Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Tameness of Pseudovariety Joins Involving ${\sf R}$

In this paper, we establish several decidability results for pseudovariety joins of the form , where is a subpseudovariety of or the pseudovariety . Here, (resp. ) denotes the pseudovariety of all -trivial (resp. -trivial) semigroups. In particular, we show that the pseudovariety is (completely) κ-tame when is a subpseudovariety of with decidable κ-word problem and is (completely) κ-tame. Moreover, if is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that is also κ-tame. In particular the joins , , , and are decidable. Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged. Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged.     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation of a discrete subgroup in and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.     

Embedding metric spaces into normed spaces and estimates of metric capacity

Let be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of as the maximal such that every m-point metric space is isometric to some subset of (with metric induced by ). We obtain that the metric capacity of lies in the range from 3 to , where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to . Research supported by the German Research Foundation, Project AV 85/1-1.     

Geometric progressions in sumsets over finite fields

Given two sets , the set of d dimensional vectors over the finite field with q elements, we show that the sumset contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector and a nonsingular d × d matrix Λ whenever . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let be an equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over . If and are torsion-free arithmetic subgroups with , the map induces a morphism : of arithmetic varieties and the rationality of is defined by using symmetries on and as well as the commensurability groups of and . An element determines a conjugate equivariant holomorphic map of which induces the conjugate morphism of . We prove that is rational if is rational.     

The Variety of Near-Rings is Generated by Its Finite Members

We show that the variety of near-rings and the variety of zero-symmetric near-rings are both generated by their finite members. We show this in a more general context: if a variety is generated by a class of algebras , then the variety of -composition algebras is generated by the class of all full function algebras on direct products of finitely many copies of algebras in .     

A Characteristic Property of Injective Holomorphic Germs

Let X ₀ be the germ at 0 of a complex variety and let be a holomorphic germ. We say that f is pseudoimmersive if for any such that , we have . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.     

Smooth Universal Taylor Series

Let be a simply connected domain in , such that is connected. If g is holomorphic in Ω and every derivative of g extends continuously on , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and we denote . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Shift coordinates,stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with boundary components and cusps (which we call generalized pairs of pants), for all possible values of and satisfying . The parametrization of by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with boundary components and cusps, for fixed and , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2. Authors’ addresses: A. Papadopoulos, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; G. Théret, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector by rational vectors of a lattice whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.     

Hypersurfaces and Codazzi tensors

In this paper we deal with the following problem. Let (M ⁿ ,〈,〉) be an n-dimensional Riemannian manifold and an isometric immersion. Find all Riemannian metrics on M ⁿ that can be realized isometrically as immersed hypersurfaces in the Euclidean space . More precisely, given another Riemannian metric on M ⁿ , find necessary and sufficient conditions such that the Riemannian manifold admits an isometric immersion into the Euclidean space . If such an isometric immersion exists, how can one describe in terms of f? Author’s address: Thomas Hasanis and Theodoros Vlachos, Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form , where is a continuous function with , runs over , the set of Farey fractions of order Q in the unit interval [0,1] and are consecutive elements of . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

Distinguishing groupwise density numbers

We show that is consistent, where is the groupwise density number and is the groupwise density number for ideals. This answers a question of Heike Mildenberger. Partially supported by Grant-in-Aid for Scientific Research (C) 17540116, Japan Society for the Promotion of Science.