Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by

for the Gaussian case,

for the Poisson case,

as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

Der. Umschwung, welchen die mathematischen Grundlagenforschungen in diesem Jahrhundert erfuhren, teilte früher einmal die Untersuchungen in drei Hauptrichtungen ab, nämlich in die intuitionistische, logistische und formalistische Auffassung der Mathematik, wonach die lebhafte Polemik über das Wesen der Mathematik sowie das Verhältnis der Logik zur Mathematik aufblütete, und zwar am heftigsten in den zwanziger Jahren dieses Jahrhunderts. Nicht in den prinzipiellen Einsichten, sondern in der Methode der Forschungen hatten doch diese drei Standpunkte viele Berührungspunkte. Erstens entlehnt der Hilbertsche Formalismus dem Logizismus das Hilfsmittel zur Formalisierung der Mathematik. Zweitens ragt die Beweisführung in der Metamathematik, welche die formalisierte Mathematik als ihren eigenen Gegenstand der Untersuchungen betrachtet, insofern nicht aus der intuitionistischen Mathematik hervor, als man die formalisierte Mathematik nur als Zeichenkombinationen ansieht. Ferner sind die formalen Regeln, welche in der intuitionistischen Denktätigkeit erfindlich sind, zuerst von Herrn Heyting formalisiert, danach von Herren Gödel, Gentzen u. a, der logistischen Untersuchungen unterworfen worden. Überdies haben diese drei Richtungen der Grundlagenforschungen heutzutage, wozu gemeinsam der erste Anstoss durch die Cantorsche Mengenlehre gegeben worden ist, auch ihre Absicht in Gemeinschaft, welche dar in besteht, die Logik in bezug auf das Unendliche zu verdeutlichen und damit der Mathematik sichere Stellung zu liefern.

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.

In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

The questions concerning the dimension of the tensor product of two K-algebras have turned out to be surprisingly difficult. In this paper we follow a method using spectral sequences (§§1-3) which in some concrete cases yields complete results (§§4-5). In particular, complete results are obtained when r is a ring of matrices, triangular matrices, polynomials or rational functions, so that in the first three cases is respectively the ring of matrices, triangular matrices or polynomials with coefficients in the arbitrary algebra A.

We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n)i be the vector space of l-tuples of n × ra-matrices. We denote by ρ: G → GL(W) a rational representation of G defined as follows:

if S ∈ G, A(i) ∈ M(n) (i = 1, 2, …, l).

It is proved that the C2-smoothly bounded pseudoconvex domains in Pn admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in Pn.

The big problem “Do Painlevé equations define new functions?”, what is called the problem of irreducibilities of Painlevé equations, was essentially solved by H. Umemura [16], [17] and K. Nishioka [9].

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

The test ideal τ(R) of a ring R of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal τ(at) associated to a given ideal a with rational exponent t ≥ 0. We first prove a key lemma of this paper (Lemma 2.1), which gives a characterization of the ideal τ(at). As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal τ(R). Moreover, we prove an analogue of so-called Skoda’s theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the “modified Briançon-Skoda theorem.”

Extending fundamental work of M. Fukushima, M. L. Silverstein, S. Carrillo Menendez, and Y. Le Jan (cf. [F71a, 80], [Si74], [Ca-Me75], [Le77]) it was recently discovered that there is a one-to-one correspondence between (equivalent classes of) all pairs of sectorial right processes and quasi-regular Dirichlet forms (see [AM91], [AM92], [AMR90], [AMR92a], [AMR92b], [MR92]). Based on the potential theory for quasi-regular Dirichlet forms, it was shown that any quasi-regular Dirichlet form on a general state space can be considered as a regular Dirichlet form on a locally compact separable metric space by “local compactification”. There are several ways to implement this local compactification. One relies on h-transformation which was mentioned in [AMR90, Remark 1.4]. A direct way using a modified Ray-Knight compactification was announced on the “5th French-German meeting: Bielefeld Encounters in Mathematics and Physics IX. Dynamics in Complex and Irregular Systems”, Bielefeld, December 16 to 21, 1991, and the “Third European Symposium on Analysis and Probability”, Paris, January 3-10, 1992, and appeared in [MR92, Chap. VI] and [AMR92b] (see also the proof of Theorem 3.7 below). One can also do this by Gelfand-transform. This way was found by the first named author independetly and announced in the “12th Seminar on Stochastic Processes”, Seattle, March 26-28, 1992. It will be discussed in Section 4.

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;

DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,

where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

Class field theory determines in a well-known way the higher dimensional cohomology groups of the idéies and idèle classes in finite Galois extensions of number fields. At the Amsterdam Congress in 1954 I announced [7] the corresponding result for the multiplicative group of the number field itself, but the proof has never been published. Meanwhile, Nakayama showed that results of this type have much broader implications than had been realized. In particular, his theorem allows us to generalize our result from the multiplicative group to the case of an arbitrary torus which is split by the given Galois extension. We also treat the case of “S-units” of the multiplicative group or torus, for a suitably large set of places S. It is a pleasure for me to publish this paper here, in recognition of Nakayama’s important contributions to our knowledge of the cohomological aspects of class field theory; his work both foreshadowed and generalized the theorem under discussion.

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

In this paper, I will prove the following theorems;

For given integers n and m such that m ≧ 2n, there exist 1-dimensional local domains which are complete intersections and have embedding dimension n + 1 and multiplicity m.

Let X be an n-dimensional connected compact complex manifold and A be an analytic subset of X. We say that the pair (X, A) is a complex analytic compactification of Cn if X − A is biholomorphic to Cn. If X admits a Kähler metric, we shall say that (X, A) is a (non-singular) Kähler compactification of Cn. For n = 1, it is easy to see that (X, A) ≃ (P1, ∞). For n = 2, Remmert-Van de Ven [17] proved that (X, A) ≃ (P2, P1) if A is irreducible, where A = P1 is linearly embedded in P2. Morrow [15] gave more detailed classifications of complex analytic compactifications of C2 For n = 3, Brenton-Morrow showed the following

THEOREM ([5]). Let (X, A) be a non-singular Kähler complex analytic compactification of C3such that the analytic subset A has only isolated singular points. Then X is projective algebraic and A is birationally equivalent to a ruled surface over an algebraic curve of genus g = b3(X)/2.

Further, Brenton [3] classified the possible types of singular points of A in the case that the canonical line bundle KA of A is not trivial.

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 .

We classify transcendental classical solutions of the third Painlevé equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painlevé equation.

We show that any bounded hyperconvex domain is Bergman complete.