Seit Herr Artin seine allgemeinen L-Funktionen, die mit Frobeniusschen Gruppencharakteren gebildet sind, entdeckt hat [1], sind die multiplikativen Relationen Dedekindscher ζ-Funktionen als additive Relationen zwischen den Frobeniusschen Gruppencharakteren mit Erfolg untersucht worden. Durch diese Methode sind gewisse Klassenzahlrelationen in den folgenden Zeilen zu betrachten.

Recently H. Hasse has given an interesting theory of Galois algebras, which generalizes the well known theory of Kummer fields; an algebra over a field Ω is called a Galois algebra with Galois group G when possesses G as a group of automorphisms and is (G, Ω)-operator-isomorphic to the group ring G(Ω) of G over Ω. On assuming that the characteristic of Ω does not divide the order of G and that absolutely irreducible representations of G lie in Ω, Hasse constructs certain Ω-basis of , called factor basis, in accord with Wedderburn decomposition of the group ring and shows that a characterization of is given by a certain matrix factor system which defines the multiplication between different parts of the factor basis belonging to different characters of G. Now the present work is to free the theory from the restriction on the characteristic. We can indeed embrace the case of non-semisimple modular group ring G(Ω).

On the euclidean plane one-parametric motion is in general a roulett motion, exceptions being a translation at each instant and a rotation with a fixed center; here we mean by a roulett motion a motion in which a certain curve rolls on another fixed curve without slipping. In this paper we extend this fact to the case of Klein spaces and investigate in detail especially the cases of the euclidean space and the projective space.

Let be the family of analytic functions F(z) which are regular and schlicht in the circle E: |z| < 1 and normalized at the origin such that F(0) = 0 and F′(0) = 1. Let be the subfamily of consisting of all functions, each of which possesses, as boundary of its image-domain., a closed Jordan curve which is supposed to be regular analytic. Then is everywhere dense in the original family . Hence, by an approximation theorem of Carathéodory on the kernel of sequence of domains, we may restrict ourselves within , so far as we concern the estimation of continuous functionals on .

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

Let q (x) be real and continuous in the infinite open interval (- ∞, ∞) and let y 1(x, λ), y 2(x, λ) be the solutions of

(1.1)

with the initial conditions

(1.2) .

This note is concerned with homotopy relations in fibre bundles over spheres, in connection with my previous note.

Let R be a fibre boundle, with the n-dimensional sphere Sn as the base space and with fibres which are connected and simple in n − 1 dimension. Let further π denote the projection of R onto Sn . We take a simplicial decomposition K of Sn which is so fine that the star of each simplex lies in a coordinate neighborhood of Sn .

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

A formulation of the differentiability of the product function of a locally compact group at the identity may be given in the following way: In a sufficiently small euclidean neighborhood of the identity product satisfies the condition:

Let D be an arbitrary connected domain and w = f(z) = u(x,y) + iv(x,y), z = x + iy, be an interior transformation in the sense of Stoïlow in D. Denote by γ a set, in D, such that D and the derived set γ′ of γ have no point in common.

The present paper is a continuation of Part I. In the introduction of Part I we have explained our main problems and sketched their results. In the present part we shall proceed to give complete proofs of them.

Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.

In connection with the class field theory a problem concerning p-groups was proposed by W. Magnus: Is there any infinite tower of p-groups G 1, G 2,…, Gn, Gn+1 ,…such that G1 is abelian and each Gn is isomorphic to Gn+1/θn(Gn+1), θn (G n+1) ≠ 1, n = 1,2,…, where θn (G n+1) denotes the n-th commutator subgroup of G n+1? The present note is, firstly, to construct indeed such a tower, to settle the problem, and also to refine an inequality for p-groups of P. Hall.

It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.