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1.
Haibo Chen Zhixiong Wen Ying Xiong 《Journal of Mathematical Analysis and Applications》2008,340(2):959-967
For level sets related to the tangential dimensions of Bernoulli measures, the Hausdorff and packing dimensions are determined. Our results extend some classical work of Besicovitch and Eggleston. 相似文献
2.
Norbert Patzschke 《Advances in Applied Mathematics》1997,19(4):486-513
A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure. 相似文献
3.
HUA SU 《数学年刊B辑(英文版)》2000,(4)
1. IntroductionThe self-similar sets (SSS) is one of the most important fractal classes, but the most properties such as dimensions, measures'' have been established upon the open set condition(OSC). It is a difficult problem to determine the structure and only a few results are knownwhen this condition is absent. On the other hand, for the graph-directed sets (GDS), ageneralization of SSS, if the OSC is satisfied, then analogous properties of the self-similarsets will hold still. The m… 相似文献
4.
5.
LOU ManLi & WU Min 《中国科学 数学(英文版)》2010,(5)
In this paper,we get the formulas of upper(lower) pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.We also obtain formulas for the dimension of the Moran measures.Our results extend the known results of some self-similar measures and Moran measures studied by Cawley and Mauldin. 相似文献
6.
Some kinds of the self-similar sets with overlapping structures are studied by introducing the graph-directed constructions
satisfying the open set condition that coincide with these sets. In this way, the dimensions and the measures are obtained.
Project supported by the National Natural Science Foundation of China and Mathematics Center of Morningside, Chinese Academy
Sciences. 相似文献
7.
Recently Lou and Wu obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.In this paper,we prove that the result is still true under the open set condition.Due to the lack of the strong separation condition,our approach is essentially different from that used by Lou and Wu.We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples. 相似文献
8.
Quasi-Monte Carlo (QMC) methods have been successfully used to compute high-dimensional integrals arising in many applications, especially in finance. To understand the success and the potential limitation of QMC, this paper focuses on quality measures of point sets in high dimensions. We introduce the order-?, superposition and truncation discrepancies, which measure the quality of selected projections of a point set on lower-dimensional spaces. These measures are more informative than the classical ones. We study their relationships with the integration errors and study the tractability issues. We present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the Sobol’ nets with that of the sets of Latin hypercube sampling and random points. Numerical results show that in high dimensions the superiority of the Sobol’ nets mainly derives from the one-dimensional projections and the projections associated with the earlier dimensions; for order-2 and higher-order projections all these point sets have similar behavior (on the average). In weighted cases with fast decaying weights, the Sobol’ nets have a better performance than the other two point sets. The investigation enables us to better understand the properties of QMC and throws new light on when and why QMC can have a better (or no better) performance than Monte Carlo for multivariate integration in high dimensions. 相似文献
9.
Jinjun Li 《Nonlinear Analysis: Real World Applications》2012,13(1):348-358
Dai and Liu obtained the formula of local dimensions of some Moran measures on Moran sets in Rd under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different to that used by Dai and Liu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples. 相似文献
10.
This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a
central subject in convex geometry and also represent an important tool in related fields. We show that these measures are
absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives
explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies
(sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions.
Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates
the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and
surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K.
Received: 15 July 1999 相似文献