A characterization of the optimal sets for self-similar measures with respect to the geometric mean error

2012 ◽  
Vol 138 (3) ◽  
pp. 201-225 ◽  
Author(s):  
Sanguo Zhu
Keyword(s):  
Geometric Mean ◽  
Mean Error ◽  
Self Similar ◽  
Geometric Mean Error

scholarly journals ASYMPTOTIC ORDER OF THE GEOMETRIC MEAN ERROR FOR SELF-AFFINE MEASURES ON BEDFORD–MCMULLEN CARPETS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050036
Author(s):  
SANGUO ZHU ◽  
SHU ZOU
Keyword(s):  
Hausdorff Dimension ◽  
Geometric Mean ◽  
Geometric Error ◽  
The Self ◽  
Separation Condition ◽  
Probability Vector ◽  
Mean Error ◽  
Affine Mappings ◽  
Geometric Mean Error ◽  
Vector Formula

Let [Formula: see text] be a Bedford–McMullen carpet associated with a set of affine mappings [Formula: see text] and let [Formula: see text] be the self-affine measure associated with [Formula: see text] and a probability vector [Formula: see text]. We study the asymptotics of the geometric mean error in the quantization for [Formula: see text]. Let [Formula: see text] be the Hausdorff dimension for [Formula: see text]. Assuming a separation condition for [Formula: see text], we prove that the [Formula: see text]th geometric error for [Formula: see text] is of the same order as [Formula: see text].

scholarly journals GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

2011 ◽  
Vol 07 (01) ◽  
pp. 173-202
Author(s):  
ROBERT CARLS
Keyword(s):  
Galois Theory ◽  
Theoretical Foundation ◽  
Abelian Varieties ◽  
Geometric Mean ◽  
Null Point ◽  
Point Counting ◽  
Characteristic 2 ◽  
Theoretic Characterization ◽  
Generalized Arithmetic

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

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