Mean Dimension for Non-autonomous Iterated Function Systems

In this paper we introduce the notions of mean dimension and metric mean dimension for non-autonomous iterated function systems(NAIFSs for short)on countably infinite alphabets which can be regarded as generalizations of the mean dimension and the Lindenstrauss metric mean dimension for non-autonomous iterated function systems.We also show the relationship between the mean topological dimension and the metric mean dimension.

Generalized Product of Iteration Function System

The definition of generalized product of fractal is first put forward for the research on the relations between original fractal and its product of fractal when the transformations of iteration function system （IFS） are incomplete. Then the representations of generalized product of IFS are discussed based on the theory of the product of fractal. Furthermore, the dimensional relations between the product of fractal and its semi-product are obtained. The dimensional relations of self-similar set are discussed. Finally, the examples for rendering fractal graphs are given. These results posses potentials in image compression and pattern recognition.

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.

Lychrel Numbers in Base 10: A Probabilistic Approach

For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.

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 Dq curves, one sees that these functional protein sequences are not completely random. The Dq of all linked functional proteins studied are multifractal-like and sufficiently smooth for the Cq （analogous to specific heat） curves to be meaningful. Furthermore, the Dq curves of the measure μ based on their CCRs for different orders to link the functional protein sequences are almost identical if q 〉 0. Finally, the Ca curves of all linked functional proteins resemble a classical phase transition at a critical point.

Particle filter based on iterated importance density function and parallel resampling

The design, analysis and parallel implementation of particle filter(PF) were investigated. Firstly, to tackle the particle degeneracy problem in the PF, an iterated importance density function(IIDF) was proposed, where a new term associating with the current measurement information(CMI) was introduced into the expression of the sampled particles. Through the repeated use of the least squares estimate, the CMI can be integrated into the sampling stage in an iterative manner, conducing to the greatly improved sampling quality. By running the IIDF, an iterated PF(IPF) can be obtained. Subsequently, a parallel resampling(PR) was proposed for the purpose of parallel implementation of IPF, whose main idea was the same as systematic resampling(SR) but performed differently. The PR directly used the integral part of the product of the particle weight and particle number as the number of times that a particle was replicated, and it simultaneously eliminated the particles with the smallest weights, which are the two key differences from the SR. The detailed implementation procedures on the graphics processing unit of IPF based on the PR were presented at last. The performance of the IPF, PR and their parallel implementations are illustrated via one-dimensional numerical simulation and practical application of passive radar target tracking.

CONSTRUCTION OF SOME KIESSWETTER-LIKE FUNCTIONS-THE CONTINUOUS BUT NON-DIFFERENT-IABLE FUNCTION DEFINED BY QUINARY DECIMAL

In this paper, we construct some continuous but non-differentiable functions defined by quinary dec-imal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of graphs of these functions and give a detailed proof.

ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

A set of contraction maps of a metric space is called an iterated function systems. Iterated function systems with condensation, can be considered infinite iterated function systems. Infinite iterated function systems on compact metric spaces were studied. Using the properties of Banach limit and uniform contractiveness, it was proved that the random iterating algorithms for infinite iterated function systems on compact metric spaces-satisfy ergodicity. So the random iterating algorithms for iterated function systems with condensation satisfy ergodicity, too.

ITERATED FUNCTION SYSTEM AND GALTON-WATSON TREE

Given a system {S1,…, SN} of N contractive similarities satisfying some strong separation condition, it has an invariant Set K for the system. In this article, the authors construct some random measure μω supported on random subset Kω of K, μω having some ＂non-standard＂ multifractal structure, which contrasts the well-knoWn multifractal formalism for the invariant measure of system {S1,.., SN} may possess. The main tool is the multifractal structures of a Galton-Watson tree, which are obtained by Liu [9], Shieh-Taylor [14], and MSrters-Shieh [12].

AFFINE TRANSFORMATION IN RANDOM ITERATED FUNCTION SYSTEMS

Random iterated function systems (IFSs) is discussed, which is one of the methods for fractal drawing. A certain figure can be reconstructed by a random IFS. One approach is presented to determine a new random IFS, that the figure reconstructed by the new random IFS is the image of the origin figure reconstructed by old IFS under a given affine transformation. Two particular examples are used to show this approach.

SOME NEW ITERATED FUNCTION SYSTEMS CONSISTING OF GENERALIZED CONTRACTIVE MAPPINGS

Iterated function systems （IFS） were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle. In 2010, D. R. Sahu and A. Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings. In this paper, following Hutchinson, D. R. Sahu and A. Chakraborty, we present some new iterated function systems by using the so-called generalized contractive mappings, which will also cover a large range of mappings. Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.

AN APPLICATION OF HARDY-BOEDEWADT'S THEOREM TO ITERATED FUNCTIONAL EQUATIONS

In this paper an iterated functional equation of polynomial type which does not possess the firt order iterative term g(x) is to be discussed. The difficulties resulted from loss of the first order term are overcome by utilization of Hardy-Boedewadt's theorem.

Upper Bound and Lower Bound Estimate of Monotone Increasing Fractal Function

Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure. In this paper we will give a numerical approximate method of upper bound and lower bound of mass distribution function f（x）（it is a monotone increasing fractal function） and its some applications.

Fixed Point Theorems of the Iterated Function Systems

In this paper, we present some fixed point theorems of iterated function systems consisting of α-ψ-contractive type mappings in Fractal space constituted by the compact subset of metric space and iterated function systems consisting of Banach contractive mappings in Fractal space constituted by the compact subset of generalized metric space, which is Mso extensively applied in topological dynamic system.

A Note on a Functional Differential Equation with State Dependent Argument

This paper is concerned with solutions of a functional differential equation. Using Krasnoselskii’s fixed point theorem, the solutions can be obtained from periodic solutions of a companion equation.

CONTINUITY OF JULIA SET FOR A FAMILY OF ENTIRE FUNCTIONS

Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.

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.

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.

Existence and Uniqueness of Analytic Solutions of Systems of Iterative Functional Equations

Let r be a given positive number. Denote by D=D r the closed disc in the complex plane C whose center is the origin and radius is r. Write A(D,D)={f: f is a continuous map from D into itself, and f|D ° is analytic}. Suppose G,H: D 2n+1 →C are continuous maps (n≥2), and G|(D 2n+1 ) °, H|(D 2n+1 ) ° are analytic. In this paper, we study the system of iterative functional equationsG(z,f(z),…,f n(z), g(z),…,g n(z))=0, H(z,f(z),…,f n(z), g(z),…,g n(z))=0, for any z∈D,and give some conditions for the system of equations to have a solution or a unique solution in A(D,D) ×A(D,D).