Comparison of Picard groups in dimension 1

We compare two Picard groups in dimension 1. Our proofs are constructive and the results generalize a theorem of J. Sands [11]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension \(\dim \mu\) of a signed Radon measure μ on the punctured closed unit ball 0?x|?≦?1 in the d-dimensional euclidean space with d?≧?2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrödinger equation (???Δ?+?μ)u?=?0 on the punctured open unit ball 0?x|?x|?=?1. If the Green function of the above equation on 0?x|?Δ?+?μ)u?=?δ _y , the Dirac measure supported by the point y, exists for every y in 0?x|?μ is referred to as being hyperbolic on 0?x|?γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon–Nikodym density dγ(x)/dx is bounded by a positive constant multiple of |x|^???2. The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures μ are invariant under basic perturbations \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.     

Quantization dimension of random self-similar measures

In this paper, we study the quantization dimension of a random self-similar measure μ supported on the random self-similar set K(ω). We establish a relationship between the quantization dimension of μ and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension.     

Beurling dimension and self-similar measures

In this paper the authors study the Beurling dimension of Bessel sets and frame spectra of some self-similar measures on ${\mathbb{R}}^d$ and obtain their exact upper bound of the dimensions, which is the same given by Dutkay et al. (2011) [8]. The upper bound is attained in usual cases and some examples are given to explain our theory.     

Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .

     

The energy of signed measures

We generalize the concept of energy to complex measures of finite variation. We show that although the energy dimension of a measure can exceed that of its total variation, it is always less than the Hausdorff dimension of the measure. As an application we prove a variant of the uncertainty principle.

     

一类椭圆方程的边界唯一延拓性

本文主要讨论了一类二阶椭圆方程Lu=-△u+Vu=0的弱解在连通凸区域边界上的唯一延拓性,并且证明了文[4]中提出的猜测在本文的情形同样成立.     

Large Deviation Principle for Additive Functionals of Brownian Motion Corresponding to Kato Measures

Let (B _t ,P ^W _x ) be the Brownian motion. Let be a Radon measure in the Kato class and A _t the additive functional associated with . We prove that A _t /t obeys the large deviation principle.     

Hausdorff dimension of fiber-coding sub-Sierpinski carpets

In this paper we study a class of subset of Sierpinski carpets for which the allowed digits in the expansions fall into each fiber set with a prescribed frequency. We calculate the Hausdorff and packing dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff and packing measures to be finite.     

Admissible measures in one dimension

In this note we show that -admissible measures in one dimension (i.e. doubling measures admitting a -Poincaré inequality) are precisely the Muckenhoupt -weights.

     

The Picard Group of Corings

We introduce and study the Picard group of a coring. We give an exact sequence relating the Picard group of a coring and its automorphisms generalizing the known exact sequences associated to an algebra and a coalgebra over a field. We also extend to corings the Aut–Pic property and we give some new corings satisfying this property.     

Elliptic points of the Picard modular group

We explicitly compute the elliptic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.      

Nonautonomous Kato classes of measures and Feynman-Kac propagators

The behavior of the Feynman-Kac propagator corresponding to a time-dependent measure on is studied. We prove the boundedness of the propagator in various function spaces on , and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation.

     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

A generalization of the little Picard theorem

We show the little Picard theorem for a map of a quasicompact manifold to a relatively compact domain of Pn whose degeneracy locus of the Kobayashi pseudodistance is contained in some algebraic hypersurface.     

On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D

(0.1)     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

关于测度的Hausdorff维数的局部化

主要研究测度的豪斯道夫维数的局部化.通过定义一个测度μx,ε,从而给出dim·Hμ在点x的局部化维数dim·Hμ(x).进而得到局部化维数dim·Hμ(x)与dim·Hμ之间的关系,并得到了一个等式关系.     

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU _L ′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.