which in ∧nF is therefore multiplication by an element of R. That factor is then defined to be the determinant of α. (A discussion of this theory may be found in [4].)

This procedure cannot be applied in general to finitely generated projective modules since, for such modules, it may happen that no homogeneous component of the exterior algebra is free of rank one.

Let C and C′ be two simple closed curves in the complex z-plane which have no point in common and surround the origin. Denote by D the annulus bounded by C and C′. Consider a family {γ} of rectifiable curves γ in D and the family P of all non-negative lower semi-continuous functions ρ = ρ(z) in D. Put

Let w = f(z) = u(x, y) + iv(x, y) be an interior transformation in the sense of Stoïlow in an arbitrary domain D, i.e. w = f(z) is continuous and single-valued in D, and unless constant takes open sets in D into open sets in the w-plane, and does not take any continuum (other than a single point) into a single point of the w-plane. Stoïlow proved that such an interior transformation can be represented as

where T(z) = ξ is a topological mapping of D onto a domain D′ and w = φ(ξ) is a meromorphic function in D′.

Problems concerning the sum of two fourth powers of integers seem to be so difficult that little has been known since long years [3]. For instance, it is an important problem to determine whether there are infinitely many prime numbers which are represented in the form p = a4+b4. But nothing is known except that the density of such prime numbers is easily proved to be 0; accordingly it is difficult to obtain a necessary and sufficient condition under which p is represented in such a form.

In this paper we prove the fundamental exact sequence in (co)homology for non-normal subgroups announced in our previous note [8]: Let H be a subgroup of a group G. If M is a G-module and if, for a natural number n, Hm(U, M) = 0 for m = 1, …, n - 1 and for every subgroup U of H which is an intersection of conjugates of H in G, then we have an exact sequence

the significance of the maps and the group Hn(H, M, M)1 will be explained below. (We have a dual result for cohomology groups).

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

Let R be an open Riemann surface. By a density P on R we mean a non-negative and continuously differentiable functions P(z) of local parameters z = x + iy such that the expression P(z)dxdy is invariant under the change of local parameters z. In this paper we always assume that P≢0 unless the contrary is explicitly mentioned. We consider an elliptic partial differential equation

which is invariantly defined on R.

Much has been studied about an almost complex structure these ten years. One of the problems about the structure is to find an affine connection which makes a given almost complex tensor field parallel. A Riemannian connection is a one without torsion for which the fundamental tensor field of a Riemannian manifold is parallel. Affine connections on the group manifold were investigated fully by E. Cartan in [1]. In this paper we treat in general some tensor fields and affine connections which make the fields parallel. Moreover some studies about certain tensor fields are given.

Ce travail est une suite du mémoire [8] paru dans ce journal. Dans [8], on a étudié la structure du groupe connexe d’isotropie d’un espace homogène de Stein G/H d’un groupe de Lie semi-simple complexe et connexe G et on a montré que le groupe H est égal au complexifié d’un sous-groupe compact maximal de H ([8], Théorème 3). Le but essentiel de ce travail est d’étendre ce résultat au cas d’un espace homogène de Stein d’un groupe de Lie complexe et connexe quelconque.

Let D be a plane domain and Γ be a component of the boundary of D consisting of a single point. According to Sario [5] we shall call Γ a weak boundary component of D if its image under any conformal mapping of D consists of a single point. A weak boundary component has been introduced by Grötzsch [2], who called it “vollkommen punktförmig”. If Γ is not weak we shall say that it is unstable (Sario [5]). We know that the weakness depends merely on the configuration of D in a neighborhood of Γ (see [4], p. 274).

Let E be a compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Suppose that E is of capacity zero. Then Ω is a domain and we shall consider a single-valued meromorphic function w = f(z) on Ω which has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at a point ζ ∈ E if there exists a neighborhood of C where the function f(z) does not take this value w.

In this paper, a formal theory of ordinal numbers is developed on an axiomatic basis whose details are described in § 1. Our primitive notions are set, class, collection, a binary relation ∈, and collection formation {!}. Sets and classes in our theory play similar roles as sets and classes respectively in Gödel [1] except the difference that an element of a class is a class but not necessarily a set. A new notion, introduced into our theory is that of collections. A collection relates to a class, just as a class relates to a set in von Neumann’s theory. That is; a set is a class and a class is a collection but the converses are not generally the case. For example, all the natural numbers, all the real numbers etc. constitute sets, the ordinal numbers which are sets constitute a proper class, and the totality of ordinal numbers as well as that of all classes are proper collections. These relations are described by axiom group (A).