In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

The infinite dimensional Laplacian operator can be discussed in connection with the infinite dimensional rotation group ([1]). Our interest centers entirely on observing how each one-parameter subgroup of the infinite dimensional rotation group contributes to the determination of the Laplacian operator.

Let E be an infinite dimensional real nuclear space and H be its completion by a continuous Hilbertian norm ║ ║ of E. Then we have the relation

where E* is the conjugate space of E. Consider a function C(ξ) on E defined by the formula

(1)

In this note we consider a finite group G which satisfies the following conditions:

• (0. 1) G is a doubly transitive permutation group on a set Ω of m + 1 letters, where m is an odd integer ≥ 3,

• (0. 2) if H is a subgroup of G and contains all the elements of G which fix two different letters α, β, then H contains unique permutation h0 ≠ 1 which fixes at least three letters,

• (0. 3) every involution of G fixes at least three letters,

• (0. 4) G is not isomorphic to one of the groups of Ree type.

Let H be a separable Hilbert space with inner product (,) and norm ║ ║. We denote by K the set of all linear operators on H. Let be a probability space and suppose we are given a family of σ-fields t≥O such that for O ≤ s ≤ t and .

In this note, we associate a foliated singular homology theory to a foliation on a compact manifold X and construct a spectral sequence which relates the foliated homology to the ordinary homology of X. Since the foliated homology is so constructed as to be related closely to a certain topological behavior of the foliation, we may expect that further study of the spectral sequence reveals some information on topology of foliation. The study and applications will be given in a sequel.

In [4], K. Ono introduced the notion of evaluations of the primitive logic LO and proved that any semi-evaluation E is an evaluation of LO if E satisfies the following conditions:

• (E1) p* → 0 = 0,

• (E2) p* → p* = 0,

• (E3) 0 → p* = p*,

• (E4) p* → (p* → q*) = p* → q*,

• (E5) p* → (q* → r*) = q* → (p* → r*),

• (E6) p* → q* = 0 implies (r* → p*) → (r* → q*) = 0,

• (E7) (x)p*(x) → p*(t) = 0 for any t, and

• (E8) if u* → p*(t) = 0 for any t, then u* → (x)p*(x) = 0.

En continuant l’étude d’un problème soulevé dans [3] et plus tard abordé par un de nous dans [1] et [2], nous avons publié deux “abstracts” (voir [6] et [7]) sur certains calculs propositionnels appellés Jn, 1 ≤ n ≤ 5, et leurs relations avec le postulat de la séparation. On développe maintenant les systèmes Jn, 1 ≤ n ≤ 5, et on esquisse la construction des calculs correspondants de prédicats de premier ordre. Enfin, on démontre la non trivialité des théories correspondantes des ensembles quand on utilise des postulats comme ceux de Zermelo-Fraenkel, mais avec le schéma de la séparation formulé sans les restrictions ad hoc pour éviter les paradoxes.

Sario’s theory of principal functions fully discussed in his research monograph [3] with Rodin stems from the principal function problem which is to find a harmonic function p on an open Riemann surface R imitating the ideal boundary behavior of the given harmonic function s in a neighborhood A of the ideal boundary δ of R.

In the author’s previous paper [3], we used Theorem 1 of the present paper to assure the existence of a signed branching Markov process with age satisfying given conditions in [3], The purpose of this paper is to give a proof of Theorem 1.

We shall discuss the sample path continuity of a stationary process assuming that the spectral distribution function F(λ) is given. Many kinds of sufficient conditions have been given in terms of the covariance function or the asymptotic behavior of the spectral distribution function.

Let L be a fixed language and K be a set of structures related to L. A sentence in L is said to be K-valid if and only if for every structure in K,

Some basis results for arithmetic, hyperarithmetic (HA) or sets which have positive measure (or which are not meager, i.e., of the second Baire category) have been obtained by several authors. For example, every non-meager set must have a recursive element (Shoenfield-Hinman, Hinman [2]) but there exists a non-meager set (as well as of measure 1) that contains no recursive element (Shoenfield [7]), and every set (i.e., arithmetic set) of positive measure contains an arithmetic element (Sacks [5], and Tanaka [12]).

The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas:

• (1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.

• (2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.

• (3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.

In the previous paper [4] we have studied the prolongations of G-structures to tangent bundles. The purpose of the present paper is to generalize the previous prolongations and to look at them from a wide view as a special case by considering the tangent bundles of higher order. In fact, in some places, the arguments and calculations in [4] are more or less simplified. Since the usual tangent bundle T(M) of a manifold M considers only the first derivatives or first contact elements of M, the previous paper contains, in most parts, only the calculation of derivatives of first order.

Let X be a Hilbert space. The topological support of a Radon probability measure P on X is the least closed subset M of X that carries the total measure 1.

In constructing various kind of mathematical theories on the basis of a common basic theory, it has been very usual to take up the set theory as the common basic theory. This approach has been already successful to a certain extent and looks like successfully developable in the future not only in constructing mathematical theories standing on the classical logic but also in constructing formal theories standing on weaker logics. In constructing mathematical theories standing on the classical logic, it has been successful in most cases only by interpreting mathematical notions in the set theory without defining any special interpretation of logical notions. In constructing any mathematical theory standing on weaker logics such as the intuitionistic logic, however, we have to give a special interpretation for logical notions, too.