Closed fractal interpolation surfaces

Based on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface.     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

Tangent measure distributions of fractal measures

Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the -dimensional tangent measure distributions at the point, which describe asymptotically the -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a Euclidean space and any dimension , at -almost every point, all -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities. Received: 27 November 1996 / Revised version: 27 February 1998     

Analyses and simulation of anisotropic fractal surfaces

The spectral densities for an anisotropic fractal surfaces are investigated. Since there is no general definition for anisotropic fractal surface, the profiles of anisotropic fractal surfaces are assumed to be fractal in two main axes. Then, the possible forms of the surface spectral densities are proposed. By using the inverse Fast Fourier Transform, anisotropic fractal surfaces can be simulated from the spectral densities.     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

Hierarchy of islands in conservative systems yields multimodal distributions of FTLEs

We investigate the motion in a chaotic layer of conservative systems using finite time Lyapunov exponents (FTLEs). For long finite time spans we find the distributions of FTLEs to be multimodal. Due to stickiness near islands of regular motion, the trajectory can spend a long time in their vicinity. The higher the order of an island in the hierarchy of islands, the smaller is the value of the largest FTLE. Using this connection, we explain the occurrence of multimodal distributions of FTLEs as a result of an overlap of individual distributions of FTLEs, each corresponding to the motion near islands of different orders.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

Dimension analysis on a class of Bush type fractal surfaces

Abstract. In this paper,the authors construct a class of fractal surfaces,Bush type surfaces,based on the Bush type functions. The Box dimension,Packing dimension and Hausdorff dimen-sion of such surfaces are investigated.     

Jessen–Wintner type random variables and fractal properties of their distributions

Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set S_ξ supporting the distribution of the random variable ξ = 2^–k τ_k, where τ_k are independent random variables taking the values 0, 1, 2 with probabilities p _0k , p _1k , p _2k , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of S_ξ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of S_ξ are found. It is also proven that for any real number a ₀ ∈ [0, 1] there exists a distribution of the r.v. ξ such that the Hausdorff–Besicovitch dimension of S_ξ is equal to a ₀. The results are applied to the study of the metric‐topological properties of the convolutions of random variables with independent binary digits, i.e., random variables ξⁱ = , where η_k are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension

Recurrent bivariate fractal interpolation surfaces (RBFISs) generalise the notion of affine fractal interpolation surfaces (FISs) in that the iterated system of transformations used to construct such a surface is non-affine. The resulting limit surface is therefore no longer self-affine nor self-similar. Exact values for the box-counting dimension of the RBFISs are obtained. Finally, a methodology to approximate any natural surface using RBFISs is outlined.     

Superfractality of the set of numbers having no frequency ofn-adic digits,and fractal probability distributions

We study the fractal properties (we find the Hausdorff-Bezikovich dimension and Hausdorff measure) of the spectrum of a random variable with independentn-adic (n2,n N digits, the infinite set of which is fixed. We prove that the set of numbers of the segment [0, 1] that have no frequency of at least onen-adic digit is superfractal.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 47, No. 7, pp. 971–975, July, 1995.     

The genus distributions for a certain type of permutation graphs in orientable surfaces

A circuit is a connected nontrivial 2-regular graph.A graph G is a permutation graph over a circuit C,if G can be obtained from two copies of C by joining these two copies with a perfect matching.In this paper,based on the joint tree method introduced by Liu,the genus polynomials for a certain type of permutation graphs in orientable surfaces are given.     

Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 \; dA -\gamma \int\limits_{\Gamma} K \; dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and ${\gamma \in [0,1]}$ is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to ${\mathcal{W}_{\gamma}}$ and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.     

Unstable Willmore surfaces of revolution subject to natural boundary conditions

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.     

Counterexamples in theory of fractal dimension for fractal structures

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.     

Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

We study the fractal properties of distributions of random variables digits of polybasic Q-representations (a generalization of n-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.