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Hausdorff dimension in exponential time

*
Proceedings 16th Annual IEEE Conference on Computational Complexity
*

*In*this paper we investigate effective versions of

*Hausdorff*

*dimension*which have been recently introduced by Lutz. We focus on

*dimension*

*in*the class E of sets computable

*in*linear

*exponential*

*time*. ... By a new general invariance theorem for resource-bounded

*dimension*we show that the class of pm-complete sets for E has

*dimension*1

*in*E. ...

*Hausdorff*

*Dimension*

*in*

*Exponential*

*Time*Definition 1 A betting strategy b is a function b : {0, 1} * → [0, 1] × {0, 1}. ...

##
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Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

2004
*
Physica D : Non-linear phenomena
*

Two-

doi:10.1016/j.physd.2003.09.009
fatcat:z7drnyiqlbcxxfnh46t6cxl7gq
*dimensional*random Lorentz gases with absorbing traps are considered*in*which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. ...*In*systems of finite spatial extension, the asymptotic decay of the survival probability is*exponential*and characterized by an escape rate, which can be related to the average positive Lyapunov exponent ... ε leads to an integer value of the*Hausdorff**dimension*D H . ...##
###
Page 5346 of Mathematical Reviews Vol. , Issue 96i
[page]

1996
*
Mathematical Reviews
*

*In*many cases, including all the examples quoted so far, the global attractor has finite

*Hausdorff*

*dimension*: this fact would allow us, ’ 34 ORDINARY DIFFERENTIAL EQUATIONS 5346

*in*principle, to reduce ... Indeed, most of the ba- sic notions and results

*in*the theory of the long-

*time*behavior of infinite-

*dimensional*dissipative dynamical systems trace their ori- gin

*in*the study of these equations, and have ...

##
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Dynamics of fractal dimension during phase ordering of a geometrical multifractal

2000
*
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
*

It is shown that at early

doi:10.1103/physreve.62.1764
pmid:11088637
fatcat:lie47gvdvje5tps2jhiv5pmgki
*times*the*Hausdorff**dimension*does not change with*time*, whereas at late*times*its dynamics follow the f(α) curve. ... A simple multifractal coarsening model is suggested that can explain the observed dynamical behavior of the fractal*dimension**in*a wide range of coarsening fractal systems. ...*In*the previous*time*step τ = m − 1 the object consists of two subsets : S m (m) with*Hausdorff**dimension*f (α(m)), and S m (m − 1) with*Hausdorff**dimension*f (α(m − 1)). ...##
###
Page 5736 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

Even more, is the

*Hausdorff**dimension*of the repeller strictly less than the*dimension*of the ambient manifold? ...*In*this article the authors consider a general abstract model, called piecewise smooth maps with holes, and show that the*Hausdorff**dimension*of the repeller of the map is strictly less than the*dimension*...##
###
ON THE NUMERICAL STUDY OF THE COMPLEXITY AND FRACTAL DIMENSION OF CMB ANISOTROPIES

1999
*
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. ... We consider the problem of numerical computation of the Kolmogorov complexity and the fractal*dimension*of the anisotropy spots of Cosmic Microwave Background (CMB) radiation. ...*Hausdorff**dimension*The association of local*exponential*instability and chaos with fractals is also well known (see e.g. [19] ). ...##
###
On the dimensions of a family of overlapping self-affine carpets

2015
*
Ergodic Theory and Dynamical Systems
*

We compute the

doi:10.1017/etds.2015.21
fatcat:gqvtz5rj45drxaywqskkydcboe
*Hausdorff*, packing and box*dimensions*outside of a small set of exceptional translations, and also for some explicit translations even*in*the presence of overlapping. ... As such, we maintain one of the key features*in*the Bedford–McMullen set-up*in*that alignment causes the*dimensions*to drop from the affinity*dimension*. ... There exists a set E ⊂ [0, 1 − 1/m] D of*Hausdorff*and packing*dimension*|D| − 1 (*in*particular of zero |D|-*dimensional*Lebesgue measure) such that ...##
###
Fractal-Based Electrolytic Capacitor Electrodes: Scaling Behavior with Respect to Fractal Order and Complexity
[article]

2018
*
arXiv
*
pre-print

An interesting

arXiv:1810.00221v1
fatcat:i4ckciimwneaxm5pj7hj2sf3aq
*exponential*relationship between capacitance and fractal order for the more open structured fractals was observed, the exponent of which was proportional to the*Hausdorff**dimension*. ... to be examined for the first*time*. ... The*Hausdorff**dimension*is shown*in*Figure 1a for each fractal investigated*in*this work. ...##
###
The Hausdorff Dimension and Capillary Imbibition

2022
*
Fractal and Fractional
*

It is established that the actual distance of fluid travel

doi:10.3390/fractalfract6060332
doaj:4ff1cb95ef214aba8f4cf4744130432c
fatcat:obuhh2qwufbqbpggevatduwns4
*in*self-avoiding fractals at the first stage of imbibition is*in*the Washburn regime, whereas at the second stage it is associated with the*Hausdorff*...*dimension*dH. ... The equilibrium height is reached at different*times*(Figure 3a ), which depends on the*Hausdorff**dimension*of the capillary (*exponential*trendline shown*in*Figure 3b ). ...##
###
Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors

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

Whereas

doi:10.1103/physreve.65.026209
pmid:11863634
fatcat:jdkdnx32vjdota23p7ft4ivybq
*in*the presence of tangencies, interactive noise yields attractor deformations, attractors with only*dimension*variability are robust, despite the fact that shadowing is grossly violated. ... Chaotic attractors arising*in*physical systems are often nonhyperbolic. ... Distances*in*the expanding ͑contracting͒ subspace grow ͑shrink͒*exponentially**in**time*on average. ͑2͒ The angle between the stable and the unstable subspaces is bounded away from zero. ͑3͒ The expanding ...##
###
Two Open Problems on Effective Dimension
[chapter]

2006
*
Lecture Notes in Computer Science
*

(Gale characterization of fractal

doi:10.1007/11780342_37
fatcat:qvyckgulfvbtjirmbv6v4o7o2i
*dimension*) Let X be a set of sequences. ... We can now define versions of*Hausdorff*and packing*dimensions*that are meaningful inside complexity classes such as*exponential**time*and*exponential*space. ... A third aspect of effective*dimension*is as a formal tool*in*Computational Complexity, allowing us to consider new working hypothesis such as "NP has positive*dimension**in**exponential**time*", that can imply ...##
###
Hausdorff dimension of exponential parameter rays and their endpoints

2007
*
Nonlinearity
*

We show that the parameter rays have

doi:10.1088/0951-7715/21/1/006
fatcat:l5cjwtfchfhdzgarmqotb72uh4
*Hausdorff**dimension*1, while the ray endpoints*in*I alone have*dimension*2. ... Analogous results were known for dynamical planes of*exponential*maps; our result shows that this also holds*in*parameter space. ... Standard Squares and Conformal Mappings The fundamental idea of our proof will be similar to*Hausdorff**dimension*estimates for dynamic rays*in*the dynamic planes of*exponential*maps as*in*[K, SZ] ; however ...##
###
Page 8544 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

the y- almost sure value of the average s-

*dimensional*density D’fi(x) = 1*dimensional**Hausdorff*measureu = (#*\F)/(#*(F)) restricted to a self-conformal set F with*Hausdorff**dimension*equal to s and satisfying ... Summary: “We prove for many self-similar, and some more gen- eral, sets E C R” that, if s is the*Hausdorff**dimension*of E and f:R”" = R" is Holder continuous with exponent m/s, then the s-*dimensional**Hausdorff*...##
###
KPZ relation does not hold for the level lines and the SLE_κ flow lines of the Gaussian free field
[article]

2014
*
arXiv
*
pre-print

*In*order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the

*exponential*moments of winding of chordal SLE curves conditioned ...

*In*this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation

*in*a natural way, shedding new light on all of them. ...

*Hausdorff*

*dimension*. 4 Almost sure

*Hausdorff*

*dimension*of the zero level line does not satisfy the KPZ relation

*In*this section we show that the expected Minkowski and almost sure

*Hausdorff*versions ...

##
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Fractal Dimension versus Process Complexity

2016
*
Advances in Mathematical Physics
*

*In*particular, a TM with three states and two colors runs

*in*at most linear

*time*, if and only if its

*dimension*is 2, and its

*dimension*is 1, if and only if it runs

*in*superpolynomial

*time*and it uses polynomial ...

*In*our setting, it makes sense to define a fractal

*dimension*for a Turing machine as the limiting fractal

*dimension*for the corresponding space-

*time*diagrams. ... This machine runs

*in*linear

*time*for even inputs and

*exponential*

*time*for odd inputs. ...

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