Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t є R} with stationary increments which has a self-similar property, that is, there exists a constant H (for the Brownian motion H = 1/2, in general 0 < H < 1 for Gaussian processes) called the exponent of self-similarity of the process, such that, for any c > 0, two processes are subject to the same law (see [10]).

Recently M. G. Crandall and P. L. Lions developed the viscosity theory on nonlinear equations in infinite dimensions and optimal control in Hilbert spaces, in two series of papers, [1], [4].

Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of any minimal basis of I, e = e(I) the multiplicity or degree of R/I, h = h(I) the height or codimension of I, i = indeg (I) the initial degree of J, i.e. the minimal degree of non zero elements of I.

This paper is mainly devoted to find bounds for v(I) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.

It is classical to characterize locally homogeneous Riemann manifolds by infinitesimal conditions. For example, [Si] asserts that the local-homogeneity is equivalent to the existence of linear isometries between tangent spaces which preserve the curvatures and their covariant derivatives up to certain orders. It is also known that the local homogeneity is equivalent to the existence of a certain tensor field of type (1, 2) (for this and a further study, see [TV]).

The present paper is concerned with the stability of diffeomorphisms of C∞ closed manifolds. Let M be a C∞ closed manifold and Diffr(M) be the space of Cr diffeomorphisms of M endowed with the Cr topology (in this paper we deal with only the case r = 0 or 1). Let us define

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

In this paper we discuss the determinism of distributions of some stable random fields which are constructed through integral-geometric method. The determinism depends on the dimension of the parameter space Rd .

The binary quadratic diophantine equation

is of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

The white noise calculus was initiated by T. Hida in 1970 in his Princeton University Mathematical Notes [3]. Recent development of the theory shows that the Laplacian plays an essential role in the analysis in question. Indeed, several kinds of Laplacians should be introduced depending on the choice of the class of white noise functionals to be analysed, as can be seen in [4], [13], [18] and so forth. Among others, we should like to emphasize the importance of the infinite dimensional Laplace-Beltrami operator, Volterra’s Laplacian and Lévy’s Laplacian (See [13], [18] and [20]).

As usual we denote by H∞(K) the Banach algebra of bounded holomorphic functions on a Riemann surface R equipped with the supremum norm ‖·‖ Consider the ideal I(f1 … fm ) of H∞(R) generated by functions f1 …fm in H∞(R). If a function g in H∞(R) belongs to I(f1 … fm or equivalently, if there exist m functions h1 …, hm in H∞(R) with

on R, then common zero points of f 1, ... f m are also zero points of g in the following strong sense:

on R for a positive constant δ > 0. The generalized corona problem asks whether the converse is valid or not. In the case g ≡ 1 on R the problem is referred to simply as the corona problem.