共查询到20条相似文献,搜索用时 203 毫秒
1.
Sanguo Zhu 《Monatshefte für Mathematik》2012,259(1):291-305
We study the quantization with respect to the geometric mean error for probability measures μ on for which there exist some constants C, η > 0 such that for all ε > 0 and all . For such measures μ, we prove that the upper quantization dimension of μ is bounded from above by its upper packing dimension and the lower one is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite. 相似文献
2.
We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all ε > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite. 相似文献
3.
SanGuo Zhu 《中国科学 数学(英文版)》2016,59(2):337-350
We study the quantization for in-homogeneous self-similar measures μ supported on self-similar sets.Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for μ of order r ∈(0, ∞) and determine its exact value ξ_r. Furthermore, we show that,the ξ_r-dimensional lower quantization coefficient for μ is always positive and the upper one can be infinite. A sufficient condition is given to ensure the finiteness of the upper quantization coefficient. 相似文献
4.
We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that νμ implies . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal. 相似文献
5.
Wolfgang Kreitmeier 《Mathematische Nachrichten》2008,281(9):1307-1327
For a large class of dyadic homogeneous Cantor distributions in ?, which are not necessarily self‐similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non‐existence of the quantization coefficient. The class contains all self‐similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
6.
Wolfgang Kreitmeier 《Journal of Mathematical Analysis and Applications》2008,342(1):571-584
For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error. 相似文献
7.
《Mathematische Nachrichten》2017,290(5-6):827-839
We study the asymptotic quantization error for Markov‐type measures μ on a class of ratio‐specified graph directed fractals E . Assuming a separation condition for E , we show that the quantization dimension for μ of order r exists and determine its exact value in terms of spectral radius of a related matrix. We prove that the ‐dimensional lower quantization coefficient for μ is always positive. Moreover, we establish a necessary and sufficient condition for the ‐dimensional upper quantization coefficient for μ to be finite. 相似文献
8.
Wei Jiaqun 《Czechoslovak Mathematical Journal》2006,56(2):773-780
In this note we show that for a *n-module, in particular, an almost n-tilting module, P over a ring R with A = EndR
P such that P
A
has finite flat dimension, the upper bound of the global dimension of A can be estimated by the global dimension of R and hence generalize the corresponding results in tilting theory and the ones in the theory of *-modules. As an application,
we show that for a finitely generated projective module over a VN regular ring R, the global dimension of its endomorphism ring is not more than the global dimension of R. 相似文献
9.
《Journal of Pure and Applied Algebra》2024,228(2):107468
Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite. 相似文献
10.
11.
We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension. 相似文献
12.
Sanguo Zhu 《Journal of Mathematical Analysis and Applications》2008,338(1):742-750
Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied. 相似文献
13.
Sanguo Zhu 《Mathematische Zeitschrift》2008,259(1):33-43
Let μ be the attracting measure of a condensation system associated with a self-similar measure ν. We determine the upper and lower quantization dimension of μ under the strong separation condition.
相似文献
14.
Sanguo Zhu 《Monatshefte für Mathematik》2011,162(3):355-374
Let??? be a self-affine measure on a general Sierpi??ski carpet E. We give a characterization for the upper and lower quantization dimension of??? in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D r (??) of??? exists for all r > 0 under an additional condition. We establish an explicit formula for D r (??) and show that it increases to the box-counting dimension ${dim_B^* \mu}$ of??? as r tends to infinity. For a class of Sierpi??ski carpets E and the uniform measures??? on E, we show that the quantization dimension of??? coincides with its box-counting dimension and that the D r (??)-dimensional upper and lower quantization coefficient of??? are both positive and finite. 相似文献
15.
16.
V. P. Maslov 《Theoretical and Mathematical Physics》2007,150(1):102-122
From the standpoint of thermodynamic averaging of fission microprocesses, we investigate the origin of radioactive release
in an NPP after an accident or after resource depletion. The genesis of the NPP release is interpreted as a new thermodynamic
phenomenon, a zeroth-order phase transition. This problem setting results in a problem in probabilistic number theory. We
prove the corresponding theorem leading to quantization of the Zipf law for the frequency of a zeroth-order phase transition
with different values of the jump of the Gibbs thermodynamic potential. We introduce the notion of hole dimension.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 118–142, January, 2007. 相似文献
17.
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. 相似文献
18.
Sanguo Zhu 《Chaos, solitons, and fractals》2012,45(11):1437-1443
Given a finite set of patterns, we consider the Moran sets determined by using each of these patterns with a prescribed frequency. For certain infinite product measures μ on such Moran sets, we determine the exact values of the quantization dimensions Dr(μ). We give various sufficient conditions for the Dr(μ)-dimensional upper quantization coefficient and the lower one to be positive and finite. We also construct an example to illustrate our main result. 相似文献
19.
M. Davoudian 《代数通讯》2013,41(9):3907-3917
We introduce and study the concept of dual perfect dimension which is a Krull-like dimension extension of the concept of acc on finitely generated submodules. We observe some basic facts for modules with this dimension, which are similar to the basic properties of modules with Noetherian dimension. For Artinian serial modules, we show that these two dimensions coincide. Consequently, we prove that the Noetherian dimension of non-Noetherian Artinian serial modules over the rings of the title is 1. 相似文献
20.
Serena Doria 《Annals of Operations Research》2012,195(1):33-48
A model of coherent upper conditional prevision for bounded random variables is proposed in a metric space. It is defined
by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff
outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero
or infinity in its Hausdorff dimension, it is defined by a 0–1 valued finitely, but not countably, additive probability. If
the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension it is proven that a coherent
upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability
defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from
below. 相似文献