Theory of real and time continuous martingales has been developed recently by P. Meyer [8, 9]. Let be a square integrable martingale on a probability space P. He showed that there exists an increasing process ‹X›t such that

Let G be a connected linear algebraic group, and p a rational representation of G on a finite-dimensional vector space V, all defined over the complex number field C.

We call such a triplet (G, p, V) a prehomogeneous vector space if V has a Zariski-dense G-orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces when p is irreducible, and to investigate their relative invariants and the regularity.

As is well known, for a real vector space V, the Fourier transformation

gives an isometry between L2(V) and L2(Vv), where Vv is the dual vector space of V and < , >: V×Vv → R is the canonical pairing.

In this note we determine the Hausdorff dimension of a family of planar sets which are generalizations of the classical Cantor set.

In his previous paper [1] the author has stated a formula concering stochastic differentials with the outline of the proof. The aim of this paper is to show this formula in details in a little more general form (Theorem 6).

Let G = G(q) be a Chevalley group defined over a field Fq of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut(G) and the centralizers of these involutions. This study was begun in the context of a different problem.

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

In [10], G.Shimura gave a method of constructing holomorphic cusp forms of even integral weight from given forms of half integral weight. In this paper, we try to present an inverse construction. To state our main result, some notational preliminaries are necessary. We denote by the complex upper half plane.

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

Let Λ be a ring with unit. If A is a left Λ-module, the dimension of A (notation: 1.dimΛA) is defined to be the least integer n for which there exists an exact sequence

0 → Xn → … → X0 → A → 0

where the left Λ-modules X0, …, Xn are projective. If no such sequence exists for any n, then 1. dimAA = ∞. The left global dimension of Λ is

1. gl. dim Λ = sup 1. dimAA

where A ranges over all left Λ-modules, The condition 1. dimA A < n is equivalent with (A, C) = 0 for all left Λ-modules C. The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ-modules.

Soit V une variété complexe et soit I le champ de tenseurs de type (1,1) de V définissant la structure complexe de V. Un champ de vecteurs ξ sur V sera dit conforme si I [ξ, η]= [ξ, Iη] pour tout champ de vecteurs η sur V. On désignera par α l’ensemble de tous les champs de vecteurs comformes sur V. α est une sous-algèbre de Lie de l’algèbre de Lie de tous les champs de vecteurs sur V. Si V est compacte, α est de dimension finie et s’identifie avec l’algèbre de Lie du groupe de Lie A(V) d’homéomorphismes analytiques de V [2] De plus, si ξ, η ∈ α, on a On peut donc définir une structure d’algèbre de Lie complexe de α en posant pour tout ξ ∈ α.

Let A be a primary algebra with unit element over a field K and Z its center. Let Ā be the simple residue class algebra of A modulo its radical. Then it is known, and can readily be seen, that there holds the inequality where t is the rank of A over its center. We call A maximally central if in particular i.e. if the rank [Z: K] takes its maximum value. Further, an algebra which is a direct sum of those primary algebras will be called maximally central too. The notion was introduced in Azumaya-Nakayama [5], as a by-product of the study of absolutely uni-serial algebras.

Let G be a finite group. Let n1, n2,…, nr, where n1>n2>… > nr = 1, be all the numbers each of which is the index of the centralizer of some element of G in G. We call the vector (n1, n2,…, nr) the conjugate type vector of G. A group with the conjugate type vector (n1, n2,…, nr) is said to be a group of type (n1, n2,…, nr).

In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.

Introducing the idea of weighted sharing of values we prove some uniqueness theorems for meromorphic functions which improve some existing results.

Let A be a noetherian ring (all the rings are supposed here to be commutative with identity), a ⊂ A a proper ideal and  the completion of A in the α-adic topology. We consider the following conditions

(WAP) Every finite system of polynomial equations over A has a solution in A iff it has one in Â.

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we define

The elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

Quasi-frobenius (= QF) rings have many interesting characterizations. One such, due to Ikeda [17] is that these rings are right (left) artinian and right (left) self-injective. Thus, if R is QF, then R is right (left) noetherian, so each direct sum of injective right R-modules is injective: in particular, each free, hence, each projective, R-module is injective. One object of this paper is to report that this property characterizes QF-rings:

The aim of this note is to study birational transformations of three-folds with nef canonical classes. One would like to write any such map as a composite of certain simple and basic transformations.

A special case was first considered by Kulikov [Ku] and later extensively studied by several authors. The next main conceptual step was Reid’s study [R2] of small resolutions of terminal threefold singularities. He obtained the elementary birational transformations as follows: find a copy of such a small resolution inside the threefold and then replace it with another resolution. This approach was extended by Kawamata [K3] to canonical singularities.