For a magnetic Schröodinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties.

In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the involutions contained in these groups and he gave characterizations of the general projective linear groups of low dimensions along these lines. The detail of the one-dimensional case has been published in [5], but the two-dimensional case has not appeared yet in detail. His work was followed by Suzuki [7], Feit [6] and Walter [11]. The present paper is a continuation of [73 and discusses a characterization of the two-dimensional projective unitary group over a finite field of characteristic 2. The precise conditions which characterize the group in question will be stated in the first section. The method employed here is similar to the one used in [7]. An application of group characters provides a formula for the order. However a difficulty comes in when one attempts to identify the group. In order to overcome this difficulty we will use a method primarily designed to study a class of doubly transitive permutation groups (cf. [9]). We need also a group theoretical characterization of a class of doubly transitive groups called (ZT)-groups. This is a generalization of a result in [8], and may be of in dependent interest.

Let C be an elliptic curve (an abelian variety of dimension one) defined over the field Q of rational numbers. A minimal Weierstrass model for C at all primes p in the sense of Néron [3] is given by a plane cubic equation of the form

where aj belongs to the ring Z of integers of Q, the zero of C being the point of infinity.

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].

This paper is a continuation of the two preceding papers [12], [13] where the classification of the de Jonquières type subgroups in the Cremona group of 3 variables is promised. However the classification of such subgroups is postponed until the article in preparation “On the maximal connected algebraic subgroups of the Cremona group II”. The purpose of this paper is to establish a general method to study algebraic subgroups in the Cremona group of n variables and to illustrate how it works and leads to the classification of Enriques (Theorem (2.25)) when applied to the 2 variable case. This method gives us also the classification of the maximal connected algebraic subgroups of the Cremona group of 3 variables.

Some complex n-tori admit holomorphic foliations of codimension one besides the flat ones. It will be shown that such nonlinear foliations, possibly with singularities, can be reduced to those on 2-tori under some topological conditions. A crucial step is an application of the Hodge theory on pseudoconvex manifolds.

A left QF-3 ring R is one in which RR, the ring considered as a left module over itself, can be embedded in a projective infective left R module Q(RR). QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5, 8, 9, 12, 13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.

The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10, 11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

In [3], M. W. Hirsch obtained some necessary conditions for the existence of an Anosov diffeomorphism on a differentiable manifold. As an application, he constructed many manifolds which do not admit Anosov diffeomorphisms.

In this paper, we will characterize the class of (conditional) hitting time distributions of single points of one dimensional generalized diffusion processes and give their tail behaviors in terms of speed measures of the generalized diffusion processes.

Koecher defined in [3] the following zeta function associated with the matrix S(n) of a positive quadratic form and one complex variable ρ

(1)

Some results for the Bergman functions in unbounded domains are shown. In particular, a class of unbounded Hartogs domains, which are Bergman complete and Bergman exhaustive, is given.

We denote a permutation group H on a set Γ by (H, Γ). (H, Γ) is called a permutation group of rank 3 if (H, Γ) is transitive and (Ha, Γ), a ∈ Γ, has exactly three orbits, where Ha is the stabiliger of a point a, namely, {α ∈ H \aα = a}

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively :

1. (1) the class number of the cyclotomic Zℓ-extension of k,

2. (2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,

3. (3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and

4. (4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.

Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define packing density of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov’s mean dimension theory.

Nous continuerons l’étude du premier théorème étoilé de Gross, faite dans le mémoire de l’auteur: Théorèmes étoilés de Gross et leurs applications, Ann. Inst. Fourier, 5 (1955), pp. 1-28; ce mémoire sera cité avec l’abréviation [A]

We investigate the uniform perfectness on a boundary point of a hyperbolic open set and distortion of a holomorphic function from the unit disk Δ into a hyperbolic domain with a uniformly perfect boundary point, especially of a universal covering map of such a domain from Δ, and we obtain similar results to celebrated Koebe’s Theorems on univalent functions.

Let be an infinitesimal group scheme, defined over an algebraically closed field of characteristic p. We employ rank varieties of -modules to study the stable Auslander-Reiten quiver of the distribution algebra of . As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case is supersolvable or a Frobenius kernel of a smooth, reductive group.

Let (R, m) be a local noetherian ring and I a proper ideal in R. Let (I) be the Rees-ring ⊕n≥0 In with respect to I. In this note we describe conditions for I and R in order that the Cohen-Macaulay property (C-M for short) of R/I can be lifted to R and (I), see Propositions 1.2, 1.3. and 1.4.