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

2022
*
arXiv
*
pre-print

We show the packing

arXiv:2204.04460v1
fatcat:qkz4dd3k75cg5lvosfesxkfudu
*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 ...##
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Hyperbolic geometry, continued fractions and classification of the finitely generated totally ordered dimension groups
[article]

2014
*
arXiv
*
pre-print

We classify the polycyclic totally ordered simple

arXiv:math/0310400v4
fatcat:t7n7pylsrzbbbdpf347gqhy4bq
*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. ...##
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Geometry of fractional spaces
[article]

2013
*
arXiv
*
pre-print

ratio

arXiv:1106.5787v3
fatcat:4q5bgefujvc5pj7few2ykhoxwm
*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. ...##
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Discrete to continuum transition in multifractal spacetimes

2011
*
Physical Review D
*

The

doi:10.1103/physrevd.84.061501
fatcat:lwboh7ixrfbehe27ugdy3vk7w4
*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*. ...##
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Why We Can Not Walk To and Fro in Time as Do it in Space? (Why the Arrow of Time is Exists?)
[article]

2000
*
arXiv
*
pre-print

Introduction 2.Universe as Time

arXiv:physics/0011036v1
fatcat:5mxqxxaiyjdp3fedvdwzplgr74
*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*. ...##
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The Theory of Gravitation in the Space - Time with Fractal Dimensions and Modified Lorents Transformations
[article]

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*. ...

##
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Weyl law for fat fractals

2010
*
Journal of Physics A: Mathematical and Theoretical
*

We find that the

doi:10.1088/1751-8113/43/39/392003
fatcat:grzw254fhfdz7ga6u4rzvi3tpi
*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). ...##
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Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

2013
*
Journal of Mathematics
*

We prove upper

doi:10.1155/2013/754134
fatcat:74pgkuy2hjfwbgegtz7cp5wvfi
*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*...##
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Geometry of fractional spaces

2012
*
Advances in Theoretical and Mathematical Physics
*

ratio

doi:10.4310/atmp.2012.v16.n2.a5
fatcat:td44nsj6grbsrhkcrh5ynv7v4e
*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. ...##
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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. ...##
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Introduction to multifractional spacetimes

2012
*
arXiv
*
pre-print

Based on

arXiv:1209.1110v1
fatcat:uzmak4mitzgkbma3tif7ju7n7y
*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. ...##
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Geometry and field theory in multi-fractional spacetime

2012
*
Journal of High Energy Physics
*

In one

doi:10.1007/jhep01(2012)065
fatcat:q3yrd3jxfbb4zpmbtumct7cecm
*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 ...##
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Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion

2008
*
Numerical Analysis and Applications
*

description

doi:10.1134/s1995423908020079
fatcat:n2akpxedivderi5z6mwkpapfbe
*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). ...##
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Scale Distributions and Fractal Dimensions in Turbulence

1996
*
Physical Review Letters
*

Level

doi:10.1103/physrevlett.77.3795
pmid:10062310
fatcat:wgr2hyjwdjfrngsrfqckds43lu
*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. ...##
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On the Hausdorff dimension of a set of complex continued fractions

1983
*
Illinois Journal of Mathematics
*

connected to the geometric structure

doi:10.1215/ijm/1256046498
fatcat:37mwyjlstvhbzmk2tusvnlr6ki
*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 ...
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