Let Rn be the n-dimensional Euclidean space, and for each point x =(x1,…, xn) we write .

It is well known that the moduli space of stable rank 2 vector bundles on ℙ2 of the fixed topological type is an irreducible smooth variety ([1], and [8]). There are also many known results on the classification of stable rank 2 vector bundles on ℙ3 with “small” Chern classes.

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

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.

The first purpose of this paper is to exhibit several families of compact manifolds that do not ad nit taut embeddings into any sphere. The second is to enumerate ths possible Z2-cohomology rings of those compact manifolds which do admit a taut embedding and whose cohomology rings satisfy certain degeneracy conditions. The first purpose is easily attained once the second has been accomplished, for it is a simple matter to present families of spaces whose cohomology rings satisfy the required degeneracy conditions, but are not on the list of those admitting a taut embedding.

In the recent years there has been a considerable effort to construct and analyze spaces of test and generalized functionals in infinite dimensional situations, cf. [3, 5, 12, 14] and literature quoted there. In particular Meyer [4, 5] has introduced a certain space of “smooth” functionals on the Wiener space, which was used by Watanabe [14] for an elegant formulation of “Malliavin’s calculus” (i.e. he proved a criterion for the existence and regularity of densities of Wiener functionals). This functional space is countably normed and one of its important properties is its algebraic structure. The proof of this property follows from an equivalence of the norms defining the space with a system of norms of Sobolev type [4, 5] (cf. also (1.5), (1.6)).

Let M be a compact boundaryless C∞-manifold, and let Diff (M) be the space of C1-diffeomorphisms of M endowed with the C1-topology. An Axiom A diffeomorphism is said to satisfy the strong transversality condition if for every x∊M, TxM = TxW3(x) + TxWu(x).

The purpose of this paper is to study the dimension formula for cusp forms of weight one, following the series of Hiramatsu [2] and Hiramatsu-Akiyama [3]. We define as usual the subgroup Γ0(N) of SL2(Z) by

.

Let K\k be a finite Galois extension of finite algebraic number fields with Galois group g. We denote by Gm the multiplicative group defined over the rational number field Q and put

.

Throughout this paper, K will be a field of characteristic zero. Let K ‹x1,…, xm › be the K-algebra in m variables x1…, xm and Im, n the T-ideal consisting of all polynomial identities satisfied by m n by n matrices. The ring R(n, m) = K ‹x1,…, xm ›/Im, n is called the ring of m generic n by n matrices.

Let Γ be a fuchsian group of the first kind and assume that Γ does not contain the element . Let S1(Γ) be the linear space of cusp forms of weight 1 on the group Γ and denote by d1 the dimension of the space S1(Γ). When the group Γ has a compact fundamental domain, we have obtained the following (Hiramatsu [3]):

(*) ,

where ς*(s) denotes the Selberg type zeta function defined by

.

Let k be an algebraic number field with the ring of integers ok = o and let G be a cyclic group of order p, an odd prime.

Let A, B be integral symmetric positive definite matrices of degree m and two, respectively, and suppose that A[X] = B is soluble over Zp for every prime p. If m ≥ 7 and min B = min B[x] (Z2 ∋ x i≠ 0) is sufficiently large, then A[X] = B is soluble over Z. We gave a conditional result for m = 6 in [7] under an assumption on the estimate from above of a kind of generalized Weyl sums. Here we give an unconditional result for special sequences of B.