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Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions [article]

Kanji Inui, Hiroki Sumi
2022 arXiv   pre-print
We show the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal and the packing measure of the limit set with respect to the packing dimension of the limit  ...  Note that the Hausdorff measure of the limit set with respect to the Hausdorff dimension is zero.  ...  equals the Hausdorff dimension of the limit set and the Hausdorff measure of the limit set with respect to the Hausdorff dimension of the limit set is zero and the packing measure of the limit set with  ... 
arXiv:2204.04460v1 fatcat:qkz4dd3k75cg5lvosfesxkfudu

Hyperbolic geometry, continued fractions and classification of the finitely generated totally ordered dimension groups [article]

Igor Nikolaev
2014 arXiv   pre-print
We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line.  ...  The result is an extension of the Effros-Shen classification of the dicyclic dimension groups.  ...  Elliott for interesting discussions and the referee for helpful comments.  ... 
arXiv:math/0310400v4 fatcat:t7n7pylsrzbbbdpf347gqhy4bq

Geometry of fractional spaces [article]

Gianluca Calcagni
2013 arXiv   pre-print
ratio of their Hausdorff and spectral dimension is greater than one.  ...  We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions.  ...  Vacaru, and especially to G. Dunne and G. Nardelli for useful discussions. G. Nardelli is also credited for the construction of the fractional transform (5.24a)-(5.24b). Open Access.  ... 
arXiv:1106.5787v3 fatcat:4q5bgefujvc5pj7few2ykhoxwm

Discrete to continuum transition in multifractal spacetimes

Gianluca Calcagni
2011 Physical Review D  
The measure in the action is characterized by a varying Hausdorff dimension and logarithmic oscillations governed by a fundamental physical length.  ...  Thanks to general arguments from fractal geometry, this scenario explicitly realizes two indirect or conjectured features of most quantum gravity models: a change of effective spacetime dimensionality  ...  This measure is natural in fractional calculus of order α + iω and well approximates many features of fractal sets.  ... 
doi:10.1103/physrevd.84.061501 fatcat:lwboh7ixrfbehe27ugdy3vk7w4

Why We Can Not Walk To and Fro in Time as Do it in Space? (Why the Arrow of Time is Exists?) [article]

L.Ya.Kobelev
2000 arXiv   pre-print
Introduction 2.Universe as Time and Space with Fractional Dimensions 3. Why Time has Direction Only to Future and Why Impossible to Walk in Time To and Fro? 4.  ...  In this paper shown possibility to walk to and fro in space and necessity of huge amount of energy for stopping time and changing direction of it in microscopic volumes. Contents: 1.  ...  In this model the time and the space are real material fields with fractional dimensions and multifractal structure (multyfractal sets) defined on sets of their carriers of measure.  ... 
arXiv:physics/0011036v1 fatcat:5mxqxxaiyjdp3fedvdwzplgr74

The Theory of Gravitation in the Space - Time with Fractal Dimensions and Modified Lorents Transformations [article]

L.Ya.Kobelev
2000 arXiv   pre-print
dimensions of time and space.  ...  So the principle of relativity (it is exact when dimensions are integer) may be treated also as a good approximation and may remain valid (but modified) in case of small fractional corrections to integer  ...  In that case in the fractal theory of time and space the measure carrier must be defined as the Riemannian sets with integer dimensions.  ... 
arXiv:physics/0006029v1 fatcat:tfg3gqfxo5fwfm4mryf5nbswsm

Weyl law for fat fractals

María E Spina, Ignacio García-Mata, Marcos Saraceno
2010 Journal of Physics A: Mathematical and Theoretical  
We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.  ...  It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure.  ...  Acknowledgment This work was partially supported by CONICET and ANPCyT (PICT 25373).  ... 
doi:10.1088/1751-8113/43/39/392003 fatcat:grzw254fhfdz7ga6u4rzvi3tpi

Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

J. Neunhäuserer
2013 Journal of Mathematics  
We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers.  ...  Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.  ...  e �rst dimensional theoretical perspective on in�nite real continued fractions can be found in the work of Jarnik [2] , who introduced upper and lower estimates on the Hausdorff dimension of sets of  ... 
doi:10.1155/2013/754134 fatcat:74pgkuy2hjfwbgegtz7cp5wvfi

Geometry of fractional spaces

Gianluca Calcagni
2012 Advances in Theoretical and Mathematical Physics  
ratio of their Hausdorff and spectral dimension is greater than one.  ...  We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions.  ...  Vacaru, and especially to G. Dunne and G. Nardelli for useful discussions. He also thanks the town of Erlangen, where this work was so pleasantly completed. Open Access.  ... 
doi:10.4310/atmp.2012.v16.n2.a5 fatcat:td44nsj6grbsrhkcrh5ynv7v4e

Page 4497 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
The author shows how it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric.  ...  This fractional dimension is analogous to the Hausdorff dimension in R.  ... 

Introduction to multifractional spacetimes

Gianluca Calcagni
2012 arXiv   pre-print
Based on fractional calculus, these continuous spacetimes have their dimension changing with the scale; they display discrete symmetries in the ultraviolet and ordinary Poincar\'e symmetries in the infrared  ...  We informally review the construction of spacetime geometries with multifractal and, more generally, multiscale properties.  ...  Therefore, dimensional flow is always measured starting from the lowest of two scales ℓ n−1 to the next ℓ n , and relatively to the latter, which sets a gauge for the rods.  ... 
arXiv:1209.1110v1 fatcat:uzmak4mitzgkbma3tif7ju7n7y

Geometry and field theory in multi-fractional spacetime

Gianluca Calcagni
2012 Journal of High Energy Physics  
In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional.  ...  Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order.  ...  This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original  ... 
doi:10.1007/jhep01(2012)065 fatcat:q3yrd3jxfbb4zpmbtumct7cecm

Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion

S. M. Prigarin, K. Hahn, G. Winkler
2008 Numerical Analysis and Applications  
description and download Programs to estimate fractal dimension of 2D sets, MS Win binaries -read a brief description and download Programs to estimate fractal dimension of 3D sets.  ...  ., and Winkler, G. (2007) Evaluation of two numerical methods to measure the Hausdorff dimension of the fractional Brownian motion, Preprint 07-01, IBB, GSF Neuherberg, 28p.  ...  Below we present realizations of fractional Brownian motion: and fractional Brownian sheet: with the Hausdorff dimensions 2.2 (on the left) and 2.8 (on the right).  ... 
doi:10.1134/s1995423908020079 fatcat:n2akpxedivderi5z6mwkpapfbe

Scale Distributions and Fractal Dimensions in Turbulence

Haris J. Catrakis, Paul E. Dimotakis
1996 Physical Review Letters  
Level sets of 2D spatial measurements of jet-fluid concentration in turbulent jets are analyzed to compute scale distributions and fractal dimensions.  ...  In turbulent mixing and combustion, in particular, such measures are useful for estimating the volume-fill fraction of isosurfaces of species composition.  ... 
doi:10.1103/physrevlett.77.3795 pmid:10062310 fatcat:wgr2hyjwdjfrngsrfqckds43lu

On the Hausdorff dimension of a set of complex continued fractions

R. J. Gardner, R. D. Mauldin
1983 Illinois Journal of Mathematics  
connected to the geometric structure of this set, Gauss' measure.  ...  This can be seen by noting that the permutations r of N induce distinct mixing measures /o h, where , is Gauss' measure and h is the natural homeomorphism of N induced by r.  ...  (14) nally, combining (11), ( 12 ), ( 13 ) and ( 14 ), we obtain (10), as required. 1/2 + i/2 Figure 5 The Set F 2 in Lemma 2.6 I-lausdorff Dimension We now estimate the measure of our set of continued  ... 
doi:10.1215/ijm/1256046498 fatcat:37mwyjlstvhbzmk2tusvnlr6ki
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