We shall denote by the Z-module of integral vectors of dimension r, by T a symmetric complex matrix with positive definite imaginary part and by g the variable vector. If we put and the fundamental theta function is expressed in the form: as a series in q and u. Other theta functions in the classical theory are derived from the fundamental theta function by translating the origin and making sums and products, so these theta functions are also expressed in the form: as series of q and u. Moreover the coefficients in the relations of theta functions are also expressed in the form: as series in q.

L’objet de ce mémoire est essentiellement la recherche et l’étude de structures algébriques permettant de traduire le Calcul propositionnel et le Calcul des prédicats restreint du premier ordre. Nous avons guidé et limité cette recherche en imposant aux ensembles algébriques introduits de satisfaire simultanément aux trois conditions:

• — donner une représentation du Calcul Logique sous sa forme la plus classique, c’est à dire avant l’obtention d’un anneau booléien par passage au quotient. Ce sera le rôle des structures “prébooléiennes”.

• — posséder un ensemble d’opérations traduisant un procédé de décision tel que celui des tableaux sémantiques, qui correspondent ici aux “tableaux de réduction”. —avoir un caractère universel dans leur construction à partir des atomes.

Für jede Riemannsche Fläche R bezeichnen wir mit M(R) die Algebra von Royden auf R und mit R* (bzw. ΔR)- die Kompaktifizierung (bzw. den idealen Rand) von Royden von R ([5]). Für jede Funktion ξ ∊ M(R) sei Jξ das kompakte Segment und X(R) der kompakte Raum . Laut der Definition von R* ist jede Funktion ξ∊M(R) auf R* (in eindeutiger Weise) stetig fortsetzbar. Bezeichnet man ebenfalls mit ξ die so fortgesetzte Funktion und mit φR die Abbildung von R*→X(R), so ist φR ein Homeomorphismus von R* auf φR(R*).

We let the notations be as in [3]. Then, in the category 6 of all bordered Riemann surfaces, the following inclusion diagram holds [3, Theorem 9]:

Let f(z) be a holomorphic function defined in the unit disk |z|<1, which we shall denote by D. Let Σ be a subset of D, whose closure has at least one point in common with C, the circumference of the unit disk. The set of all values a such that the equation f(z) = a has infinitely many solutions in Σ is called the range of f(z) in Σ, and is denoted by R(f, Σ). Let τ be a point of C, and let {zn) be a sequence of points in D with the properties: . The non-Euclidean (hyperbolic) distance ρ(zn, zn+1) between two points zn and zn+1 of the sequence is defined to be equal to

(cf.[3], Ch. II).

The theory of mathematical objects, developed in this work, is a trial system intended to be a prototype of set theory. It concerns, with respect to the only one primitive notion “proto-membership”, with a field of mathematical objects which we shall hereafter simply call objects, it is a very simple system, because it assumes only one axiom scheme which is formally similar to the aussonderung axiom of set theory. We shall show that in our object theory we can construct a theory of sets which is stronger than the Zermelo set-theory [1] without the axiom of choice.

As defined by Arens and Kaplansky [2] a ring A is strongly regular (s.r.) in case to each a∊ A there corresponds x = xa∊A depending on a such that a2x = a. In the present article a ring A is defined to be a s.r. extension of a subring B in case each a>∊A satisfies a2x-a∊B with x = xa∊A. S.r. rings are, then, s.r. extensions of each subring. A ring A which is a s.r. extension of the center has been called a ξ-ring (see Utumi [13], Drazin [3], Martindale [11], and their bibliographies).

On the basis of Prof. R. Brauer’s fundamental work, certain orthogonality relations for characters of finite groups have recently been studied by Brauer himself, M. Osima, and one of the present writers; see Iizuka [7] and the references there. In the present short note some general remarks on orthogonality relations, dealing with “blocks” and “sections” of general type, are given first. They are of elementary, and often formal, nature and their proofs are merely combinations of known arguments. So, no deep significance is claimed on them, in comparison with the above alluded results based on deeper arithmetico-group-theoretical considerations. However, applied to blocks and sections of such deeper nature, our remarks give some rather useful informations on them. Thus, for instance, the “maximality” feature of 77-blocks is given a formulation (Prop. 5 below) finer than the one given in [71 Further, some new types of blocks and sections can be constructed, again in application of our remarks to such classical ones. These new blocks and sections give thus new orthogonality relations and we hope that some of them may turn to have some significance. There arize also several problems, which are stated at the end of the present note and to some of which we wish to come back elsewhere.

The motivation for the results in this note comes from a theorem of Macaulay. Let f1, …, fn be elements of a polynomial ring R over a field, and let I be the ideal they generate. Assume IR and rank (I) = n. Then the theorem of Lasker and Macaulay asserts that I is unmixed (all prime ideals belonging to I have rank n). Macaulay [1, p. 51] proved further that any power of I is unmixed.

Recently, W. Feit [2] obtained some results on factorizable groups of odd order. By using his procedure and applying the theory of R. Brauer [1], we can prove the following theorem similar to that of W. Feit [2]: