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.     

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.     

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.     

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     

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.     

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.     

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:

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.     

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     

Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L ¹), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.     

Topological dimensions and multifractal Rényi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension and the lower and upper q-Rényi dimensions and of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al. Author’s address: Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

A variational problem for submanifolds in a sphere

Let be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S ^{n + p} , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional , where is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S ^{n + p} . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.     

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.     

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For and , we show that any minimizing biharmonic map from to S^k is smooth off a closed set whose Hausdorff dimension is at most n-5. When n = 5 and k = 4, for a parameter we introduce a -relaxed energy of the Hessian energy for maps in so that each minimizer of is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of for .Received: 5 April 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

A <Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript>-Regularity Theorem for Nondegenerate CR Mappings

We prove the following regularity result: If , are smooth generic submanifolds and M is minimal, then every C^k-CR-map from M into M which is k-nondegenerate is smooth. As an application, every CR diffeomorphism of k-nondegenerate minimal submanifolds in of class C^k is smooth.     

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.     

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 the Mean Value of the Index of Composition of an Integer

For each integer n 2, let be the index of composition of n, where . For convenience, we write (1)=(1)=1. We obtain sharp estimates for and , as well as for and . Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

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.