The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:

in particular ε(0) is the Fock vacuum.

Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD (z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and ∂D.

The flow of Bingham type through a domain Ω in the d-th dimensional space Rd (d ≥ 2) during the time (0, T) is a flow of an incompressible visco-plastic fluid governed by the equations for a velocity vector u = (u1,…,ud) and a stress tensor

Let Ω ⊂ Rm be a bounded domain with smooth boundary, where m ≥ 2. We consider the exponential energy functional

for u: Ω→ Rn, where n ≥ 2.

Let S be a minimal algebraic surface of general type defined over the complex number field C, and let K denote the canonical bundle. According to [10], we call S a canonical surface if the rational map ФK associated with | K | induces a birational map of S onto the image X. We denote by Q (X) the intersection of all hyperquadrics through X.

Let S be a normal surface of degree n in , where (n, k) = (4,3), (6,4) or (8,5). People try to describe all possible combinations of singularities on such surfaces. The case (4,3) is already very complicated. Using properties of K3 surface and elementary transformations of Dynkin Graphs effectively, Urabe [17] was able to solve the problem partially when all singularities are rational double points.

Let A = k[X, Y, Z] and k[U] be polynomial rings over a field k and let l, m and n be positive integers with gcd(l, m, n) = 1. We denote by p the defining ideal of the space monomial curve x = ul, y = um, and z = un.