Recently I. Gelfand and M. Neumark [2] have determined the types of irreducible unitary representations of the group G1 of linear transformations of the straight line. The analogous result is obtained for the group G2 of transformations z → az + b in the complex-number plane , where a and b run over all complex numbers with the exception of a = 0, which may be considered as the group of all sense-preserving similar transformations in the two-dimensional euclidean space E2.

Die Struktur diskret bewerteter perfekter Körper ist bisher von H. Hasse, F. K. Schmidt, O. Teichmüller und E, Witt eingehend untersucht worden. Es ist schon von diesen Autoren bewiesen worden, dass der Restklassenkörper eines diskret perfekten Körpers K stets in K ein mit multiplikativ isomorphes Repräsentantensystem besitzt, und sogar, dass es im charakteristikgleichen Fall ein Reprasentantensystem R von gibt, welches einen mit isomorphen Körper bildet. Dabei lässt sich K ais Potenzreihenkörper eines Primelementes aus K mit Koeffizienten aus R darstellen.

Previously W. Krull conjectured that every completely integrally closed primary domain of integrity is a valuation ring, The main purpose of the present paper is to construct in §1 a counter example against this conjecture. In § 2 we show a necessary and sufficient condition that a field is a quotient field of a suitable completely integrally closed primary domain of integrity which is not a valuation ring.

The rotational part of the holonomy group of a Riemann space is called its homogeneous holonomy group. A Riemann space, whose homogeneous holonomy group is one-parametric, was investigated by Liber and an alternative treatment of the same problem was given by S. Sasaki [1]. I will treat here a Riemann space with two-parametric homogeneous holonomy group and prove the following theorem by the method analogous to that of [1]. I thank Prof. T. Ootuki for his kind advice.

The present paper is concerned with the classification and corresponding extension theorem of mappings of the (n+-2)-complex Kn+2 (n>2) into the space Y whose homotopy groups πi(Y) vanish for i < n and i = n+1, and the n-th homotopy group πn(Y) of which has a finite number of generators. Our methods followed here are essentially analogous to those of Steenrod [2].

An element a of a ring R is called regular, if there exists an element x of R such that a×a = a, and a two-sided ideal a in R is said to be regular if each of its elements is regular B. Brown and N. H. McCoy [1] has recently proved that every ring R has a unique maximal regular two-sided ideal M(R), and that M(R) has the following radical-like property: (i) M(R/M(R)) = 0; (ii) if a is a two-sided ideal of R, then M(a) = a ∩ M(R); (iii) M(Rn) = (M(R))n, where Rn denotes a full matrix ring of order n over R. Arens and Kaplansky [2] has defined an element a of R to be strongly regular when there exists an element x of R such that a2x = a. We shall prove in this note that replacing “regularity” by “strong regularity,” we have also a unique maximal strongly regular ideal N(R), and shall investigate some of its properties.

Let G and H be finite groups. If a group G̅ has an invariant subgroup H̅, which is isomorphic with H, such that the factor group G̅/H̅ is isomorphic with G. then we say that G̅ is an extension of H by G. Now let G be the Galois group of a normal extension K over an algebraic number field k of finite degree. The imbedding problem concerns us with the question, under what conditions K can be imbedded in a normal extension L over k such that the Galois group of L over k is isomorphic with G̅ and K corresponds to H̅. Brauer connected this problem with the structure of algebras over k, whose splitting fields are isomorphic with K.

We consider the differential operator

in an n-dimensional (n≧2), orientiable, compact Riemann space R with the metric ds2 = gij(x)dxidxj. Here bij(x) is a contravariant tensor such that the quadratic form and ai(x) changes, by the coordininates transformation x → x̅, as follows:

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).

Let us consider soluble groups whose Sylow subgroups are all abelian. Such groups we call A-groups, following P. Hall. A-groups were investigated thoroughly by P. Hall and D. R. Taunt from the view point of the structure theory. In this note, we want to give some remarks concerning representation theoretical properties of A-groups.

Let be the Cayley algebra of dimension 8 over the field R of real numbers and let be the set of all 3 × 3 Hermitian matrices

Let G be a locally compact topological group and let U be a neighborhood of the identity in G. A curve g(λ) (|λ| ≦ 1) in G, which satisfies the conditions,

is called a one-parameter subgroup of G. If there exists a neighborhood U1 of the identity in G such that for every element x of U1 there exists a unique one-parameter subgroup g(λ) which is contained in U and g(1) =x, we shall call, for the sake of simplicity, that U has the property (S). It is well known that the neighborhoods of the identity in a Lie group have the property (S). More generally it is proved that if G is finite dimensional, locally connected, and is without small subgroups, G has the same property. In this note, these theorems will be generalized to the case when G is unite dimensional and without small subgroups.

About the behavior of brownian motion at time point oo, there are many results by P. Lévy, A. Khintchine etc. P. Lévy cited a theorem by A. Kolmogoroff as the most precise result in his famous book “Processus stochastiques et mouvement brownian” without proof. In this paper we shall prove this theorem, using the similar result about the random sequence by W. Feller, and then, applying the theorem of projective invariance by P. Lévy, we shall find also the behavior of brownian motion at time point 0 from the above theorem.

Let D be an open set in the 2-plane, C its boundary, z0 a point on C, and f(z) a one-valued meromorphic function in D.

1. Let be an abstract Riemann surface in the sense of Weyl-Radó, and an open covering surface over . If a curve C = {P(t);0≦t<1} on tends to the ideal boundary of but its projection terminates at an inner point of as t→1, we shall say that C determines an accessible boundary point (which will be abbreviated by A.B.P.) of relatively to . The set of all the A.B.P.S of relative to will be called accessible boundary (relative to ) and denoted by 3i() or by (, ). Throughout in this paper () will be supposed to be non-empty.

In this note, we give a characterization of inner automorphisms in the set of automorphisms of a compact connected group, and then apply it to give a proof of a theorem due to K. Iwasawa on the group of automorphisms of a compact group.

In a recent paper [3] Tannaka gave an interesting ordering theorem for subfields of a p-adic number field, The purpose of the present note is firstly to observe, on modifying Tannaka’s argument a little, that his restriction to those subfields over which the original field is abelian may be removed and in fact the theorem holds for arbitrary fields which are not p-adic number fields, indeed in a much refined form, and secondly to formulate a similar ordering theorem for algebraic number fields in terms of idèle-class groups in place of element groups.