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1.
2.
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. 相似文献
3.
Manfred Stoll 《Monatshefte für Mathematik》2005,144(2):131-139
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. 相似文献
4.
J. P. Moreno 《Monatshefte für Mathematik》2007,152(3):255-263
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. 相似文献
5.
Min Ho Lee 《Monatshefte für Mathematik》2007,152(4):321-336
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. 相似文献
6.
We prove effective upper bounds for the almost periodicity of polynomial Euler products
in the half-plane of absolute convergence. From this we deduce estimates for the roots of the equation
, where c is any non-zero complex number which is attained by
. The method relies mainly on effective diophantine approximation.The first author was supported by a grant of the Humboldt Foundation. 相似文献
7.
In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if
is a Gabor frame for
with frame bounds A and B, then the following two inequalities hold:
and
. In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form
, where Δ
k
and Λ
k
are arbitrary sequences of points in
and
, 1 ≤ k ≤ r.
Corresponding author for second author
Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China 相似文献
8.
Min Ho Lee 《Monatshefte für Mathematik》2004,141(3):187-196
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. 相似文献
9.
For fixed k ≥ 3, let Ek(x) denote the error term of the sum
, where
1. It is proved that if the Riemann hypothesis is true, then
,
. A short interval result is also obtained. 相似文献
10.
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 ⋯ xryω+1ztω = x1 ⋯ xryztω, 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. 相似文献
11.
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. 相似文献
12.
Wojciech Jaworski 《Monatshefte für Mathematik》2008,155(2):135-144
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 相似文献
13.
Let f be a cusp form of the Hecke space
and let L
f
be the normalized L-function associated to f. Recently it has been proved that L
f
belongs to an axiomatically defined class of functions
. We prove that when λ ≤ 2, L
f
is always almost primitive, i.e., that if L
f
is written as product of functions in
, then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if
then L
f
is also primitive, i.e., that if L
f
= F
1
F
2 then F
1 (or F
2) is constant; for
the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset
of functions f for which L
f
belongs to the more familiar extended Selberg class
is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in
. 相似文献
14.
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. 相似文献
15.
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. 相似文献
16.
Emmanuel Preissmann 《Monatshefte für Mathematik》2007,150(3):233-239
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. 相似文献
17.
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 相似文献
18.
Min-Chun Hong Changyou Wang 《Calculus of Variations and Partial Differential Equations》2005,23(4):425-450
For
and
, we show that any minimizing biharmonic map from
to Sk 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 相似文献
19.
Klaus Langmann 《Monatshefte für Mathematik》2004,143(3):205-227
We show that the unsolvability of the Diophantine equation
is equivalent to a good uniform distribution of the set
. The proof depends on the asymptotic evaluation of the Gauss sum
. 相似文献
20.
H. Rindler 《Monatshefte für Mathematik》2006,147(3):265-272
For
, let E(λ*, λ*) be the set
It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that
where dim denotes the Hausdorff dimension. 相似文献