Hausdorff Fractal Dimension Research Articles

Sparse block-Jacobi matrices with arbitrarily accurate Hausdorff dimension

We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i.e. nonrandom, block-Jacobi matrices may be determined with any degree of precision, improving a result of Zlatoš [Andrej Zlatoš, Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004) 216–252].

Finite-frequency dynamics of vortex loops at the 4He superfluid phase transition

The finite-frequency dynamics of the 4He superfluid phase transition can be formulated in terms of the response of thermally excited vortex loops to an oscillating flow field. The key parameter is the Hausdorff fractal dimension dH of the loops, which affects the dynamics because the frictional force on a loop is proportional to the total perimeter P of the loop, which varies as P ∼ adH where a is the loop diameter. Solving the 3D Fokker-Planck equation for the loop response at frequency ω yields a superfluid density which varies at Tλ as ω1/(dH-1). This power-law variation with ω agrees with the scaling form found by Fisher, Fisher, and Huse, since the dynamic exponent z is identified as z = dH -1. Flory scaling for the self-avoiding loops gives a fractal dimension in terms of the space dimension d as dH = (d + 2)/2, yielding z = d/2 = 3/2 for d = 3, in complete agreement with dynamic scaling.

Refined Bounds on Kolmogorov Complexity for ω-Languages

The paper investigates bounds on various notions of complexity for ω–languages. We understand the complexity of an ω–languages as the complexity of the most complex strings contained in it. There have been shown bounds on simple and prefix complexity using fractal Hausdorff dimension. Here these bounds are refined by using general Hausdorff measure originally introduced by Felix Hausdorff. Furthermore a lower bound for a priori complexity is shown.

Exceptional sets for self-affine fractals

AbstractUnder certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

A HIERARCHY OF MOTION IN ELECTROSPINNING PROCESS AND Ε-INFINITY NANOTECHNOLOGY

Electrospun nanofibers have a wide range of potential applications such as photonic structures, microfluid channels (nanofluidics), catalysis, sensors, medicine, pharmacy, drug delivery, invisibility devices (e.g. stealth planes, stealth clothes), radioprotection, tissue engineering, and nanometer-sized semiconductor devices. A hierarchy of motion in electrospinning process is identified, and the E-infinity theory is applied to explain the phenomenon. These materials with hierarchical geometrical structures always display quantum-like properties and have many fascinating nano-effects, such as a remarkable increase in strength, high surface energy and surface reactivity, excellent thermal and electric conductivity. Here we also show that nanotechnology could be seen as a link between the deterministic classic mechanics and the indeterministic quantum mechanics, and E-infinity theory could help in understanding various nano effects. We also predict that the Hausdorff dimension for the hierarchy of motion should be close to the average Hausdorff dimension of E-infinity spacetime, and the characteristics of electrospun nanofibers depend mainly upon the Hausdorff-fractal dimension. An experiment is performed to verify our predictions.

Fractal Dimension and Size Scaling of Domains in Thin Films of MultiferroicBiFeO3

Domains in ferroelectric films are usually smooth, stripelike, very thin compared with magnetic ones, and satisfy the Landau-Lifshitz-Kittel scaling law (width proportional to square root of film thickness). However, the ferroelectric domains in very thin films of multiferroic BiFeO3 have irregular domain walls characterized by a roughness exponent 0.5-0.6 and in-plane fractal Hausdorff dimension H||=1.4+/-0.1, and the domain size scales with an exponent 0.59+/-0.08 rather than 1/2. The domains are significantly larger than those of other ferroelectrics of the same thickness, and closer in size to those of magnetic materials, which is consistent with a strong magnetoelectric coupling at the walls. A general model is proposed for ferroelectrics, ferroelastics or ferromagnetic domains which relates the fractal dimension of the walls to domain size scaling.

Impulsive current-induced superconducting junction in spatially broken symmetry of fluxons

Embarking upon McCumber–Stewart model, the temporal evolution of a superconducting Josephson junction in both overdamped and underdamped regimes is examined. The junction is driven by a pulse train current, and the potential of the tunnelling fluxons is assumed to lack spatial symmetry such that associated with a ratchet potential. In this respect, Langevin and Fokker–Plank-like equations specifying the junction are solved for which, time-dependent phase solutions are obtained and discussed on the basis of the potential broken symmetry and pulse signal properties by which, an ad hoc value of the frequency equals to the fractal Hausdorff dimension ω = D H = ln(2)/ln(3) = 0.630929753 of a normal Cantor set is ascribed.

Application of E-infinity theory to turbulence

Abstract El-Naschie’s E-infinity theory is applied to turbulence. The Hausdorff-fractal dimension for turbulent flow is defined, the critical values for laminar flow (D = 3.98) and turbulent flow (D = 4.23) are obtained, and the fractal dimension for fully developed turbulent flow is D > 6.8. It is also shown that the Navier–Stokes equations are invalid for an exact model of turbulent flow and that two-dimensional planar turbulence does not exist in nature.

Unstable manifolds and Hölder structures associated with noninvertible maps

We study the case of a smooth noninvertible map $f$ with Axiom A, inhigher dimension. In this paper, we look firstat the unstable dimension (i.e the Hausdorff dimension of the intersection betweenlocal unstable manifolds and a basic set $\Lambda$), and prove that it is givenby the zeroof the pressure function of the unstable potential, considered on the naturalextension $\hat\Lambda$ of the basic set $\Lambda$; as a consequence,the unstable dimension is independent of the prehistory $\hat x$.Then we take a closer look at the theorem of construction for the localunstablemanifolds of a perturbation $g$ of $f$, and for the conjugacy $\Phi_g$ definedon $\hat \Lambda$.If the map $g$ is holomorphic, one can prove some special estimates of theHölderexponent of $\Phi_g$ on the liftings of the local unstable manifolds.In this way we obtain a new estimate of the speed of convergence of the unstabledimension of $g$, when $g \rightarrow f$.Afterwards we prove the real analyticity of the unstable dimension whenthe map $f$ depends on a real analytic parameter.In the end we show that there exist Gibbs measures on the intersections betweenlocal unstable manifolds and basic sets,and that they are in fact geometric measures; using this, the unstable dimensionturns out to be equal to the upper box dimension. We notice also that in thenoninvertible case, the Hausdorff dimension of basic sets does not vary continuously with respect to the perturbation $g$of $f$. In the case of noninvertible Axiom A maps on $\mathbb P^2$,there can exist an infinite number of local unstable manifolds passingthrough thesame point $x$ of the basic set $\Lambda$, thus there is no unstablelamination. Therefore many of the methods used in the case of diffeomorphisms break down and new phenomena and methods of proof must appear. The results in this paper answer to some questions of Urbanski([21]) about the extensionof one dimensional theory of Hausdorff dimension of fractals to thehigher dimensional case. They also improve some results andestimates from [7].

The fractal dimension of ionization cascades in the glow discharge

The glow discharge's main ionization breakdown processes have been understood for about one hundred years. The glow discharge, however, still remains an area of active research in relation to pattern formation and far-from-equilibrium systems. The primary and secondary ionization processes can be mathematically modelled as general branching processes. Not only is the Townsend breakdown criterion obtained but the ionization avalanche can be characterized as a branching set with a unique Hausdorff fractal dimension. These fractal dimensions can be utilized in applications using similarity principles and Paschen's Law.

Fractal dimension of transport coefficients in a deterministic dynamical system

In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and investigate the dependence of transport coefficients on the slope of the map. We present analytical arguments, supported by numerical calculations, showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of the graphs of these functions is 1 with a logarithmic correction, and find that the exponent $\gamma$ controlling this correction is bounded from above by 1 or 2, depending on some detailed properties of the system. Using numerical techniques we show local self-similarity of the graphs. The local self-similarity scaling transformations turn out to depend (irregularly) on the values of the system control parameters.

Hamiltonian fractals and chaotic scattering of passive particles by a topographical vortex and an alternating current

We investigate the dynamics of passive particles in a two-dimensional incompressible open flow composed of a fixed topographical point vortex and a background current with a periodic component, the model inspired by the phenomenon of topographic vortices over mountains in the ocean and atmosphere. The tracer dynamics is found to be typically chaotic in a mixing region and regular in far upstream and downstream regions of the flow. Chaotic advection of tracers is proven to be of a homoclinic nature with transversal intersections of stable and unstable manifolds of the saddle point. In spite of simplicity of the flow, chaotic trajectories are very complicated alternatively sticking nearby boundaries of the vortex core and islands of regular motion and wandering in the mixing region. The boundaries act as dynamical traps for advected particles with a broad distribution of trapping times. This implies the appearance of fractal-like scattering function: dependence of the trapping time on initial positions of the tracers. It is confirmed numerically by computing a trapping map and trapping time distribution which is found to be initially Poissonian with a crossover to a power-law at the PDF tail. The mechanism of generating the fractal is shown to resemble that of the Cantor set with the Hausdorff fractal dimension of the scattering function to be equal to d≃1.84.

Sparse potentials with fractional Hausdorff dimension

We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.

Geometry of fluctuating vortex loops at superfluid phase transitions

The geometrical properties of thermally excited vortex loops near a superfluid phase transition are deduced from an analytic vortex-renormalization theory. The fractal Hausdorff dimension of the loops is d H=2.5, and the corresponding ‘anomalous’ dimensionality exponent is η φ =−0.5. As the temperature is increased towards T c the density distribution of loops of average diameter a crosses over from exponential to algebraic decay in the loop diameter. Just at T c the distribution falls off algebraically as a − λ , where λ= d+1=4.0, in exact agreement with a cosmic-string prediction of Vachaspati and Vilenkin.

Hausdorff dimension of random fractals with overlaps

This paper considers random fractals generated by contractive transformations which satisfy a separation condition weaker than the open set condition. This condition allows overlaps in the iterations. Estimates of the Hausdorff dimension of this kind of random fractals are obtained.

The Hausdorff and box dimension of fractals with disjoint projections in two dimensions

In this paper, we obtain an exact formula for the Hausdorff and box dimensions of a class of self-affine sets in two dimensions, namely those with disjoint projections. We prove, in particular, that fractals in this class have a Hausdorff and box dimension that is equal to the maximum Hausdorff and box dimension of one of their projections.

An improved method to estimate the fractal dimension of physical fractals based on the Hausdorff definition

In this paper we present an algorithm to estimate the Hausdorff fractal dimension. The algorithm uses a recursive formula with a fast enough convergence. The accuracy of results is independent on the size, i.e., degree of definition of the fractal set. This fact is particularly useful when studying real physical fractals with a low definition, such as colloidal aggregates of small size. The different tests reveal no dependence of the results on the irregularities of the fractal. Thus, self-similarity or statistical similarity of the fractal set does not affect results. The proposed algorithm gives correct values for all the fractal dimension of the tested sets. Finally, the algorithm was used to evaluate the Hénon attractor fractal dimension and was applied to an experimental system obtained from a two-dimensional aggregation of latex colloidal particles.

FRACTONS AND FRACTAL STATISTICS

Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by [Formula: see text] shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

Percolation in sign-symmetric random fields: Topological aspects and numerical modeling

The topology of percolation in random scalar fields psi(x) with sign symmetry [i.e., that the statistical properties of the functions psi(x) and -psi(x) are identical] is analyzed. Based on methods of general topology, we show that the zero set psi(x)=0 of the n-dimensional (n>/=2) sign-symmetric random field psi(x) contains a (connected) percolating subset under the condition |nablapsi(x)| not equal0 everywhere except in domains of negligible measure. The fractal geometry of percolation is analyzed in more detail in the particular case of the two-dimensional (n=2) fields psi(x). The improved Alexander-Orbach conjecture [Phys. Rev. E 56, 2437 (1997)] is applied analytically to obtain estimates of the main fractal characteristics of the percolating fractal sets generated by the horizontal "cuts," psi(x)=h, of the field psi(x). These characteristics are the Hausdorff fractal dimension of the set, D, and the index of connectivity, straight theta. We advocate an unconventional approach to studying the geometric properties of fractals, which involves methods of homotopic topology. It is shown that the index of connectivity, straight theta, of a fractal set is the topological invariant of this set, i.e., it remains unchanged under the homeomorphic deformations of the fractal. This issue is explicitly used in our study to find the Hausdorff fractal dimension of the single isolevels of the field psi(x), as well as the related geometric quantities. The results obtained are analyzed numerically in the particular case when the random field psi(x) is given by a fractional Brownian surface whose topological properties recover well the main assumptions of our consideration.

Thick points for spatial Brownian motion: multifractal analysis of occupation measure

Let $\mathscr{T}(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for Brownian motion in $\mathbb{R}^3$. We prove that $\sup_{|x|\leq1}\mathscr{T}(x,r)/(r^{2}|\log r|)\rightarrow16/\pi^2$ a.s. as $r\rightarrow0$, thus solving a problem posed by Taylor in 1974. Furthermore, for any $a \in(0,16/\pi^2)$, the Hausdorff dimension of the set of “thick points” $x$ for which $\lim\sup_{r \to 0}\mathscr{T}(x,r)/(r^2|\log r|) = a$ is almost surely $2-a\pi^2 /8$; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension $d \ge 3$. These results are related to the LIL of Ciesielski and Taylor for the Brownian occupation measure of small balls in the same way that Lévy’s uniform modulus of continuity, and the formula of Orey and Taylor for the dimension of “fast points ”are related to the usual LIL. We also show that the lim inf scaling of $\mathscr{T}(x,r)$ is quite different: we exhibit nonrandom $c_1,c_2 \ge 0$, such that $c_1 < \sup_x\lim \inf _{r \to 0}\mathscr{T}(x,y)/r^2 < c_2$ a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of “limsup type.”