Let K be a commutative ring with identity element and let Λi, i = 1, 2 be two K-projective associative algebras with identity element such that the map k → k · l of K into Λi is a monomorphism of K onto a K-direct summand of Λi , i = 1, 2. Let Λ = Λ1 ⊗ Λ2 and A be a two sided (Λ1, Λ2 )-bimodule.

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

Consider a linear transformation T whose domain D(T) and range R(T) lie in a complex Banach space X. We denote by [X] the class of all linear transformations T for which D(T) = X and T is continuous (bounded). Also, we denote the resolvent set of T by ρ(T) and the spectrum of T by σ(T).

Let D denote the open unit disk in the complex plane and let C be the boundary of D. If, for a given complex-valued function f(z) defined in D, the existence of a subset M of C is known, with the linear measure of M equal to 2 π, as well as an estimate on the growth of |f/(z)| I on sequences in D which tends to a point of M, then such a result will be called a “statistical” result on order. This terminology is due to Lelong-Ferrand [3].

Let Γ be a ring with unit element and let Λ be the Ore extension of Γ with respect to a derivation d of Γ [4, 3]. It is shown in [3] that l.gl. dim Λ ≦ 1 + l.gl. dim Γ. It is not in general possible to replace this inequality by equality.

In der vorliegenden Arbeit wird untersucht, was für eine Struktur eine Untergruppe von endlicher p-Gruppe habe, wenn mit jeder Untergruppe von als ganzes vertauschbar ist. In diesen tritt natürlich ein nicht nur aus E bestehender Normalteiler von auf und er enthält ein Zentrumselement von der Ordnung p. Wir fragen: Gilt das Ähnliche auch, falls kein Normalteiler ist? Die gestellte Frage läßt sich für regülare beantworten, deren Klasse 2 ist, aber für andere bleibt sie noch offen. Bevor wir diese Frage untersuchen, stellen wir einige Definitionen und nötige Sätze und Hilfssätze zusammen.

L’objet du présent travail est d’étudier, dans un espace de Green Ω, une généralisation des lignes de Green, à savoir les trajectoires orthogonales des surfaces de niveau de où G désigne la fonction de Green de pôle fixé y0 ∈ Ω, et h une fonction harmonique > 0 fixée dans Ω. Le cas h = 1, qui est celui des lignes de Green ordinaires, a été étudié en détail par M. Brelot et G. Choquet dans [4]. On montrera ici, moyennant l’introduction de notions convenablement appropriées, que les propriétés de ce cas initial s’étendent.

Let G be an algebraic linear group over a field F of characteristic 0, and let H be an algebraic subgroup of G. Let A, M be rational G-modules. In [4], we defined (A, M), and, in particular, relative cohomology groups Hn(G, H, M) were defined as (F, M).

The Grothendieck ring of a finite group has been studied by Swan ([5], [6]). At the end of [6] he determined completely the structure of the Grothendieck ring G(Z) of a cyclic p-group over the ring of rational integers Z.

Let x = (xl,…xn) be a point in the n-dimensional Euclidean space and let be the unit sphere In the (n + 1)-dimensional Euclidean space with coordinate (x, t), we put

and

where denotes the boundary of . We also use the following notation:

Let f be a nonconstant function meromorphic in the unit disc , with circumference C, and let Ez be a subset of C with positive (linear) measure. Suppose that at each ζ ∈ Ez f has an angular limit aζ and let It is known that Ew contains a closed set with positive harmonic measure (see Priwalow [6, p. 210] or Tsuji [7, p. 339]).

In our paper Cl] we showed that there exist Schottky groups whose singular sets have positive 1-dimensional measure. Since the example was very complicated, it is natural to seek for simpler examples. Further the problem how about the singular sets of more general groups occurs.

In this note we shall extend, for arbitrary n, Pesin’s [11] bidimensional definition for quasiconformal mappings and establish its equivalence with Gehring’s [7] and Väisälä’s [15] definitions.

The intuitionistic logic LJ and Curry’s LD (cf. [1], [2]) are logics stronger than Johansson’s minimal logic LM (cf. [3]) by the axiom schemes ⋏→x and y ∨ (y→⋏), respectively. However, LM can not be taken literally as the intersection of these two logics LJ and LD, which is stronger than LM by the axiom scheme (⋏ → x) VyV (y→⋏). In pointing out this situation, Prof. K. Ono suggested me to investigate the general feature of the intersection of any pair of logics. In this paper, I will show that the same situation occurs in general. I wish to express my thanks to Prof. K. Ono for his kind guidance.

Récemment J. L. Doob [8] a trouvé quelques théorèmes intéressants sur l’allure des fonctions méromorphes au voisinage d’un point singulier essentiel isolé en utilisant la topologie fine c’est-à-dire la topologie la moins fine qui fait toutes les fonctions sous-harmoniques continues. A l’égard de ces résultats, on étudie dans ce mémoire quelques aspects de fonctions méromorphes au voisinage d’un point singulier essentiel isolé ou d’un point-frontière irrégulier.

We consider in this paper eleven conditions on a commutative ring R. The first of these is that R contains an identity. It is well known that each of the other properties is a consequence of the first condition. This paper considers other relations which exist between these properties. A complete diagram of all simple implications which exist between the eleven properties, together with proof of these implications, is given in section 3. Examples illustrating simple implications which do not hold are presented in section 4.

The universal character of the primitive logic LO in the sense that popular logics such as the lower classical predicate logic LK, the intuitionistic predicate logic LJ, Johansson’s minimal predicate logic LM, etc. can be faithfully interpreted in LO is very remarkable even from the view point of mechanical proof-checking. Since LO is very simple, deductions in LO could be mechanized in a simple form if a suitable formalism for LO is found out. Main purpose of this paper is to introduce a practical formalism for LO, practical in the sense that it is suitable at least for mechanical proof-checking business.

M. Riesz [8] introduced the notion of α-superharmonic functions in n(≥1)-dimensional Euclidean space Rn in connection with the potential of order α. In this paper, we shall first define the α-superharmonic and α-harmonic functions in a domain D. In case α = 2, they coincide with ones in the usual sense.