E. Cartan has proved that the prolonged system of a Pfaffian system in involution is also in involution. But he has treated only the case of a Pfaffian system of some special type. On the other hand in his book [3] he has reduced the solution of any differential system to the solution of a Pfaffian system of the type mentioned above, and this reduction is precisely the method of the prolongation of a differential system. Therefore it seems to be disirable to establish the above mentioned theorem in the case of an arbitrary differential system.

Let G be a finite group. Let n1, n2,…, nr, where n1>n2>… > nr = 1, be all the numbers each of which is the index of the centralizer of some element of G in G. We call the vector (n1, n2,…, nr) the conjugate type vector of G. A group with the conjugate type vector (n1, n2,…, nr) is said to be a group of type (n1, n2,…, nr).

The Poisson process , as is well-known, is a temporally and spatially homogeneous Markoff process satisfying

(1)

(2)

where k is a non-negative integer and λ is a positive constant. In this note we consider the random variable Lm(ω) which denotes the length of t-interval such that X(t, w) = m (m = 0, 1, 2,…) and some of other properties concerning them.

A differential form co on a complete variety Un is said to be of the first kind if it is finite at every simple point of any variety which is birationally equivalent to U. Let k be a common field of definition for Uand ω, and let Pbe a generic point of Uover k. If ω is of the first kind, then ω(P) is of course a differential form of the first kind belonging to the extension k(P) of k.

The aim of this article is to investigate algebraic nature of systems of one-valued mappings of given group into given field and to apply it to the theory of Galois algebras and duality of compact T0-groups. The results obtained in the following are those; factor systems of Galois algebras with finite Galois groups are defined without any restrictions on the orders of Galois groups and the coefficient fields, a necessary and sufficient condition for them to be associated with Galois fields is obtained, dualities of finite groups are obtained very simply without any restrictions for coefficient field of representations, and Tannaka’s duality of compact T0-groups is proved without the use of the compactness of Tannaka representation groups of representations of compact T- groups and the use of Kampen’s theorem.

Previously C. Chevalley [1] proved the followings:

1. Let x1,…, Xn be algebraically independent elements over a field t which has infinitely many elements. Then :

a) If y is an element of [x1,…, xn] and if y is not in , then there exist elements y2…, yn of [x1,…, xn] such that [x1,…, xn] is integral over [y1, y2,…, yn]

Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :

I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.

II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.

Cohomology groups of (associative) algebras have been introduced (for higher dimensions) and studied by G. Hochschild in his papers [2], [3] and [4]. 1-, 2-, and 3-dimensional cohomology groups are in closest connection with some classical properties of algebras. In particular, an algebra is absolutely segregated. if and only if its 2-dimensional cohomology groups are all trivial. It is thus of use and importance to determine the structure of algebras with universally vanishing 2-cohomology groups, i.e. absolutely segregated algebras; they form a class which is wider than the class of all algebras with universally vanishing 1-cohomology groups, i.e. separable algebras in the sense of the Dickson-Wed-derburn theorem.

As L. Sario [8] and the others have shown, an important method to investigate the properties of an open Rlemann surface F is the use of an exhaustion of F consisting of compact domains on F. But, for the same purpose, the investigation of properties of a non-compact region on F is also important.

The cohomology theory of associative algebras has been developed by G. liochschild [1], [2], [3], and the 1-, 2-, and 3-dimensional cohomology groups have been interpreted with reference to classical notions of structure in his papers. Recently M. Ikeda has obtained, by a detailed analysis of Hochschild’s modules, an interesting structural characterization of the class of algebras whose 2-dimensional cohomology groups are all zero [5].

Eine wichtige Aufgabe der Darstellungstheorie ist die Aufstellung eines vollstandigen Systems von inäquivalenten irreduziblen Darstellungen einer endlichen Gruppe Obwohl man Verfahren kennt, die dies theoretisch leisten (etwa die Reduktion der regularen Darstellung), ist die praktische Durchführung nur in wenigen Fallen möglich. Die Aufgabe vereinfacht sich jedoch wesentlich, wenn sich alle irreduziblen Darstellungen von auf monomiale Gestalt transfor-mieren lassen; denti alle monomialen Darstellungen einer endlichen Gruppe kann man nach einem sehr einfachen Verfahren gewinnen (siehe [4], S. 140).

About the behaviour of brownian motion at time point ∞ there are many results by P. Levy and A. Khintchine etc. The method of W. Feller is applicable to a similar discussion about a homogeneous differential process. In this paper we shall study, applying his method, the properties of a homogeneous differential process.

We prove in this paper some integral formulas about volume and surface area which are the extensions of the classical formulas such as Guldin-Pappus’s theorem about the solid of rotation and the surface of rotation and Holditch’s theorem about the area of the domains bounded by the loci of three points on a segment that moves on the euclidean plane. The formulas we prove are so elementary that they may be found in some literature, but the proofs here given are very simple by the use of moving frames and I assume that they are of some interest.

Mit Hilfe der Dedekindschen ζ-Funktion erhält man bekanntlich Klassenzahlrelationen gewisser algebraischen Zahlkörper. Ais ein Spezialfall gilt der

Satz l.1) Es sei K ein bizyklischer biquadratischer Zahlkörper und seien k0, k1und k2die drei quadratischen Teilkörper von K. Ferner seien h, h0, h1und h2die Klassenzahlen von K, k0, k1bzw. k3. Dann besteht die Relation

Hier ist Q der Index der Grupps der Einheiten , welche von der Einheiten von ki (i = 0, 1, 2) erzeugt wird, in der vollen Gruppe der Einheiten E von K: d.h. .

The purpose of the present paper is to give a new proof to the triad excision theorem of Blakers and Massey [1], in case m≦2 and n≦2, by the aid of path spaces and in connection with a recent work of J. P. Serre [2].

In the former paper, I have developped a theory of meromorphic functions in a neighbourhood of a bounded closed set E of logarithmic capacity zero, by means of Evans’ potential fnnction u(z), which tends to ∞, when z tends to any point of E. It is not known, whether such a potential function exists on an open Riemann surface with null boundary, but by a substitute of Evans’ function. we shall develop the similar theory of meromorphic functions on an open Riemann surface with null boundary.

In a Bourbaki seminary note, La Théorie des Fonctions Thêta, A. Weil has discussed two fundamental theorems of the general theory of Theta functions. The first, due to H. Poincaré, was proved very skilfully in the note by means of harmonic integrals on a torus and the second, due to Frobenius, was treated by the systematic use of the notion of analytic structure.

Recently the Hahn-Banach theorem for normed spaces over non-archimedean valued fields was treated by A. F. Monna [1], L S. Cohen [2], A. W. Ingleton [3], and the writer [4]. In [3] and [4] very essential use was made of an idea of L. Nachbin [5]

Recently M. Ikeda [1] succeeded in determining the structure of absolutely segregated algebras, i.e. algebras whose 2-cohomology groups all vanish. His beautiful result reads : an algebra A, of finite rank over its ground field, is absolutely segregated if and only if i) the residue-algebra A/N modulo the radical N is separable and, moreover, ii) the A-left-module N is an (Mo)-module. A. simplification was given by H. Nagao [5], who obtained, besides an interesting-result on algebras with vanishing 3-(or higher) cohomology groups, an elegant short proof to the fact that under the assumption of i), the property ii) is necessary, and sufficient, for the absolute segregation of A.

Recently Mr. Seizô Itô kindly called the author’s attention to the fact that the derivation of (4.15) in the above referred paper is insufficient since the paper does not contain the proof of the compactness condition:

(1)

uniformaly in a and b