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1.
Given a collection S of subsets of some set
and
the set cover problem is to find the smallest subcollection
that covers
that is,
where
denotes
We assume of course that S covers
While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually
Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance
exist, under a certain general condition: that for a random subset
and nondecreasing function f(·), there is a decomposition of the complement
into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|))
can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that
are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family
of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects,
and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in
and for guarding an x-monotone polygonal chain. 相似文献
2.
An affine pseudo-plane X is a smooth affine surface defined over
which is endowed with an
-fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic
-action on X and X is an
-surface, we shall show that the universal covering
is isomorphic to an affine hypersurface
in the affine 3-space
and X is the quotient of
by the cyclic group
via the action
where
and
It is also shown that a
-homology plane X with
and a nontrivial
-action is an affine pseudo-plane. The automorphism group
is determined in the last section. 相似文献
3.
In this paper we study the worst-case error (of numerical integration) on the unit sphere
for all functions in the unit ball of the Sobolev space
where
More precisely, we consider infinite sequences
of m(n)-point numerical integration rules
where: (i)
is exact for all spherical polynomials of degree
and (ii)
has positive weights or, alternatively to (ii), the sequence
satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration)
in
has the upper bound
where the constant c depends on s and d (and possibly the sequence
This extends the recent results for the sphere
by K. Hesse and I.H. Sloan to spheres
of arbitrary dimension
by using an alternative representation of the worst-case error. If the sequence
of numerical integration rules satisfies
an order-optimal rate of convergence is achieved. 相似文献
4.
Frame Decomposition of Decomposition Spaces 总被引:3,自引:0,他引:3
A new construction of tight frames for
with flexible time-frequency localization
is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on
a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition
on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space,
anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type
tight frames are constructed on
相似文献
5.
Daniel A. Klain 《Discrete and Computational Geometry》2006,36(3):457-477
Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in
We then show that continuous isometry-invariant simple valuations on polytopes in
for
are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in
hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant
valuations on convex polytopes and convex bodies in the hyperbolic plane
a partial characterization in
and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas,
containment theorems, and isoperimetric and Bonnesen-type inequalities. 相似文献
6.
7.
Let
denote the linear space over
spanned by
. Define the (real) inner product
, where V satisfies: (i) V is real analytic on
; (ii)
; and (iii)
. Orthogonalisation of the (ordered) base
with respect to
yields the even degree and odd degree orthonormal Laurent polynomials
, and
. Define the even degree and odd degree monic orthogonal Laurent polynomials:
and
. Asymptotics in the double-scaling limit
such that
of
(in the entire complex plane),
, and
(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a
matrix Riemann-Hilbert problem on
, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developed in [2],[3]. 相似文献
8.
Sadahiro Saeki 《Journal of Fourier Analysis and Applications》1995,2(1):15-28
Let
and
Under certain conditions on
we shall prove that
converges nontangentially to
at
for
相似文献
9.
A.J.E.M. Janssen 《Journal of Fourier Analysis and Applications》1994,1(4):403-436
Let
and let
In this paper we investigate the relation between the frame operator
and the matrix
whose entries
are given by
for
Here
, for any
We show that
is bounded as a mapping of
into
if and only if
is bounded as a mapping of
into
Also we show that
if and
only if
where
denotes the identity operator of
and
respectively, and
Next, when
generates a frame, we have that
has an upper frame bound, and the minimal dual function
can be computed as
The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and
Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case.
Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a
generating a frame are inherited by
In particular, we show that
when
generates a frame
Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr
for relating frame bound questions on complementary lattices by means of the Poisson summation formula. 相似文献
10.
Earl Berkson Oscar Blasco María J. Carro T.A. Gillespie 《Journal of Fourier Analysis and Applications》2006,12(4):447-481
We develop a general condition for automatically discretizing strong type bisublinear maximal estimates that arise in the
context of the real line. In particular, this method applies directly to Michael Lacey’s strong type boundedness results for
the bisublinear maximal Hilbert transform and for the bisublinear Hardy-Littlewood maximal operator, furnishing the counterpart
of each of these two results (without changes to the range of exponents) for the sequence spaces
We then take up some transference applications of discretized maximal bisublinear operators to maximal estimates and almost
everywhere convergence in Lebesgue spaces of abstract measures. We also broaden the scope of such applications, which are
based on transference from
by developing general methods for transplanting bisublinear maximal estimates from arbitrary locally compact abelian groups. 相似文献
11.
Pedro J. Miana 《Semigroup Forum》2006,73(1):61-74
In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but
not in groups) are given. We also give several applications in particular cases
and
相似文献
12.
We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights
and
on
, for which the a priori inequality
holds with constant independent of
. Here
is the Laplace-Beltrami operator on the sphere
. Due to the relation between
and the tangential component of the gradient,
, we obtain some "Morawetz-type" estimates for
on
. As a consequence we establish some new estimates for the free Schr?dinger propagator
, which may be viewed as certain refinements of the
-(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem
for certain time dependent first order spherical perturbations of the
dimensional Schr?dinger equation. 相似文献
13.
This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set
The energy density is of the same order as
where
is periodic, u is a vector-valued function in
and
The conductivity
is equal to 1 in the "hard" phases composed by
two by two disjoint-closure periodic sets while
tends uniformly to 0 in the "soft" phases composed by periodic thin layers which separate the hard phases. We prove that
the limit energy, according to γ-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of
order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases
is less than or equal to N and is obtained by evaluating the γ-limit of the rescaled energy of density
in the torus. Therefore, the homogenization result is achieved by a double γ-convergence procedure since the cell problem
depends on ε. 相似文献
14.
James H. Schmerl 《Discrete and Computational Geometry》2007,38(1):155-167
Suppose n ≥ 2 and
are such that X is infinite and Y is the set of vertices of an n-simplex. Then there is blue/red coloring of
such that no set similar to X is monochromatically blue and no set similar to Y is monochromatically red. 相似文献
15.
On the Least Median Square Problem 总被引:1,自引:0,他引:1
Jeff Erickson Sariel Har-Peled David M. Mount 《Discrete and Computational Geometry》2006,36(4):593-607
We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression
estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in
and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains
k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide
nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding
whether n given points in
are affinely non-degenerate requires Ω(nd) time. 相似文献
16.
C. Carton-Lebrun 《Journal of Fourier Analysis and Applications》1995,2(1):49-64
For
define
where
Pointwise estimates and weighted inequalities describing the local Lipschitz continuity
of
are established. Sufficient conditions are found
for the boundedness of
from
into
and a spherical restriction property is proved. A study of the moment subspaces of
is next developed in the one-variable case, for
locally integrable,
a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in
Characterizations are also given for each class. Applications related to the approximation and decomposition of
are discussed. 相似文献
17.
António M. Caetano Hans-Gerd Leopold 《Journal of Fourier Analysis and Applications》2006,12(4):427-445
The concept of local growth envelope
of the quasi-normed function space
is applied to the Triebel-Lizorkin spaces of generalized smoothness
In order to achieve this, a standardization result for these and corresponding Besov spaces is derived. 相似文献
18.
Rafal Latala Piotr Mankiewicz Krzysztof Oleszkiewicz Nicole Tomczak-Jaegermann 《Discrete and Computational Geometry》2007,38(1):29-50
We consider polytopes in
that are generated by N vectors in
whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random
polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur
distance between them is essentially of a maximal order n. This result is an analogue of the well-known Gluskin's result
for spherical vectors. We also study the norms of projections on such polytopes and prove an analogue of Gluskin's and Szarek's
results on basis constants. The proofs are based on a version of "small ball" estimates for linear images of random subgaussian
vectors. 相似文献
19.
A compact set
is staircase connected if every two points
can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone.
denotes the smallest number of edges of such a path.
is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the
r-th kernel is nonempty, compact and staircase connected provided
. 相似文献
20.
The explicit formulas for the sums of positive powers of the
integers
unrepresentable by the triple of integers
, are derived. 相似文献