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1.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function
. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient
, for certain external subspaces
of the hyperfinite dimensional Banach space
, with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and
are isometrically isomorphic as Banach algebras.
Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic. 相似文献
2.
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
3.
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. 相似文献
4.
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
5.
R. G. Novikov 《Selecta Mathematica, New Series》1997,3(2):245-302
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. 相似文献
6.
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
7.
8.
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
9.
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
10.
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
11.
We prove the meromorphic version of the Weil–Oka approximation theorem in a reduced Stein space X and give some characterizations of meromorphically
-convex open sets of X. As an application we prove that for every meromorphically
-convex open set D of a reduced Stein space X with no isolated points there exists a family
of holomorphic functions on X such that the normality domain
of
coincides with D. Mathematics Subject Classification (2000) 32E10, 32C15, 32E30, 32A19 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.
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
14.
J. Nielsen Clelland 《Selecta Mathematica, New Series》1997,3(1):1-77
I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation
for one function of three independent variables. The PDE is formulated as an exterior differential system on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of
equivalence to study the structure of the space of conservation laws. My main result is: Theorem.
Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by
a closed 3-form in the differential ideal
${\cal I}$
on the original 12-manifold M. I show that if a nontrivial conservation law exists, then has a deprolongation to an equivalent system on a 7-manifold N, and any conservation law for can be expressed as a closed 3-form on N that lies in . Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic)
equation of the formA (u
xx
u
yy
-u
2
xy
)+Bu
xx
+2Cu
xy
+Du
yy
+E = 0 where A, B, C, D, E are functions of x, y, t, u, u
x
, u
y
, u
t
. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using
Cartan's method of equivalence. I show that the non-linearizable equation has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant
and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a
linear equation, i.e., is linearizable. 相似文献
15.
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
16.
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
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.
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
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.
We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate
statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product . 相似文献