Before stating the purpose, we explain some propositional logics treated in this paper. The logical symbols we use are: implication →, conjunction ∧, disjunction ∨, and the propositional constant ⋏ denoting contradiction.

The idea of multivariate wide-sense Markov processes has been recently used by F.J. Beutler [1], In his paper, he shows that the solution of a linear vector stochastic differential equation in a wide-sense Markov process. We obtain here a characterization of such processes and as its consequence obtain the conditions under which it satisfies Beutler’s equation. Furthermore, in stationary Gaussian case we show that these are precisely stationary Gaussian Markov processes studied by J. Doob [5].

In the previous paper [6] we have discussed Markov processes in Rd with the Green function G (x, y) satisfying are positive constants), and showed that the regular points of its process are the same as those of α-stable process. The present article is closely related to the previous one.

The main objective of this paper is to prove the following theorem, which generalizes some results of [1], [2], [6]. Our theorem is also suggested by the work of Novikov [5].

The Riemann’s theta functions associated with a closed Riemann surface are absolutely convergent. In the present paper, we shall show an example of an hyperelliptic Riemann surface of infinite genus such that the Riemann’s theta functions associated with are absolutely convergent.

After one introduces the theory of normal families in a course in complex analysis, the usual pattern is to give an example of a non-normal family. One of the simplest, of course, is the sequence fn(z) = nz, n = 1,2, ···. The very devastating effect of multiplying by zero insures the required abnormality!

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

In [5] we defined the notion of separable extensions of a ring as a generalization of that of separable algebras. In this paper we try to introduce the notion of ‘central’ separable algebras to separable extensions. Such extensions are viewed in §2. But we do not know that it is suitable or not to call them central separable extensions. As the Morita theorem for projective generators is fundamental to examine the central separable algebras we generalize it in §1. Let R be a ring and let A and B be left R-modules. We assume that R-module B is isomorphic to a direct summand of a finite direct sum of copies isomorphic to A.

A doubly transitive permutation group on the set of symbols Ω is called a Zassenhaus group if satisfies the following condition: the identity is the only element leaving three distinct symbols fixed.

Let K be an extension of a field k and p denotes the characteristic. In [4] and [5], we proved that if K is inseparable algebraic over k, then considered as an algebra over k, K is not rigid.

Let K be the universal domain. Let G be a finite additive group of odd order |G| and Xa(a ∈ G) be indeterminates indexed by the elements in G. We mean by PG the projective space Projk(K[(Xa)a ∈ G]).

p. 187, line 20: Delete “—s” from the formula.

p. 228, line 4: Delete “—1” from the formula.

p. 228, lines 18 and 21: Delete “—s” from the formulas.

Let Q be the rational number field, and let K = (D > 0 a rational integer) be a real quadratic field. Then, throughout this paper, we shall understand by the fundamental unit εD of the normalized fundamental unit εD > 1.

We denote by D the unit disc {z; |z| < 1} and by the totality of one to one conformal mappings z′ = s(z) of D onto itself. A meromorphic function f(z) in D is normal if and only if the family is a normal family in D in the sense of Montel. We denote by the totality of the normal meromorphic functions in D.

Let M be a finitely generated module over a regular local ring R. It is well known that the sum of homological dimension and codimension of M is equal to the global dimension of R. For modules over an arbitrary ring, this is in general not true. The purpose of this paper is to investigate the properties of such sums in the semi-local case.

In the previous paper [6] we have shown the examples of hyperelliptic Riemann surfaces of infinite genus such that the Riemann’s theta functions associated with them are absolutely convergent.