Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

In the late sixties and early seventies the operation of weak normalization was formally introduced first in the case of analytic spaces and later in the abstract scheme setting (cf. [6] & [4]). The notion arose from a classification problem. An unfortunate phenomenon in this area occurs when one tries to parametrize algebraic objects associated with a space by an algebraic variety; the resulting variety is, in general, not uniquely determined and may, for example, depend on the choice of coordinates. Under certain conditions one does know that the normalization of the parameter variety is unique. The price one pays for passing to the normalization is that often this variety no longer parametrizes what it was intended to; one point on the original parameter variety may split into several in the normalization. This problem is avoided if one passes instead to the weak normalization of the original variety. One then obtains a variety homeomorphic to the original variety with universal mapping properties that guarantee uniqueness.

The notion of fields of moduli introduced first by Matsusaka [8] has been developed by Shimura [12] exclusively in the area of polarized abelian varieties. Later Koizumi [7] gave an axiomatic treatment for the notion.

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.

In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

Our aim is to prove

THEOREM. Let L be a positive lattice of E-type such that [L: L̃] < ∞ and L̃ is indecomposable.

• (i) If L ≅ L1 ⊗ L2for positive lattices L1, L2, then L1, L2 are of E-type and [L1: L̃1], [L2:L̃2] < ∞ and L̃1, L̃2 are indecomposable.

• (ii) If L is indecomposable with respect to tensor product, then for each indecomposable positive lattice X we have

• (1) L ⊗ X ≅ L ⊗ Y implies X ≅ Y for a positive lattice Y,

• (2) If X= ⊗t L ⊗ X′ where X′ is not divided by L, then O(L ⊗ X) is generated by O(L), O(X′) and interchanges of L’s, and

• (3) L ⊗ X is indecomposable.

One of the main problems in complex analysis has been to determine when two open sets D1, D2 in Cn are biholomorphically equivalent. In [26] Poincaré studied perturbations of the unit ball B2 in C2 of a particular kind, and found necessary and sufficient conditions on a first order perturbation that the perturbed domain be biholomorphically equivalent to B2. Recently Burns, Shnider and Wells [7] (cf. also Chern-Moser [9]) have studied the deformations of strongly pseudoconvex manifolds. They proved that there is no finite-dimensional deformation theory for M if one keeps track of the boundary.

P. Lévy introduced a generalized notion of Brownian motion in his monograph “Processus stochastiques et mouvement brownien” by taking the time parameter space to be a general metric space. Let (M, d) be a metric space and let O be a fixed point of M called the origin. Following his definition, a Brownian motion parametrized with the metric space (M, d) is a Gaussian system ℬ = {B(m); m ∈ M} such that the difference B(m) − B(m′) is a random variable with mean zero and variance d(m, m′), and that B(O) = 0.

The aim of this paper is to establish the principle of limiting absorption for uniformly propagative systems Λ(x, Dx) = E(x)-1AjDj, Dj = — i∂/∂xj, with perturbations of long-range class, where the perturbation of long-range class, roughly speaking, means that E(x) approaches to E0, E0 being the N × N identity matrix, as | x | → ∞ with order O(| x |-δ), 0 < δ ≦ 1. (The more precise assumptions will be stated below and we require some additional assumptions on the derivatives of E(x).) The spectral and scattering problem for uniformly propagative systems was first formulated by Wilcox [10]. Since then, the principle of limiting absorption has been proved by many authors ([5], [7], [8], [11] etc.). The perturbations discussed in their works belong to the short-range class with δ > 1.

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

Let be the Schwartz space of all rapidly decreasing functions on Rn, be the topological dual space of and for each positive integer p, be the space of all elements of which are continuous in the p-th norm defining the nuclear Fréchet topology of . The main purpose of the present paper is to show that if {Xt, t ∈ [0, + ∞]} is an -valued Gaussian process and for any fixed φ ∈ the real Gaussian process {Xt(φ), t ∈ [0, + ∞)} has a continuous version, then for any fixed T > 0 there is a positive integer p such that {Xt, t ∈ [0, T]} has a version which is continuous in the norm topology of .