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1.
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
2.
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
3.
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
4.
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
5.
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
6.
L. Olsen 《Monatshefte für Mathematik》2005,146(2):143-157
For a probability measure μ on a subset of
, the lower and upper Lq-dimensions of order
are defined by
We study the typical behaviour (in the sense of Baire’s category) of the Lq-dimensions
and
. We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower Lq-dimension
attains the smallest possible value and the upper Lq-dimension
attains the largest possible value. 相似文献
7.
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:
相似文献
i) | There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ and every l ∈ {0, 1, 2, …} we have |
ii) | For every compact set with and Kc connected and every function continuous on K and holomorphic in K0, there exists a subsequence of , such that, for every compact set we have |
8.
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
9.
Franki Dillen Johan Fastenakels Joeri Van der Veken Luc Vrancken 《Monatshefte für Mathematik》2007,152(2):89-96
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. 相似文献
10.
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. 相似文献
11.
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
12.
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
13.
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
14.
B. Kuzma 《Monatshefte für Mathematik》2006,149(2):119-128
Nondegenerate mappings that preserve Jordan triple product on
are characterized. Here, n ≥ 3 and
is an arbitrary field.
This work was supported by grants from the Ministry of Science of Slovenia. 相似文献
15.
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 相似文献
16.
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
17.
Bernhard Lamel 《Monatshefte für Mathematik》2004,142(4):315-326
We prove the following regularity result: If
,
are smooth generic submanifolds and M is minimal, then every Ck-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 Ck is smooth. 相似文献
18.
Let
be a univariate, separable polynomial of degree n with roots x
1,…,x
n
in some algebraic closure
of the ground field
. It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all
such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on
by permutation of the coordinates and each
defines a relation P − P(x
1,…,x
n
) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an
H-basis of G-invariant relations if and only if the algebra of coinvariants
has dimension ‖G‖ over
. To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite
group G has dimension ‖G‖ if and only if G = Σ
n
acting via the tautological permutation representation. 相似文献
19.
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
20.
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. 相似文献