Topic
Singular measure
About: Singular measure is a research topic. Over the lifetime, 138 publications have been published within this topic receiving 1569 citations.
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TL;DR: In this article, the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami, has been considered and it has been shown that general nonlinear functions do not operate on the domain of Δ.
152 citations
TL;DR: In this article, lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models were established, and it was shown that there are close connections between manifold estimation and the problem of deconvolving a singular measure.
Abstract: We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.
78 citations
TL;DR: Lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models are found and it is shown that there are close connections between manifold estimation and the problem of deconvolving a singular measure.
Abstract: We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.
68 citations
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/legal.html) implique l'accord avec les conditions generales d’utilisation, i.e., usage commerciale ou impression systématique, constitutive of an infraction pénale.
Abstract: © Bulletin de la S. M. F., 1978, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
65 citations
TL;DR: An analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure and it is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system.
Abstract: In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Henon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.
61 citations