We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion

In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.

Let Rd be the d-dimensional Euclidean space where each point is expressed by a column vector. Let | x | and ‹x, y› denote the norm and the inner product in Rd. Let Q = (Qjk) be a real d × d-matrix of which all eigenvalues have positive real parts. Let X be a process of Ornstein-Uhlenbeck type (OU type process) on Rd associated with a Levy process {Z: t ≥ 0} and the matrix Q. Main purpose of this paper is to give a recurrence-transience criterion for the process X when Q is a Jordan cell matrix and to compare it with the case when Q is diagonalizable. Here by a Levy process we mean a stochastically continuous process with stationary independent increments, starting at 0.

The purpose of this article is to prove the following theorem:

Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.

This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

Let U be a smooth quasi-projective variety over C and let f be a regular function on U. Let DU be the sheaf of algebraic differential operators on U and let M be a regular holonomic DU-module: here, regular means that there exists some smooth compactification X of U and some extension of M as a DX-module which is regular holonomic on X (one also may avoid the use of a smooth compactification to define regularity, see [17]).

Let k be a positive integer greater than 1, and let X(n) be a real primitive character modulo k, The series

can be divided into groups of k consecutive terms.

It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.