A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

For an integer N let X1(N) be the modular curve defined over Q which corresponds to the modular group Γ1(N) To each primitive cusp form f ═ Σ amqm, a1═1, (Γ normalized new form in the sense of [1]) on Γ1(N) of weight 2, there corresponds a factor Jf of the jacobian variety of X1(N) (cf. Shimura [19]).

Let f═(f0, f1, …, fn) (n ≧ 1) be a transcendental system in ∣z∣ < ∞. That is, f0, f1,…, fn are entire functions without common zeros and the characteristic function of f defined by H.

Modular forms of half-integral weight are of intrinsic interest: many of the functions of classical number theory transform under a matrix group with half-integral weight. The aim of this paper is to refine some results and techniques which have been introduced to study these functions and the arithmetic information which they contain.

This is a continuation of Part I, which appeared in this journal.

In the previous paper I we have defined the following notions: orthogonal Jordan triple system (OJTS), orthogonal symmetric graded Lie algebra (OSGLA), orthogonal Jordan algebra (OJA), Hermitian symmetric graded Lie algebra (HSGLA). And we have shown that equivalent classes of OJTS naturally correspond to equivalent classes of OSGLA and through this correspondence we have naturally constructed HSGLA’s from the OJTS’s associated with OJA’s with unity.

Let H(m) denote the class number of the field where Q is the rational number field and ζm is a primitive m-th root of unity for a positive rational integer m.

The purpose of this paper is to study threefolds X, with negative Kodaira dimension k(X) and positive irregularity q(X), defined over the complex field C.

We shall discuss in this paper some problems in non-linear prediction theory. An Ornstein-Uhlenbeck process {U(t)} is taken to be a basic process, and we shall deal with stochastic processes X(t) that are transformed by functions f satisfying certain condition. Actually, observed processes are expressed in the form X(t) = f(U(t)). Our main problem is to obtain the best non-linear predictor X̂(t, τ) for X(t + τ), τ > 0, assuming that X(s), s ≤t, are observed. The predictor is therefore a non-linear functional of the values X(s), s ≤ t.