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Lowness for effective Hausdorff dimension

Steffen Lempp, Joseph S. Miller, Keng Meng Ng, Daniel D. Turetsky, Rebecca Weber
2014 Journal of Mathematical Logic  
We examine the sequences A that are 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)].  ... 
doi:10.1142/s0219061314500111 fatcat:yqpp7nx6urhm3gj7xcxs542noa

Spheres, Polytopes, Neighbours, Paths and Trees: Ecological and Morphometric Aspects of Multidimensional Space

J. T. Temple
1996 The Paleontological Society Special Publications  
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,  ...  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,  ... 
doi:10.1017/s2475262200003890 fatcat:pcff2knbhrbirjra5yllwxwe4e

The Strength of the Besicovitch-Davies Theorem [chapter]

Bjørn Kjos-Hanssen, Jan Reimann
2010 Lecture Notes in Computer Science  
A theorem of Besicovitch and Davies implies 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.  ... 
doi:10.1007/978-3-642-13962-8_26 fatcat:3vm2icfh7neztcc4cbuqzcau3m

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

Wellington da Cruz
2003 Chaos, Solitons & Fractals  
We consider Farey series of rational numbers in terms of fractal sets labeled by the 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.  ... 
doi:10.1016/s0960-0779(02)00580-5 fatcat:gjutaht5grcklobsig6aj7cge4

The detection of cracks in beams using chaotic excitations

J. Ryue, P.R. White
2007 Journal of Sound and Vibration  
The measures introduced in this work are the halfspace correlation 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.  ... 
doi:10.1016/j.jsv.2007.06.043 fatcat:zzltv35wszbpbaa436ywk3hxzy

Dimension Estimation Using Weighted Correlation Dimension Method

Yuanhong Liu, Zhiwei Yu, Ming Zeng, Shun Wang
2015 Discrete Dynamics in Nature and Society  
Among all the 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  ... 
doi:10.1155/2015/837185 fatcat:ojdqwmyhoberxainzkoksvvpyu

Use the information dimension, not the Hausdorff [article]

A. J. Roberts
2005 arXiv   pre-print
In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised 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 .  ... 
arXiv:nlin/0512014v1 fatcat:ol77yzlborffvipvreo4nuomyq

Hausdorff dimension of repellors in low sensitive systems

A.E. Motter, P.S. Letelier
2000 Physics Letters A  
Methods to estimate the 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.  ... 
doi:10.1016/s0375-9601(00)00674-5 fatcat:cvw5efzqq5b6vjailusg4gsdze

Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors

Holger Kantz, Celso Grebogi, Awadhesh Prasad, Ying-Cheng Lai, Erik Sinde
2002 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We study the 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  ... 
doi:10.1103/physreve.65.026209 pmid:11863634 fatcat:jdkdnx32vjdota23p7ft4ivybq

Hausdorff dimension of a particle path in a quantum manifold

Piero Nicolini, Benjamin Niedner
2011 Physical Review D  
We show that the 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.  ... 
doi:10.1103/physrevd.83.024017 fatcat:mcibyla6ybfztah7f3sisxkglq

Modified Hausdorff Fractal Dimension (MHFD) [article]

Reza Farrahi Moghaddam, Mohamed Cheriet
2015 arXiv   pre-print
The 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) .  ... 
arXiv:1505.03493v2 fatcat:n5o73k23vvg7vor2aytxryc4hm

ON THE NUMERICAL STUDY OF THE COMPLEXITY AND FRACTAL DIMENSION OF CMB ANISOTROPIES

A. E. ALLAHVERDYAN, V. G. GURZADYAN, A. A. SOGHOYAN
1999 International Journal of Modern Physics D  
The correlation of complexity of the anisotropy spots with their fractal 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.  ... 
doi:10.1142/s0218271899000298 fatcat:pdrnyjhmyzdiddrwdruetkcfne

Connectedness properties of dimension level sets

Daniel Turetsky
2011 Theoretical Computer Science  
We prove that the set of all points of 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 .  ... 
doi:10.1016/j.tcs.2011.03.006 fatcat:zzbnsg5ez5brrpgtuj6bvxvbzu

Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future

A.A. Potapov
2017 Universal Journal of Physics and Application  
The author has been investigating these issues 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  ... 
doi:10.13189/ujpa.2017.110103 fatcat:ek2fw54gyjg7hlocjrlj4ztfee

Effective Fractal Dimension in Algorithmic Information Theory [chapter]

Elvira Mayordomo
2008 New Computational Paradigms  
I thank an anonymous referee, Philippe Moser, and David Doty 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.  ... 
doi:10.1007/978-0-387-68546-5_12 fatcat:s2dznipmm5cn5oc7siyfl3u2i4
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