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Lowness for effective Hausdorff dimension

2014
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Journal of Mathematical Logic
*

We examine the sequences A that are

doi:10.1142/s0219061314500111
fatcat:yqpp7nx6urhm3gj7xcxs542noa
*low**for**dimension*, i.e., those*for*which the*effective*(*Hausdorff*)*dimension*relative to A is the same as the unrelativized*effective**dimension*. ...*Lowness**for**dimension*is a weakening of*lowness**for*randomness, a central notion in*effective*randomness. ... We say that A ∈ 2 ω is*low**for*(*effective**Hausdorff*)*dimension*if (∀X ∈ 2 ω )[dim A (X) ≥ dim(X)]. ...##
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Spheres, Polytopes, Neighbours, Paths and Trees: Ecological and Morphometric Aspects of Multidimensional Space

1996
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The Paleontological Society Special Publications
*

The size of such polytopes depends on the

doi:10.1017/s2475262200003890
fatcat:pcff2knbhrbirjra5yllwxwe4e
*effective**dimensionality*of the environment: in ecospace of*low**dimensions*(rigorous or peripheral environments) the niches are large, their centres far apart, ... Thẽ earest-neighbour configuration of the*Hausdorff*metric is the most stable during increase in*dimensions*. ... The size of such polytopes depends on the*effective**dimensionality*of the environment: in ecospace of*low**dimensions*(rigorous or peripheral environments) the niches are large, their centres far apart, ...##
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The Strength of the Besicovitch-Davies Theorem
[chapter]

2010
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Lecture Notes in Computer Science
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A theorem of Besicovitch and Davies implies

doi:10.1007/978-3-642-13962-8_26
fatcat:3vm2icfh7neztcc4cbuqzcau3m
*for*Cantor space 2 ω that each Σ 1 1 (analytic) class of positive*Hausdorff**dimension*contains a Π 0 1 (closed) subclass of positive*dimension*. ...*dimension*. ... One way to conceive of weak randomness is in terms of*effective**Hausdorff**dimension*. ...##
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The Hausdorff dimension of fractal sets and fractional quantum Hall effect

2003
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Chaos, Solitons & Fractals
*

We consider Farey series of rational numbers in terms of fractal sets labeled by the

doi:10.1016/s0960-0779(02)00580-5
fatcat:gjutaht5grcklobsig6aj7cge4
*Hausdorff**dimension*with values defined in the interval 1 < h < 2 and associated with fractal curves. ... Our results come from the observation that the fractional quantum Hall*effect*-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. ... We have also the possibility of fractal sets with irrational values*for*the*Hausdorff**dimension*. ...##
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The detection of cracks in beams using chaotic excitations

2007
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Journal of Sound and Vibration
*

The measures introduced in this work are the halfspace correlation

doi:10.1016/j.jsv.2007.06.043
fatcat:zzltv35wszbpbaa436ywk3hxzy
*dimension*, which is the correlation*dimension*measured on one-half of the attractor, and the*Hausdorff*distance between the two attractors ...*For*these two attractor-based measures, their variations are investigated with respect to crack size to find whether they are appropriate crack indicators or not. r ... In Fig. 6 , the*Hausdorff*distance decreases in the*low*-frequency region. ...##
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Dimension Estimation Using Weighted Correlation Dimension Method

2015
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Discrete Dynamics in Nature and Society
*

Among all the

doi:10.1155/2015/837185
fatcat:ojdqwmyhoberxainzkoksvvpyu
*dimension*estimation methods, correlation*dimension*(CD) method is one of the most popular ones, which always assumes that the*effect*of every point on the intrinsic*dimension*estimation ... Intrinsic*dimension*estimated by the high density area is more reliable than the ones estimated by the*low*density or boundary area. ...*For*projection approach, the first step is to extract a*low*-*dimensional*representation from a high-*dimensional*space; then the representation is analyzed and the*dimension*is estimated 2 Discrete Dynamics ...##
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Use the information dimension, not the Hausdorff
[article]

2005
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arXiv
*
pre-print

In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised

arXiv:nlin/0512014v1
fatcat:ol77yzlborffvipvreo4nuomyq
*dimensions*D_q of order q≤0, including the*Hausdorff**dimension*, are*effectively*... The reason is that these*dimensions*are extraordinarily sensitive to regions of*low*density in the multi-fractal data. ... Observe that the predicted*dimensions**for*positive q (*low*α) are quite good*for*all realisations, especially near the information*dimension*, D 1 . ...##
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Hausdorff dimension of repellors in low sensitive systems

2000
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Physics Letters A
*

Methods to estimate the

doi:10.1016/s0375-9601(00)00674-5
fatcat:cvw5efzqq5b6vjailusg4gsdze
*Hausdorff**dimension*of invariant sets of scattering systems are presented. ... The d−*dimensional**Hausdorff*measure K(d) is supposed to be the limit of K n (d)*for*n → ∞. K(d) is infinite*for*d less than the*Hausdorff**dimension*D H , and is zero*for*d greater than D H [1] . ... In computing the uncertainty*dimension*of the singularities, the uncertainty method*effectively*computes the box-counting*dimension*of the set of all discontinuities. ...##
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Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors

2002
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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
*

We study the

doi:10.1103/physreve.65.026209
pmid:11863634
fatcat:jdkdnx32vjdota23p7ft4ivybq
*effects*of noise on chaotic attractors with these nonhyperbolic behaviors by investigating the scaling laws*for*the*Hausdorff*distance between the noisy and the deterministic attractors. ... We compare two sources of nonhyperbolicity: ͑1͒ tangencies between stable and unstable manifolds, and ͑2͒ unstable*dimension*variability. ...*For**low*-*dimensional*chaotic systems ͑systems with only one unstable direction͒ that have quadratic tangencies, the shadowing time T(⑀) scales algebraically with the noise amplitude ⑀ as T(⑀)ϳ⑀ Ϫ1/2 ͓1͔ ...##
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Hausdorff dimension of a particle path in a quantum manifold

2011
*
Physical Review D
*

We show that the

doi:10.1103/physrevd.83.024017
fatcat:mcibyla6ybfztah7f3sisxkglq
*Hausdorff**dimension*accounts*for*both the quantum mechanics uncertainty and manifold fluctuations. ... After recalling the concept of the*Hausdorff**dimension*, we study the fractal properties of a quantum particle path. ... P.N. and B.N. would like to thank the Perimeter Institute*for*Theoretical Physics, Waterloo, ON, Canada*for*the kind hospitality during the final period of work on this project. ...##
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Modified Hausdorff Fractal Dimension (MHFD)
[article]

2015
*
arXiv
*
pre-print

The

arXiv:1505.03493v2
fatcat:n5o73k23vvg7vor2aytxryc4hm
*Hausdorff*fractal*dimension*has been a fast-to-calculate method to estimate complexity of fractal shapes. ... The modified*Hausdorff*fractal*dimension*stands on two features that weaken the requirement of presence of a shape and also reduce the impact of the noise possibly presented in the input image. ... A less sever*effect*is expected*for*lower resolutions. This phenomenon, which can be imagined as a 'deflection'*effect*and shown inFigure 4, would weaken applicability of the*Hausdorff*model (1) . ...##
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ON THE NUMERICAL STUDY OF THE COMPLEXITY AND FRACTAL DIMENSION OF CMB ANISOTROPIES

1999
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International Journal of Modern Physics D
*

The correlation of complexity of the anisotropy spots with their fractal

doi:10.1142/s0218271899000298
fatcat:pdrnyjhmyzdiddrwdruetkcfne
*dimension*is revealed as well. ... Thus, we demonstrate the calculability of such an abstract descriptor as the Kolmogorov complexity*for*CMB digitized maps. ... We start from the brief account of*effect*of geodesic mixing, the concepts of Kolmogorov complexity and the*Hausdorff**dimension*. ...##
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Connectedness properties of dimension level sets

2011
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Theoretical Computer Science
*

We prove that the set of all points of

doi:10.1016/j.tcs.2011.03.006
fatcat:zzbnsg5ez5brrpgtuj6bvxvbzu
*effective**Hausdorff**dimension*1 in R n (n ≥ 2) is connected, and simultaneously that the complement of this set is not path-connected when n = 2. ... The author also thanks Joseph Miller*for*insightful conversation, the anonymous referee*for*corrections and comments, and Victoria University of Wellington*for*its hospitality. ...*For*example, as shown in [1] , if X is an arbitrary union of Π 0 1 classes, then the classical*Hausdorff**dimension*of X is the supremum of the*effective**Hausdorff**dimensions*of the elements of X . ...##
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Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future

2017
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Universal Journal of Physics and Application
*

The author has been investigating these issues

doi:10.13189/ujpa.2017.110103
fatcat:ek2fw54gyjg7hlocjrlj4ztfee
*for*exactly 35 years and has obtained results of the big scientific and practical worth. ... The main ideas and strategic directions in synthesis of fundamentally new topological radar detectors of*low*-contrast targets / objects have been considered. ... Acknowledgements This work was supported in part by the project of International Science and Technology Center No. 0847.2 (2000 -2005, USA), Russian Foundation*for*Basic Research (projects №№ 05-07-90349 ...##
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Effective Fractal Dimension in Algorithmic Information Theory
[chapter]

2008
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New Computational Paradigms
*

I thank an anonymous referee, Philippe Moser, and David Doty

doi:10.1007/978-0-387-68546-5_12
fatcat:s2dznipmm5cn5oc7siyfl3u2i4
*for*many helpful suggestions. ... But in fact much more is true*for*certain classes, as Hitchcock shows in [24] .*For*sets that are*low*in the arithmetical hierarchy, constructive*dimension*and*Hausdorff**dimension*coincide. ... The quantities H s (X) and P s (X) -which may be infinite -are called the s-*dimensional**Hausdorff*(outer) ball measure and the s-*dimensional*packing (outer) ball measure of X, respectively. ...
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