各向异性扩散DLA的标度性质

本文利用坐标变换的方法讨论了各向异性扩散对DLA集团标度行为的影响。对二维正方形点阵,解析计算了各向异性扩散DLA集团的Hausdorff维数。结果表明,此时生长集困的Hausdorff维数D随各向异性扩散概率p连续变化,其最大值为5/3,最小值为3/2,还将这里所得到的理论结果与Jullien等人的数值结果进行了比较,发现它们符合得很好。最后,还解析讨论并计算了各向异性扩散DLA的广义维数D_q(p)。     

各向异性扩散DLA集团的豪斯道夫维数与标度性质

讨论了DLA集团的各向异性扩散效应.计算机模拟证实了具有各向异性扩散规则的DLA集团有严格的菱形结构.导出了一个粒子的各向异性扩散的新方程，计算了各向异性扩散DLA集团的豪斯道夫维数，结果表明，有效外半角β_eff=min(β_ix,β_iy).讨论了各向异性扩散DLA集团的广义维数，使用修改的楔模型得到了广义维数D_q的表达式. 关键词：     

A practical method to experimentally evaluate the Hausdorff dimension: an alternative phase-transition-based methodology

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.     

Scaling behavior of threshold epidemics

We numerically simulate the dynamics of atomic clusters aggregation deposited on a surface interacting with the growing island. We make use of the well-known DLA model but replace the underlying diffusion equation by the Smoluchowski equation which results in a drifted DLA model and anisotropic jump probabilities. The shape of the structures resulting from their aggregation-limited random walk is affected by the presence of a Laplacian potential due to, for instance, the surface stress field. We characterize the morphologies we obtain by their Hausdorff fractal dimension as well as the so-called external fractal dimension. We compare our results to previously published experimental results for antimony and silver clusters deposited onto graphite surface.     

KPZ in One Dimensional Random Geometry of Multiplicative Cascades

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity [KPZ88] appears. This note was inspired by the recent work of Duplantier and Sheffield [DS08] proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.     

各向异性扩散控制聚集集团的豪斯道夫维数

本文通过解析计算,给出了各向异性扩散控制聚集集团有效外半角β=min{β_x,β_y},修正了文献[1]关于有效外半角β=max{β_x,β_y}的假设,对各向异性扩散控制聚集集团的豪斯道夫维数D随各向异性扩散几率p的变化公式也作了相应的修正,并进行了计算机模拟实验,发现修正后的公式与实验符合较好,与Jullien等人的计算机模拟结果进行了比较,在p≥1/4时两者的结果是一致的。 关键词：     

Hausdorff Dimension of Non-Hyperbolic Repellers. I: Maps with Holes

This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps. Here we consider a general abstract model, that we call piecewise smooth maps with holes. We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold. Our approach also provides information on escape rates and dynamical dimension of the repeller.     

Diffusion with restrictions

A nonlinear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additional repulsive and attractive forces leading to a changed local mobility. The stationary and the time dependent behaviour of the system are studied based upon the master equation approach. Different to conventional diffusion it results in a time dependent bump the position of which increases with time described by an anomalous diffusion exponent. The fractal dimension of this random walk is exclusively determined by the space dimension. The applicability of the model to describe glasses is discussed.     

Measures of Maximal Dimension for Hyperbolic Diffeomorphisms

We establish the existence of ergodic measures of maximal Hausdorff dimension for hyperbolic sets of surface diffeomorphisms. This is a dimension-theoretical version of the existence of ergodic measures of maximal entropy. The crucial difference is that while the entropy map is upper-semicontinuous, the map dim_H is neither upper-semicontinuous nor lower-semicontinuous. This forces us to develop a new approach, which is based on the thermodynamic formalism. Remarkably, for a generic diffeomorphism with a hyperbolic set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures.Partially supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT's Funding Program.     

Fractal dimension of Siegel disc boundaries

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalization fixed point and expressing the Siegel disc boundary as the attractor of an associated Iterated Function System. Received: 26 January 1998 / Received in final form: 5 June 1998 / Accepted: 11 June 1998     

SIMULATION OF MULTIPLE FRACTAL AND COMPACT GROWTH OF ULTRA-THIN FILMS ON HEXAGONAL SUBSTRATE

The multiple cluster growth of ultra-thin films with different deposition rate and different substrate temperature has been studied by kinetic Monte-Carlo simulation. With increasing diffusion rate along cluster edges (corresponding to an increasing substrate temperature), pattern structures change smoothly from fractal islands, compact islands with random shapes, to regular islands, and the average branch width of clusters increases continuously up to some constant value in the compact island limit. The formation of the multiple fractal and compact clusters can be described quantitatively by multifractal. The results of multifractal analysis show that with pattern change from fractal to compact islands, the Hausdorff dimension D₀, the information dimension D₁, and the correlation dimension D₂ decrease, while the width and height of the multifractal spectra increase.     

Spectral Dimension of Trees with a Unique Infinite Spine

Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.     

Scaling properties of randomly triangulated planar random surfaces: A numerical study

Results of Monte Carlo simulations of a model of random surfaces based on planar random triangulations with gaussian embedding in D-dimensional euclidean space are presented, for various positive and negative values of D and various forms for the action. Estimates are given for the fractal dimension (Hausdorff dimension of the embedding) and the spreading dimension (intrinsic Hausdorff dimension). The scaling properties appear to depend on the short-distance properties of the triangulations and seem to be nonuniversal, at least for positive D.     

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).     

Dimensions of Attractors in Pinched Skew Products

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.     

A Lower Estimation of the Hausdorff Dimension for Attractors with Overlaps

We give a lower estimate of the Hausdorff dimension for attractors which can be obtained by an overlapping construction.     

Polyhedra formation and transient cone ejection of a resonant microdrop forced by an ac electric field

New deformation or fission phenomena are reported for microdrops driven by an ac electric field at their resonant frequencies. The Maxwell forces that pull out the vertices from a drop can be enhanced when the ac frequency is comparable to both the drop resonant frequency and the inverse charge relaxation time of the diffuse layer. The selected polyhedra possess symmetries that ensure a global force balance of the Maxwell forces and a linear dimension consistent with a sphere whose nth harmonic (n is up to six in the observation) coincides with the applied ac frequency. At high voltages, the resonant focusing of charges by the vibration modes produces evenly distributed and transient Taylor cones that can eject charged nanodrops.     

Elliptic Genera of 2d $${\mathcal{N}}$$ N = 2 Gauge Theories

We consider the spectrum of discrete Schrödinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.     

On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions

We consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc. We also introduce different ways to define and study the generalized spectrum for dimensions—a one-parameter family of characteristics of dimension type.     

Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces

We describe and implement a technique for extracting forces from the relaxation of an overdamped thermal system with normal modes. At sufficiently short time intervals, the evolution of a normal mode is well described by a one-dimensional Smoluchowski equation with constant drift velocity v, and diffusion coefficent D. By virtue of fluctuation dissipation, these transport coefficients are simply related to conservative forces, F, acting on the normal mode: F=kBTv/D. This relationship implicitly accounts for hydrodynamic interactions, requires no mechanical calibration, makes no assumptions about the form of conservative forces, and requires no prior knowledge of material properties. We apply this method to measure the electrostatic interactions of polymer microspheres suspended in nonpolar microemulsions.