The purpose of this paper is to investigate the micro-local structure and to calculate, by constructing the holonomy diagrams, the b-functions (See [2]) of irreducible regular prehomogeneous vector spaces (See [1]).

In this paper it is proved that every locally strictly localizable Radon measure of type (ℋ) is strictly localizable, from where it follows immediately the existence of lifting for these measures.

We are concerned with the behaviour of Fourier series in an interval [0, 2K) and, in particular, interested in that of so-called lacunary series. The spectrum of a Taylor series is defined by Spec (F) = , where Z + denotes the semi-group of positive integers.

This paper is concerned with the cohomology at “infinity” (in the sense of Harder [4], [5]) of a congruence subgroup of the symplectic group G = Sp(2l, R). G is the subgroup of GL(2l, R) consisting of matrices g satisfying tgJg = J where

We consider the continuous valuation of logic, where the certainty of a statement is measured with a number in the closed unit interval I =[0, 1].

The idea originates in continuous logics, which have been investigated from various standpoints, in [1] ~ [4] and [8] ~ [10], for example. More comprehensive information can be seen in [4] and others.

Let X be a complex space. Let Dx be the Douady space of compact complex subspaces of X [6] and px : Zx→ Dx the corresponding universal family of subspaces of X.

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0 . But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

This paper is a sequel to Motohashi [4]. In [4], a series of theorems named “elimination theorems of uniqueness conditions” was shown to hold in the classical predicate calculus LK. But, these results have the following two defects : one is that they do not hold in the intuitionistic predicate calculus LJ, and the other is that they give no nice axiomatizations of some sets of sentences concerned. In order to explain these facts more explicitly, let us introduce some necessary notations and definitions. Let L be a first order classical predicate calculus LK or a first order intuitionistic predicate calculus LJ. n-ary formulas in L are formulas F(ā) in L with a sequence ā of distinct free variables of length n such that every free variable in F occurs in ā.

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

Throughout, ε is any small positive number, θ any real number, n, nj, k, N some positive integers and p, pj any primes. By ‖θ‖ we mean the distance from θ to the nearest integer. Write C(ε), C(ε, k) for positive constants which may depend on the quantities indicated inside the parentheses.

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.