In 1969, Mumford [8] proved that, for a complete non-singular algebraic surface F over the complex number field C, the dimension of the Chow group of zero-cycles on F is infinite if the geometric genus of F is positive. To this end, he defined a regular 2-form ηf on a non-singular variety S for a regular 2-form η on F and for a morphism f: S → SnF, where SnF is the 72-th symmetric product of F, and he showed that ηf vanishes if all 0-cycles f(s), s ∈ S, are rationally equivalent. Roitman [9] later generalized this to a higher dimensional smooth projective variety V.

By the coadjoint representation of a connected Lie group G with the Lie algebra g we mean the representation CoAd(g) = tAd(g-1) in the dual space g*. Imitating Chevalley’s argument for complex semi-simple Lie algebras, we shall show that the CoAd (G)-invariant polynomial algebra on g* is finitely generated by algebraically independent polynomials when G is the inhomogeneous linear group IU(n) or ISO(n). In view of a well-known theorem [8, p. 183] our results imply that the centers of the enveloping algebras for the (or the complexified) Lie algebras of these groups are also finitely generated. Recently much more inhomogeneous groups have been studied in a similar context [2]. Our results, however, are further reaching as far as the groups IU(n) and ISO(n) are concerned [cf. 3, 4, 6, 7, 9].

This paper is a continuation to the previous one [3], We shall show that, for the inhomogeneous linear group ISL(n + 1, R) (resp. ISp(n, R)), the coadjoint invariant polynomial algebra is generated by one (resp. n) algebraically independent element. We shall state our results more precisely.

Let C be an affine curve, contained on a non-singular surface X as a closed 1-dimensional subscheme. If P is a closed point on C, the blowing-up C′ of C with center P (induced by the blowing-up of X with center P) is an affine curve. It is known that there is a sequence:

where C is the normalization of C, and each Ci + 1 is the blowing-up of Ci with center a singular point Pt on Ci (i = 0, …, k – 1).

In [7], O. Lehto and K. I. Virtanen introduced the concept of normal meromorphic functions in connection with the study of boundary behaviour of meromorphic functions of one complex variable.

In this paper, we generalize the theory of normal meromorphic functions to the case of holomorphic mappings into higher dimensional complex spaces in connection with the theory of hyperbolic manifolds and Nevanlinna theory.

The main concern of this paper is the generalizations of the big Picard theorem and Lindelöf’s theorem which appear in the classical function theory.

We shall discuss regularities and related topics on weak solutions of the system of the following quasi-linear elliptic differential equations (a combination of almost single equations)

in a bounded domain Ω in Rn (n ≧ 2), where A1j … (A1j …, Anj) are given vector functions of (x, u, ▽uj), Bj are scalar functions of the same variables, and ▽uj = (∂uj/∂x1, …, ∂uj/∂xj denote the gradients of the uj = uj(x) (j = 1, …, m).

Let X = {X(A); A ∈ Q} be a Lévy’s Brownian motion with the basic time parameter space Q, where Q is taken to be the n-dimensional metric space Qn,k of constant curvature (2 ≤ n ≤ ∞, — ∞ < k: < ∞), i.e., Q is one of

(a) Euclidean space for k = 0, (b) sphere for k > 0 and

(c) real hyperbolic space for K < 0.

The increment X(A) — X(B) is, by definition, Gaussian in distribution and has mean 0 and variance d(A, B), the distance between A and B. The existence of such a Gaussian random field is well known ([3], [4], [16] and [23]).

In this paper, we shall study a generalization of the Maillet determinant. hetp be an odd prime, and G = (Z/pZ)*. We shall identify any integer and its image in G if there is no fear of confusion. For any integer a, let R(a) denote an integer satisfying

This is the first in a series of papers concerned with a relation between a representation in the polynomial ring of additive groups and its translation invariant operators. The present study is to observe several properties of a polynomial sequence pa(x) satisfying the binomial identity:

by means of some translation invariant operators.