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1.

On a question of Brezis and Marcus

**总被引：3，自引：0，他引：3** S. Filippas V. Maz'ya A. Tertikas 《Calculus of Variations and Partial Differential Equations》2006,25(4):491-501

Motivated by a question of Brezis and Marcus, we show that the

*L*^{p}–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an*L*^{q}norm*q*≥*p*, with a constant depending on the interior diameter of Ω. 相似文献2.

Stability of the Picard Bundle

**总被引：2，自引：0，他引：2** Biswas I.; Brambila-Paz L.; Gomez T. L.; Newstead P. E. 《Bulletin London Mathematical Society》2002,34(5):561-568

Let

*X*be a non-singular algebraic curve of genus g 2,*n*2an integer, a line bundle over*X*of degree*d*> 2*n*(*g*–1) with (*n*,*d*) = 1 and*M*_{}the moduli space of stable bundles ofrank*n*and determinant over*X*. It is proved that the Picardbundle*W*_{}is stable with respect to the unique polarisation of*M*_{}. 2000*Mathematics Subject Classification*14H60, 14J60. 相似文献3.

The paper gives a condition for the expressible set of a sequence to have Lebesgue measure zero. 相似文献

4.

Approximation algorithms for Hamming clustering problems

**总被引：1，自引：0，他引：1**We study Hamming versions of two classical clustering problems. The

time a set

*Hamming radius**p*-*clustering problem*(HRC) for a set*S*of*k*binary strings, each of length*n*, is to find*p*binary strings of length*n*that minimize the maximum Hamming distance between a string in*S*and the closest of the*p*strings; this minimum value is termed the*p*-*radius of**S*and is denoted by . The related*Hamming diameter**p**-clustering problem*(HDC) is to split*S*into*p*groups so that the maximum of the Hamming group diameters is minimized; this latter value is called the*p*-*diameter of**S*.We provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever*k*is constant. We also observe that HDC admits straightforward polynomial-time solutions when*k*=O(log*n*) and*p*=O(1), or when*p*=2. Next, by reduction from the corresponding geometric*p*-clustering problems in the plane under the*L*_{1}metric, we show that neither HRC nor HDC can be approximated within any constant factor smaller than two unless P=NP. We also prove that for any >0 it is NP-hard to split*S*into at most*pk*^{1/7−}clusters whose Hamming diameter does not exceed the*p*-diameter, and that solving HDC exactly is an NP-complete problem already for*p*=3. Furthermore, we note that by adapting Gonzalez' farthest-point clustering algorithm [T. Gonzalez, Theoret. Comput. Sci. 38 (1985) 293–306], HRC and HDC can be approximated within a factor of two in time O(*pkn*). Next, we describe a 2^{O(p/)}*k*^{O(p/)}*n*^{2}-time (1+)-approximation algorithm for HRC. In particular, it runs in polynomial time when*p*=O(1) and =O(log(*k*+*n*)). Finally, we show how to find inFull-size image

*L*of O(*p*log*k*) strings of length*n*such that for each string in*S*there is at least one string in*L*within distance (1+), for any constant 0<<1. 相似文献5.

We study a number of natural families of binary differential equations (BDE's) on a smooth surface in . One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE.

More generally, we consider a natural class of BDE's on such a surface , and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

6.

The Teichmüller space of a finite-type surface is considered.It is shown that Teichmüller distance is not

*C*^{2 + }forany > 0. Furthermore, Teichmüller distance is not*C*^{2+ g}for any gauge function*g*with . 2000*Mathematics Subject Classification*30F60. 相似文献7.

8.

The variational partial differential equation (PDE) approach for image denoising restoration leads to PDEs with nonlinear
and highly non-smooth coefficients. Such PDEs present convergence difficulties for standard multigrid methods. Recent work
on algebraic multigrid methods (AMGs) has shown that robustness can be achieved in general but AMGs are well known to be expensive
to apply. This paper proposes an accelerated algebraic multigrid algorithm that offers fast speed as well as robustness for
image PDEs. Experiments are shown to demonstrate the improvements obtained. 相似文献

9.

We consider a single machine scheduling problem with two min-sum objective functions: the sum of completion times and the sum of weighted completion times. We propose a simple polynomial time (1+(1/

*γ*),1+*γ*)-approximation algorithm, and show that for*γ*>1, there is no (*x*,*y*)-approximation with 1<*x*<1+(1/*γ*) and 1<*y*<1+(*γ*-1)/(2+*γ*). 相似文献10.

I. Biswas 《Topology》2006,45(2):403-419

Let

*X*be a nonsingular algebraic curve of genus*g*?3, and let*M*_{ξ}denote the moduli space of stable vector bundles of rank*n*?2 and degree*d*with fixed determinant*ξ*over*X*such that*n*and*d*are coprime. We assume that if*g*=3 then*n*?4 and if*g*=4 then*n*?3, and suppose further that*n*_{0},*d*_{0}are integers such that*n*_{0}?1 and*nd*_{0}+*n*_{0}*d*>*nn*_{0}(2*g*-2). Let*E*be a semistable vector bundle over*X*of rank*n*_{0}and degree*d*_{0}. The generalised Picard bundle*W*_{ξ}(*E*) is by definition the vector bundle over*M*_{ξ}defined by the direct image where*U*_{ξ}is a universal vector bundle over*X*×*M*_{ξ}. We obtain an inversion formula allowing us to recover*E*from*W*_{ξ}(*E*) and show that the space of infinitesimal deformations of*W*_{ξ}(*E*) is isomorphic to*H*^{1}(*X*,*End*(*E*)). This construction gives a locally complete family of vector bundles over*M*_{ξ}parametrised by the moduli space*M*(*n*_{0},*d*_{0}) of stable bundles of rank*n*_{0}and degree*d*_{0}over*X*. If (*n*_{0},*d*_{0})=1 and*W*_{ξ}(*E*) is stable for all*E*∈*M*(*n*_{0},*d*_{0}), the construction determines an isomorphism from*M*(*n*_{0},*d*_{0}) to a connected component*M*^{0}of a moduli space of stable sheaves over*M*_{ξ}. This applies in particular when*n*_{0}=1, in which case*M*^{0}is isomorphic to the Jacobian*J*of*X*as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over*J*, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles. 相似文献