Let G be a group and H a subgroup of G. With a left G-module M, relative cohomology groups Hn(G, H, M) of G on M, relative to H, have been defined by Adamson [1] and may be expressed as (Z, M) in the notation of relative homological algebra of Hochschild [2], where Z denotes the G-module of rational integers (acted by G trivially). Regarding M as a right G-module, (M, Z) are similarly relative homology groups Hn(G, H, M). In case H is of finite index in G, Hochschild [2] defines further negative-dimensional relative homology and cohomology groups.

Let k be an algebraic number field of degree n = r1 + 2r2 with r1 real conjugates k(l) (1 ≦ l ≦ r1) and r2 pairs of complex conjugates k(m), k(m+r2)) (r1 + 1 ≦ m ≦ r1 + r2). Let o be the integral domain consisting of all integers in k.

The problem of extending a continuous map is one of the most important problems in algebraic topology. Many topologists have contributed for the solution of this problem. One of the most powerful methods in the extension problem is the obstruction theory defined first by S. Eilenberg [1] and developed by many others. N. E. Steenrod worked on the primary obstruction and showed that there is a strong connection between obstruction theory and cohomology operations [7].

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

1. Introduction. In this paper order will always mean linear or total order, and, unless otherwise stated, the composition of any semigroup will be denoted by +. A semigroup S is an ordered semigroup (notation o.s.) if S is an ordered set and for all a, b, c in S

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

In this note, we denote by Q the rational number field, by EΩ the whole unit group of an arbitrary number field Ω of finite degree, and by rΩ the rank of where generally G* for an arbitrary abelian group G means a maximal torsion-free subgroup of G. (NK/ΩEK)* is shortly denoted by and (G1 : G2) is, as usual, the index of a subgroup G2 in G1.

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are:

• (I) identical representation,

• (II) rationally irreducible representation of degree l – 1,

• (III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

Let Ω be a locally compact separable metric space and let Ф be a positive symmetric kernel. Then the inner and outer capacities of subsets of Ω are defined by means of Ф-potentials of positive measures in the following manner. We define the capacity c(K) of a compact set K in a certain manner by means of Ф-potentials.

1. Introduction. M. A. Lavrentiev made use of a relative distance function to establish some important results concerning the correspondence between the frontiers under a conformal mapping of a simply connected domain onto the unit circle. The purpose of this note is to show that some of these results are valid for the boundary correspondences induced by the more general class of quasi-conformal mappings.

In [1] I investigated some logics intermediate between intuitionistic and classical predicate logics. The purpose of this paper is to show the possibility of applying some intermediate logics to mathematics namely, to show that some mathematical theorems which are provable in the classical logic but not provable in the intuitionistic logic are provable in some intermediate logics. Let LZ be the logical system obtained from LJ′ a variant of Gentzen’s LJ [2], by adding as axioms all those sequents which can be obtained from a sequent scheme Z by substitution for propositional, predicate, or individual variables.

A stochastic process X(A, ω) is called Brownian motion with an N-dimensional parameter when it satisfies the following conditions:

1) For any positive integer n and any set of points A1, A2, …, An in an N-dimensional Euclidian space EN, the joint variable is subject to an n-dimensional Gaussian distribution having the vector 0 as its mean vector.

1. Various strides have been done to characterize the conformal structure of Riemann surfaces by the algebraic structure of some appropriate function algebras on them (cf. Bers [2], Rudin [29], Royden [26], [28], Heins [7], Kakutani [12], Wermer [33] etc.). In this paper we discuss, corresponding to the above, the problem to determine the quasiconformal structure of Riemann surfaces by the algebraic structure of some function algebras on them.

Let w = f(z) be a quasiconformal mapping, in the sense of Pfluger [5]-Ahlfors [1], with maximal dilatation K, which will be simply referred to a K-QC mapping. As is well known, any K-QC mapping w = f(z) of Im z > 0 onto Im w > 0 can be extended to a homeomorphism from Im z ≧ 0 onto Im w ≧ 0 and hence it transforms any set of logarithmic capacity zero on Im z = 0 into a set with the same property on Im w = 0.

Consider circles cν(ν = ±1, ±2, …) with centers ξν on the real axis of the z-plane such that they are disjoint from each other and cluster to infinity z = ∞ from the both sides of the real axis. Here, without loss of generality, we may assume that for every positive integer ν.

In spite of invaluable importance of the fundamental circuit logic of performing the basic logical operators AND, OR, and NOT in achieving complicated computations for electronic computing machines, the research of planning and programming of proving a given mathematical or logical assertion for electronic computing machines is incomparatively far behind. This is of course partly because of little practical demand for solving such a theoretical problem, but there is also the essential reason for it that the proving procedures are much more difficult than the computing ones.

A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G. Soient g et les algèbres de Lie de G et K respectivement. est une sous-algèbre réelle de l’algèbre complexe g. Soit (i2 = –1). Alors est une sous-algèbre complexe de g.