Let p be a prime > 2 and m an arbitrary positive integer; define

where (r/p) is the Legendre symbol. We consider the problem of finding the highest power of p dividing Sm. A little more generally, if we put

where a is an arbitrary integer, we seek the highest power of p dividing Sm(a). Clearly Sm = Sm(0), and Sm(a) = Sm(b) when a ≡ b (mod p).

The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.

Let P be a principal fibre bundle over M with group G and with projection π : P → M. By definition of a principal fibre bundle, G acts on P on the right. We shall denote this transformation law by ρ

In the theory of functions meromorphic in |z| < + ∞, Iversen [4] proved the following: If w = f(z) is meromorphic in |z| < + ∞ and has an essential singularity at z = ∞, then any inverse function-element of this function with the centre w0 can be continued analytically to any point w ≠ wo, except possibly this point w, in any disc having the centre at the point w and containing the point w0.

There is a problem called the chain problem of prime ideals, which asks, when 0 is a Noetherian local integral domain, whether the length of an arbitrary maximal chain of prime ideals in 0 is equal to rank 0 or not.

Unter einem bizyklischen biquadratischen Zahlkörper verstehen wir einen absolut abelschen Zahlkörper vierten Grades, der von zwei verschiedenen quadratischen Zahlkörpern zusammengesetzt wird. Wir behandeln im folgenden die Einheiten- und Idealklassengruppe dieses Körpers, indem wir die Ergebnisse von [6] und [7] weiterführen.

A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary then

The common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

In this note we show in § 1, as the main result, that any connected Lie subgroup of the general linear group GL(n, R) can be realized as the holonomy group of a linear connection, i.e. the homogeneous holonomy group of the associeted affine connection, defined on an affine space of dimension n (n ≧ 2).

MM. Ozeki et Hano ont récemment démontré [3] que tout sous-groupe connexe de Lie du groupe linéaire GL(n, R), n≧2, peut être réalisé comme groupe d’holonomie d’une certaine connexion linéaire dans un espace affine de dimension n.

In Introduction In differential geometry of linear connections, A. Nijenhuis has introduced the concepts of local holonomy group and infinitesimal holonomy group and obtained many interesting results [6].

The purpose of the present note is to generalize his results to the case of connections in arbitrary principal fiber bundles with Lie structure groups. The concept of local holonomy group can be immediately generalized and has been already utilized by S. Kobayashi [4]. Our main results are Theorems 4 and 5 on infinitesimal holonomy groups. The proofs depend on a little sharpened form of a theorem of Ambrose-Singer [1]. In the case of linear connections, our infinitesimal holonomy group coincides with that of Nijenhuis, as we shall show in Section 6.

In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 by

LEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.

Montel [10] proved in 1912 the following theorem: Let w = f(z) be an analytic function in the horizontal strip B : 0 < x < + ∞, 0 < y < 1 (z = x + iy) which is continuous on 0 < x < + ∞, 0 ≦ y < 1 and omits at least two values. If f (x) converges to a value w0 as x → + ∞, then f(z) converges to W0 as z tends to in 0 < x < + ∞, 0 ≦ y < 1 − ε for any ε such that 0 < ε < 1.

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined by

Theorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

Let f(z) be a nonconstant analytic transformation of the unit circle U : ∣z∣ < 1 into a Riemann surface ℜ. As an extension of a classical theorem of F. and M. Riesz, the author proved in Theorem 3.4 of [2] that if the image of U is relatively compact in ℜ and has universal covering surface of hyperbolic type, and if, at every point of a set on ∣z∣ = 1 of positive inner linear measure, there terminates a curve along which f(z) has limit, then the set of such limits has positive inner logarithmic capacity. This theorem was followed by the first proposition in Kuramochi [1], which asserts that, if ℜ has a null boundary and the image of U excludes a set of positive logarithmic capacity on ℜ and if, at every point of a set E on ∣z∣ = 1, there terminates a curve along which f(z) has limit in the union of a set of inner logarithmic capacity zero on ℜ and the boundary components of ℜ then the inner linear measure of E is zero.