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1.
Christophe Dupont 《Israel Journal of Mathematics》2012,192(2):505-533
Let f be a holomorphic endomorphism of ?? k . We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to log d k . We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of f. They in particular provide lower bounds for the Hausdorff dimension of the Julia set. 相似文献
2.
We shall present here results concerning the metric entropy of
spaces of linear and nonlinear approximation under very general conditions. Our
first result computes the metric entropy of the linear and m-terms
approximation classes according to a quasi-greedy basis verifying the
Temlyakov property. This theorem shows that the second index r is not visible
throughout the behavior of the metric entropy. However, metric entropy does
discriminate between linear and nonlinear approximation.
Our second result extends and refines a result
obtained in a Hilbertian framework by Donoho, proving that under orthosymmetry
conditions, m-terms approximation classes are characterized by the metric
entropy. Since these theorems are given under the general context of quasi-greedy
bases verifying the Temlyakov property, they have a large spectrum of
applications. For instance, it is proved in the last section that they can be
applied in the case of
L
p
norms for
R
d
for 1 < p < \infty.
We show that the lower bounds needed for this paper in fact follow
from quite simple
large deviation inequalities concerning hypergeometric or binomial distributions.
To prove the upper bounds, we provide a very simple universal coding
based on a thresholding-quantizing constructive procedure. 相似文献
3.
Yu. V. Malykhin 《Mathematical Notes》2012,91(5-6):800-807
We study the relationship between two characteristics of functional classes, pseudodimension and bracketing entropy. (Pseudodimension is a generalization of VC-dimension to classes of functions. Bracket entropy characterizes the L 1-error of one-sided approximation of a class by finite sets.) It is shown that classes of continuous functions with finite pseudodimension possess a finite bracketing ?-entropy for any ? > 0. We establish a general result concerning the relationship between the VC-dimension of classes of sets and their bracketing entropy. 相似文献
4.
In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d-dimensional unit cube. 相似文献
5.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(3):347-352
Let Td : L2([0, 1]d) → C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k−1 (log k)d− 1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(−Cɛ−2¦ log ɛ¦2d−1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms. 相似文献
6.
Ferenc Weisz 《Analysis Mathematica》2013,39(4):297-320
Under some weak conditions on θ, it was verified in [21, 17] that the maximal operator of the ? 1-θ-means of a tempered distribution is bounded from H p (? d ) to L p (? d ) for all d/(d + α) < p ≤ ∞, where 0 < α ≤ 1 depends only on θ. In this paper, we prove that the maximal operator is bounded from H d/(d+α)(? d ) to the weak L d/(d+α)(? d ) space. The analogous result is given for Fourier series, as well. Some special cases of the ? 1-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski and Riesz summations. 相似文献
7.
《Journal of Complexity》2001,17(2):467-492
We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the Lq(Td)-sense, where Td is the d-dimensional torus. The functions to be approximated are in the unit ball SBrp, θ of the mixed smoothness Besov space or in the unit ball SWrp of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)+ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SBrp, θ are both n−r(log n)(d−1)(r+1/2−1/θ). 相似文献
8.
The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form ???u =?cu p , with 0 < p < p s = (d + 2)/(d - 2), defined on bounded domains of ${{\mathbb{R}^d}, d \geq 3}$ , without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constants, as well as gradient bounds. 相似文献
9.
We give characterizations of radial Fourier multipliers as acting on radial L
p
functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of
corresponding results for general Hankel multipliers. Besides L
p
− L
q
bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include
results on interpolation of multiplier spaces.
G. Garrigós partially supported by grant “MTM2007-60952” and Programa Ramón y Cajal, MCyT (Spain). A. Seeger partially supported by NSF grant DMS 0652890. 相似文献
10.
Lin Feng Wang 《Annali di Matematica Pura ed Applicata》2012,191(3):539-550
Let M be an n-dimensional closed manifold with metric g, dμ = e h(x) dV(x) be the weighted measure and ? μ, p be the weighted p-Laplacian. In this article, we get the lower bound estimate of the first nonzero eigenvalue for the weighted p-Laplacian when the m-dimensional Bakry-émery curvature has a positive lower bound. 相似文献
11.
K. J. Swanepoel 《Discrete and Computational Geometry》1999,21(3):437-447
We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) in the d -dimensional Banach spaces
p
d
independent of d . This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d [19]. Our upper bounds follow from characterizations of singularities of SMTs due to Lawlor and Morgan [14], which we extend,
and certain
p
-inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d -dimensional Banach space (i.e. Minkowski space); the same upper bound for Steiner points having been found by Lawlor and
Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1 -summing norms.
Received April 22, 1997, and in revised form October 1, 1997. 相似文献
12.
We obtain a Möbius characterization of the n-dimensional spheres S n endowed with the chordal metric d 0. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S n , d 0) for some n ≥ 1. 相似文献
13.
Ferenc Weisz 《Journal of Mathematical Analysis and Applications》2011,379(2):910-929
A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L1(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations. 相似文献
14.
In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu 0; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW 0 1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular. 相似文献
15.
Cristian Conde 《Journal of Mathematical Analysis and Applications》2009,356(2):664-673
We study the geometry of the set Δp, with 1<p<∞, which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of B(H). These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920-931], we introduced the metric dp and exposed several results about this metric space. The aim of this work is to prove that the space (Δp,dp) behaves in many senses like a nonpositive curvature metric space. 相似文献
16.
I. Ya. Aref’eva 《Proceedings of the Steklov Institute of Mathematics》2014,285(1):26-29
We consider two holographically related theories. As the first (d + 1)-dimensional theory, we consider a model in which the (d + 1)-dimensional space is the direct product of ? d and the half-axis ?+ and in which the kinetic operator has a nonlocal term induced by the nonlocal kinetic operator of the p-adic effective action. It turns out that the kinetic operator in the second, holographically related, d-dimensional theory is the kinetic operator of the string field theory effective action. 相似文献
17.
Let S
d-1 denote the (d − 1)-dimensional unit sphere centered at the origin of the d-dimensional Euclidean space. Let 0 < α < π. A set P of points in S
d-1 is called almost α-equidistant if among any three points of P there is at least one pair lying at spherical distance α. In
this note we prove upper bounds on the cardinality of P depending only on d.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
18.
Christian Borgs Jennifer T. Chayes Prasad Tetali 《Probability Theory and Related Fields》2012,152(3-4):509-557
We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice ${\mathbb{Z}^{d}}$ —heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2. 相似文献
19.
Huaiqian Li 《中国科学 数学(英文版)》2018,61(3):487-510
We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, N)(equivalently, RCD~*(0, N), condition with N∈N\ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the(minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al.(2016). 相似文献
20.
Ian Doust 《Journal of Functional Analysis》2008,254(9):2336-2364
A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non-trivial 1-negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree (T,d) must have strict p-negative type for all p in an open interval (1−ζ,1+ζ), where ζ>0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T. In particular, if the edges of the tree are not weighted, then it follows that ζ depends only upon the number of vertices in the tree.We also give an example of an infinite metric tree that has strict 1-negative type but does not have p-negative type for any p>1. This shows that the maximal p-negative type of a metric space can be strict. 相似文献