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Hausdorff dimension in exponential time

K. Ambos-Spies, W. Merkle, J. Reimann, F. Stephan
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}.  ... 
doi:10.1109/ccc.2001.933888 dblp:conf/coco/Ambos-SpiesMRS01 fatcat:evwb22hd3bca7b6tsds7rloxly

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

I. Claus, P. Gaspard, H. van Beijeren
2004 Physica D : Non-linear phenomena  
Two-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 .  ... 
doi:10.1016/j.physd.2003.09.009 fatcat:z7drnyiqlbcxxfnh46t6cxl7gq

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  ... 

Dynamics of fractal dimension during phase ordering of a geometrical multifractal

Avner Peleg, Baruch Meerson
2000 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
It is shown that at early 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)).  ... 
doi:10.1103/physreve.62.1764 pmid:11088637 fatcat:lie47gvdvje5tps2jhiv5pmgki

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

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.  ...  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] ).  ... 
doi:10.1142/s0218271899000298 fatcat:pdrnyjhmyzdiddrwdruetkcfne

On the dimensions of a family of overlapping self-affine carpets

JONATHAN M. FRASER, PABLO SHMERKIN
2015 Ergodic Theory and Dynamical Systems  
We compute the 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  ... 
doi:10.1017/etds.2015.21 fatcat:gqvtz5rj45drxaywqskkydcboe

Fractal-Based Electrolytic Capacitor Electrodes: Scaling Behavior with Respect to Fractal Order and Complexity [article]

Benjamin Barnes, Othman Suleiman, JeanPaul Badjo, Kausik S Das
2018 arXiv   pre-print
An interesting 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.  ... 
arXiv:1810.00221v1 fatcat:i4ckciimwneaxm5pj7hj2sf3aq

The Hausdorff Dimension and Capillary Imbibition

Didier Samayoa, Ernesto Pineda León, Lucero Damián Adame, Eduardo Reyes de Luna, Andriy Kryvko
2022 Fractal and Fractional  
It is established that the actual distance of fluid travel 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 ).  ... 
doi:10.3390/fractalfract6060332 doaj:4ff1cb95ef214aba8f4cf4744130432c fatcat:obuhh2qwufbqbpggevatduwns4

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  
Whereas 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  ... 
doi:10.1103/physreve.65.026209 pmid:11863634 fatcat:jdkdnx32vjdota23p7ft4ivybq

Two Open Problems on Effective Dimension [chapter]

Elvira Mayordomo
2006 Lecture Notes in Computer Science  
(Gale characterization of fractal 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  ... 
doi:10.1007/11780342_37 fatcat:qvyckgulfvbtjirmbv6v4o7o2i

Hausdorff dimension of exponential parameter rays and their endpoints

Mihai Bailesteanu, Horia Vlad Balan, Dierk Schleicher
2007 Nonlinearity  
We show that the parameter rays have 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  ... 
doi:10.1088/0951-7715/21/1/006 fatcat:l5cjwtfchfhdzgarmqotb72uh4

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]

Juhan Aru
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  ... 
arXiv:1312.1324v2 fatcat:oglko4zhkvgf5ifauxcrykvr6m

Fractal Dimension versus Process Complexity

Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil
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.  ... 
doi:10.1155/2016/5030593 fatcat:g6ptxsbn3jfqfaaayeedmk7twm
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