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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.
Erhard Aichinger 《Monatshefte für Mathematik》2004,143(2):89-103
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
. 相似文献
3.
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
4.
5.
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
6.
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
7.
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
8.
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
9.
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 相似文献
10.
Marilyn Breen 《Monatshefte für Mathematik》2006,148(2):91-100
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. 相似文献
11.
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
12.
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
13.
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 相似文献
14.
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 |
15.
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
16.
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
17.
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
18.
L. Olsen 《Monatshefte für Mathematik》2008,155(2):191-203
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 相似文献
19.
Jorge J. Betancor Claudio Jerez Sandra M. Molina Lourdes Rodríguez-Mesa 《Monatshefte für Mathematik》2008,153(2):89-103
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 相似文献
20.
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