Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

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 .     

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.     

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.     

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.     

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     

Porosity and diametrically maximal sets in C(K)

This paper is motivated by recent attempts to investigate classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let be the family of all closed, convex and bounded subsets of C(K) endowed with the Hausdorff metric. A completion of is a diametrically maximal set satisfying A ⊂ D and diam A = diam D. Using properties of semicontinuous functions and an earlier result by Papini, Phelps and the author [12], we characterize the family γ(A) of all possible completions of . We give also a formula which calculates diam γ(A) and prove finally that, if K is a Hausdorff compact space with card K > 1, then the family of those elements of having a unique completion is uniformly very porous in with a constant of lower porosity greater than or equal to 1/3.     

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.     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let B denote the unit ball in ⁿ, n 1, and let and denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces , , and weighted Bergman spaces , , , of holomorphic functions f on B for which and respectively are finite, where and The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and .(a) If for some , then for all p, , with .(b) If for some p, , then for all with . Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose is holomorphic in B. If for some p, , and , then . Conversely, if for some p, , then the series in * converges.     

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:

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.     

New spaces of functions and hyperfunctions for Hankel transforms and convolutions

In this paper we study the Hankel transformation and convolution on certain spaces of entire functions and its dual that is a space of hyperfunctions and contains the (even)-Schwartz space S _e ′. We prove that the Hankel transform is an automorphism of . Also the Hankel convolutors of are investigated. Authors’ addresses: Jorge J. Betancor, Claudio Jerez and Lourdes Rodríguez-Mesa, Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271 La Laguna (Sta. Cruz de Tenerife), Espa?a; Sandra M. Molina, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350 (7600), Mar del Plata, Argentina     

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.     

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.     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

Homomorphisms on Function Lattices

In this paper we study real lattice homomorphisms on a unital vector lattice , where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.     

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.     

A new example of singular height transition for capillary surfaces

Finn and Kosmodemyanskii, Jr. gave an example of a domain containing a disk , and of a family of domains converging to as , such that the heights u ^t of capillary surfaces in vertical tubes with the sections in a gravity field g satisfy for every , but for which u ¹< u ⁰ over for all g > 0. In subsequent work, Finn and Lee characterized the most general convex that leads to such a discontinuous transition when is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains are disks concentric to . Thus, the limited smoothness in the original example of the convergence to of the approxim ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.Received: 3 September 2002, Accepted: 17 February 2003, Published online: 1 July 2003Mathematics Subject Classification: 76B45, 53A10, 49Q10     

Minkowski Arrangements of Spheres

Let be a non-negative number not greater than 1. Consider an arrangement of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of associate a concentric sphere of radius times the radius of S. We call this sphere the -kernel of S. The arrangement is said to be a Minkowski arrangement of order if no sphere of overlaps the -kernel of another sphere. The problem is to find the greatest possible density of n-dimensional Minkowski sphere arrangements of order . In this paper we give upper bounds on for .     

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