Let

be a parabolic differential operator defined in Ω = Rn + (0, ∞), where Rn is the n-dimensional Euclidean space, the point x ∈ Rn is represented by its coordinates (x1,...,xn) and aij(=aji), bi and c are functions in (x,t)∈Ω.

Let Rn be the n-dimensional Euclidean space, each point of which is denoted by its coordinate x = (x1,...,xn). The variable t is in the real half line [0, ∞).

It is a privilege to share in the honoring of Professor Katuzi Ono on this occasion, It would be enough to be simply putting in evidence my high regard for a good and dear friend. But there is so very much more than that here. It is an opportunity for me to declare my esteem for the kind of mathematician who looks at his field from the outside as well as the inside; who sees the richness of all experience and seeks its instruction to know better and better the meaning of mathematics ; who by the example of his own life urges young, coming scientists to the more productive path of an uncompartmentalized personality. Katuzi Ono is all of these.

In the pairwise spectral analysis of two projections F and G (see [1, §4] of the References) there are determined two resolutions of the identity, EF-and EG, which are spectral measures on the Borel interval [0, 1], and which are, in fact, the spectral measures of the Hermitian operators SF and SG of (1. 6) below.

Let Ω be the set of symbols 1,2,..., n. Let be a doubly transitive group on Ω of order 6n(n — 1) not containing a regular normal subgroup and let be the stabilizer of the set of symbols 1 and 2.

Soit X un espace séparé localement compact et dénombrable à l’infini. Désignons par l’espace des fonctions numériques réelles continues sur X et à support compact et par l’espace complété de par la norme uniforme, et par et les sous-ensembles de et constitués par des fonctions non-négatives.

We consider in the 2 dimensional space with the coordinate (x,y). Let Γ be a segment of the y-axis containing the origin in its interior and let Ω be a domain whose boundary contains Γ.

L’étude sur les directions de Julia des fonctions méromorphes dans le plan fini est fait présque complètement par Ostrowski [5] et après Rauch [6] etc. l’ont développé.

The aim of this paper is to prove the following main theorem:

By explicit studying on theta-multipliers (i.e. the multipliers which appear in theta-transformation formulas under general modular substitutions), we can naturally get the reciprocity law of Gauss sums or quadratic residue symbols. This remarkable fact, by Cauchy, Kronecker, Hecke and others, is very classical, but its theoretical meaning has not been sufficiently clear yet.

Soit Ω un ouvert de l’espace euclidien Rn à n(≧2) dimensions. Les recherches des fonctions harmoniques dans Ω sont essentielles dans la théorie classique du potentiel. A ce moment-là, il est bien connu que le balayage pour le noyau de Green dans Ω joue un grand rôle (par example, cf. M. Brelot [1]).

In [8], A. Morimoto proved that the automorphism group of a holomorphic principal fiber bundle over a compact complex manifold has a structure of a complex Lie group with the compact-open topology. The purpose of this paper is to get similar results on the automorphism groups of more general types of locally trivial fiber spaces over complex spaces. We study automorphisms of a holomorphic fiber bundle over a complex space which has a complex space Y as the fiber and a (not necessarily complex Lie) group G of holomorphic automorphisms of Y as the structure group (see Definition 3. l).

A systematic treatment of analytical dynamics was given by E. Cartan in [1], where the 1-form plays the fundamental role. We give here a further investigation. One of our main purposes is to clarify relations between dynamical systems and Finsler spaces and the other is to formulate an intrinsic bundle structure of the systems. This paper is closely related to my previous papers [4] [5].

Let be a stochastic process, where T is a finite interval. The ε-entropy is defined as the following quantity:

(1)

In this short note, is shown a necessary and sufficient condition for a logic to be an intermediate propositional logic in Umezawa’s sense (see the reference), under such an assumption that any logic in consideration (as a subclass of LK-provable propositions) contains at least the axioms of the positive propositional logic LPS (Curry’s LA) as its axioms and is closed with respect to the rules of detachment and substitution.

In the present paper, we would like to show a theorem concerning with reducibility of provability in the primitive logic. This theorem seems to suggest a procedure to find the proof-note of a given proposition which is provable in the primitive logic.

D. Mumford has shown an excelent algebralization of theory of theta constants and theta functions in his papers: On the equations defining abelian varieties I, II, III (Invent. Math. 1. 237-354 (1966), 3. 75-135 (1967), 3. 215-244) (1967). Our starting point and idea, however, are something different from those of Mumford; we begin our study at characterizing abelian addition formulae among all the possible addition formulae, and we want to give expressions to everything in words of matric notations.

The purpose of this paper is to get a certain explicit expression of automorphic factors, formulated rather differently than usual, of the classically well known theta function

(1)

Let E be a non-singular projective curve of genus g ≥ 0, P the projective line and let F be the surface E×P. Then it is well known that a ruled surface F* which is birational to F is biregular to a surface which is obtained by successive elementary transformations from F (for the notion of an elementary transformation, see [3]).

Für einen relativ-galoischen Zahlkörper K über k bezeichnen wir mit G(K/k) die zugehörige galoische Gruppe und mit (K: k) dem Erweiterungsgrade. Ferner seien bzw. K* die gröBten, unverzweigten Erweiterungskörper von K, die beziehungsweise die folgenden Eigenschaften besitzen: die galoische Gruppe abelsch; die galoische Gruppe ist im Zentrum von enthalten; K* ist das Kompositum von K und einem über k abelschen Erweiterungskörper.

Let p be a prime and let Ω be the set of p symbols 1,2, ..., p, called points. Let be a transitive permutation group on Ω such that

(I) contains a subgroup of index p which is not the stabilizer of a point.