1. Let E be a totally disconnected compact set in the 2-plane and Ω its complement with respect to the extended z-plane. Then Ω is a region. Let be an exhaustion of Ω satisfying the following conditions:

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

Let f be of class U in Seidel’s sense ([4, p. 32], = “inner function” in [3, p. 62]) in the open unit disk D. Then f has, by definition, the radial limit f(eiθ) of modulus one a.e. on the unit circle K. As a consequence of Smirnov’s theorem [5, p. 64] we know that the function

We have given in [3] the definition of the quasi-flow and discussed the representation and the time-change of the quasi-flow. Further we have defined the TQ-system and studied some properties of them using the time-change and the representation of the quasi-flow. These properties are very useful to study the ergodicity, the entropy and increasing partitions of the automorphism. We are now going to extend the definition of the TQ-system and to study the similar problems as the above.

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

In the previous papers [1], [2] we have studied the prolongations of G-structures to tangent bundles of arbitrary order, and in [3] we have considered the prolongations of connections to the tangential fibre bundles of higher order. The purpose of the present paper is to study the liftings of tensor fields and affine connections to tangent bundles of higher order. In fact, most of results in [4], [5] will be generalized in a natural fashion and some of the formulas concerning vertical and complete lifts in [5] will be unified and generalized in our formulas concerning ‹Λ›-lifts (cf. §3).

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

1. Introduction. Let M, N be complex manifolds and G be a group of holomorphic automorphisms of N. In [3] (c.f. p. 74) W. Kaup introduced the notion of holomorphic maps into a family of holomorphic maps between complex spaces. By definition, a map g: M→G is holomorphic if and only if the induced map g̃(x, y): = g(x) (y) (x∈M, y∈N) of M×N into N is holomorphic in the usual sense. The purpose of this note is to give a description of a holomorphic map of a connected complex manifold M into G. We show first the existence of the maximum connected Lie subgroup G0 of G which is a complex Lie transformation group of N.

Let G be a simply connected simple Lie group and C the center of G, which is isomorphic with the fundamental group of the adjoint group of G. For an element c of C, an element x of the Lie algebra g of G is called a representative of c in g if exp x = c. Sirota-Solodovnikov [7] found a complete set of representatives of the center C in g and studied the group structure of C, and using their results Goto-Kobayashi [1] classified subgroups of the center C with respect to automorphisms of G. The group structure of C was also studied in Takeuchi [8],

Let us consider transporting particle in the n-dimensional Euclidian space Rn. It is assumed that a particle originating at a point x∈Rn moves in a straight line with constant speed c and continues to move until it suffers a collision. The probability that the particle has a collision between t and t + Δ is kΔ + o(Δ), where k is constant. When a particle has a collision, say at y in Rn, it moves afresh from y with an isotropic choice of direction independent of past history.

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

1. Nous avons un théorème, ce qu’on appelle le théorème de Cartan-Dieudonné, sur les générateurs du groupe orthogonal: Toute transformation orthogonale à n variables sur un corps de caractéristique ≠2 est un produit de n symetries au plus, et sur un corps de caractéristique 2 elle est un produit au plus de n transvections orthogonales, sauf un seul cas, [1] ou [3]. C’est une généralisation d’un résultat obtenu par E. Cartan, relatif au corps des nombres réels ou au corps des nombres complexes [4].