The purpose of this paper is to give a proof to the result announced in [3]. Let X be an algebraic surface defined over C. X is called a K3 surface if its canonical line bundle Kx is trivial and dim H1(X, ϕX) = 0. It is known that the automorphism group Aut (X) of X is isomorphic, up to a finite group, to the factor group O(Sx)/Wx, where O(Sx) is the automorphism group of the Picard lattice of X (i.e. Sx is the Picard group of X together with the intersection form) and Wx is its subgroup generated by all reflections associated with elements with square (–2) of Sx ([11]). Recently Nikulin [8], [10] has completely classified the Picard lattices of algebraic K3 surfaces with finite automorphism groups.

A connected complex Lie group without non-constant holomorphic functions is called a toroidal group ([5]) or an (H, C)-group ([9]). Let X be an n-dimensional toroidal group. Since a toroidal group is commutative ([5], [9] and [10]), X is isomorphic to the quotient group Cn/Γ by a lattice of Cn. A complex torus is a compact toroidal group. Cousin first studied a non-compact toroidal group ([2]).

Let L be a holomorphic line bundle over X. L is said to be homogeneous if is isomorphic to L for all x ε X, where Tx is the translation defined by x ε X. It is well-known that if X is a complex torus, then the following assertions are equivalent:

In this paper we shall study elastic waves in two isotropic media with different densities and Lamè’s constants. In seismology elastic waves are studied when the border of two media is a hyperplane, however there are no results on elastic waves in the non-flat border case. First in Section 1 we shall show an existence theorem of the solutions of an initial boundary value problem which is satisfied by the displacements of two media. Next we shall discuss about propagation of singularities of the solutions, for Hörmander and Lax-Nirenberg showed that an appearance of propagation of singularities is similar to one of propagation of waves.

Infinitely divisible probability measures on discrete spaces admitting a commutative convolution have been studied in various frameworks. For generalized convolutions related to Delphic structures an important contribution was made by Gilewski and Urbanik in [5]. In [11] Schwartz based his detailed analysis on convolutions arising from orthogonal series. Both of these approaches can be included into the framework of discrete hypergroups chosen f.e. by Gallardo and Gebuhrer in [4]. The main result common to these sources is the fact that, roughly speaking, all infinitely divisible probability measures are of Poisson type. Although the double coset spaces of Gelfand pairs are commutative hypergroups admitting an extended harmonic anafysis, the analytic methods developed in the theory of Gelfand pairs (see Dieudonné [3] and Heyer [7]) provide a more direct access to specific results like the characterization of divisible (idempotent, infinitely divisible) measures. For discrete Gelfand pairs (and their double coset spaces) Letac and his school have achieved remarkable results (see Letac [8], [9] and the references therein). The still unpublished thesis [1] of S. Ben Mansoor contains much information about divisible probability measures on cubes. In our exposition we reprove Ben Mansoor’s main theorem 3.5.4 with additional care, and discuss the special case of the ra-dimensional cube which was the basic object of study already in Letac, Takács [10]. It turns out that the main obstacle to be overcome in establishing the Poisson representation of infinitely divisible probability measures is the presence of idempotent factors, a problem that has been excluded f.e. in the work [4] of Gallardo and Gebuhrer.

Consider a Riemann surface X equipped with a projective structure, that is, a covering of X with coordinate neighborhoods U and corresponding (holomorphic) local coordinates {t} such that in the intersection U ∩ U′ of any two such coordinate neighborhoods U and U′ change of local coordinates is mediated by a fractional linear transformation

In the theory of automorphic forms on covering groups of the general linear group, a central role is played by certain local representations which have unique Whittaker models. A representation with this property is called distinguished. In the case of the 2-sheeted cover of GL2, these representations arise as the the local components of generalizations of the classical θ-function. They have been studied thoroughly in [GPS]. The Weil representation provides these representations with a very nice realization, and the local factors attached to these representations can be computed using this realization. It has been shown [KP] that only in the case of a certain 3-sheeted cover do we find other principal series of covering groups of GL2 which have a unique Whittaker model. It is natural to ask if these distinguished representations also have a realization analgous to the Weil representation.

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.

The existence of the asymptotic probability measure of the Riemann zeta-function was proved in Bohr-Jessen’s classical paper [3] [4].

Let s = σ + it be a complex variable, ζ(s) the Riemann zeta-function, and R an arbitrary rectangle with the edges parallel to the axes. Then, for any σ0 > ½ and T > 0, the set

is Jordan measurable, and we denote the Jordan measure of this set by V(T, R; ξ). Then, Bohr-Jessen’s main result asserts the existence of the limit

The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].

Fix a complex number q with |q| < 1. Let T1…, Tn be n-commuting q-difference operators defined by

for a function f(x), x = (x1,…,xn) ε (C*)n. Consider a system of linear q-difference equations in several variables for a matrix valued function on (C*)n as follows:

Let Rn+1 = Rn X R denote the (n + l)-dimensional Euclidian space (n ≧ 1). For X ε Rn+1 we write X = (x, t) with x ε Rn and t ε R. For an α with 0 < α < 1, we write

where Δ is the Laplacian on Rn and (– Δ)α is the α-fractional power of - Δ on R*

In this paper we study a connection between global and local properties of Kaehler manifolds, more specifically we study a connection between the volumes of small geodesic balls of a manifold M and some generalized Chern numbers. We use the standard power series expansion for Vm(r).

In Theorem 3.1 we give characterizations of a flat compact Kaehler manifold in terms of the volumes of small geodesic balls and generalized Chern numbers ωn-1c1(M) and ωn-2c12(M). In Theorem 4.1 similar questions for complex space forms are considered. So we prove one particular case of the Conjecture (IV) stated by Gray and Vanhecke [6].