Cantor Type Fixed Sets of Iterated Multifunction Systems Corresponding to Self-Similar Networks

We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.

Activation Functions Effect on Fractal Coding Using Neural Networks

Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results,since without this nonlinearity,the results of the network will be less accurate.Nonlinearity is the mission of all nonlinear functions,except for polynomials.The activation function must be dif-ferentiable for backpropagation learning.This study’s objective is to determine the best activation functions for the approximation of each fractal image.Different results have been attained using Matlab and Visual Basic programs,which indi-cate that the bounded function is more helpful than other functions.The non-lin-earity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals.The most commonly cho-sen activation function is the sigmoidal function,which produces a positive value.Other functions,such as tansh or arctan,whose values can be positive or negative depending on the network input,tend to train neural networks faster.The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape.In this paper,we have provided the appro-priate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.

Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.

Chaos game representation of functional protein sequences,and simulation and multifractal analysis of induced measures

Investigating the biological function of proteins is a key aspect of protein studies.Bioinformatic methods become important for studying the biological function of proteins.In this paper,we first give the chaos game representation (CGR) of randomly-linked functional protein sequences,then propose the use of the recurrent iterated function systems (RIFS) in fractal theory to simulate the measure based on their chaos game representations.This method helps to extract some features of functional protein sequences,and furthermore the biological functions of these proteins.Then multifractal analysis of the measures based on the CGRs of randomly-linked functional protein sequences are performed.We find that the CGRs have clear fractal patterns.The numerical results show that the RIFS can simulate the measure based on the CGR very well.The relative standard error and the estimated probability matrix in the RIFS do not depend on the order to link the functional protein sequences.The estimated probability matrices in the RIFS with different biological functions are evidently different.Hence the estimated probability matrices in the RIFS can be used to characterise the difference among linked functional protein sequences with different biological functions.From the values of the D q curves,one sees that these functional protein sequences are not completely random.The D q of all linked functional proteins studied are multifractal-like and sufficiently smooth for the C q (analogous to specific heat) curves to be meaningful.Furthermore,the D q curves of the measure μ based on their CGRs for different orders to link the functional protein sequences are almost identical if q ≥ 0.Finally,the C q curves of all linked functional proteins resemble a classical phase transition at a critical point.

Three-Dimensional Modeling of the Retinal Vascular Tree via Fractal Interpolation

In recent years,the three dimensional reconstruction of vascular structures in the field of medical research has been extensively developed.Several studies describe the various numerical methods to numerical modeling of vascular structures in near-reality.However,the current approaches remain too expensive in terms of storage capacity.Therefore,it is necessary to find the right balance between the relevance of information and storage space.This article adopts two sets of human retinal blood vessel data in 3D to proceed with data reduction in the first part and then via 3D fractal reconstruction,recreate them in a second part.The results show that the reduction rate obtained is between 66%and 95%as a function of the tolerance rate.Depending on the number of iterations used,the 3D blood vessel model is successful at reconstruction with an average error of 0.19 to 5.73 percent between the original picture and the reconstructed image.

Non-Spectral Problem of Self-Affine Measures in R³

The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D). In the present paper, we establish some relations inside the zero set by the Fourier transform of the self-affine measure μM,D. Based on these facts, we show that μM,D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM,D), where the number 4 is the best possible. This extends several known conclusions.

Chung’s functional law of the iterated logarithm for the Brownian sheet

In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.

Connectedness of Invariant Sets of Graph-Directed IFS

In this paper, we study the connectedness of the invariant sets of a graph-directed iterated function system(IFS). For a graph-directed IFS with N states, we construct N graphs. We prove that all the invariant sets are connected, if and only if all the N graphs are connected; in this case, the invariant sets are all locally connected and path connected. Our result extends the results on the connectedness of the self-similar sets.

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.