Some Research In Fractal Interpolation And Fractal Networks

Fractal interpolation is one of the important research topics in fractal geom-etry theory.In 2015,Michael F.B arnsley and Peter R.Massopust used the bilinear function to study the fractal interpolation curves on the plane.In this dissertation,a generalization of the above results is given in the three-dimensional case.A class of fractal interpolation surfaces is obtained by using bilinear functions.Research on network problems with fractal structure characteristics has received great at-tention in recent decades.In this dissertation,we mainly consider a class of networks con-structed by recursive division methods.In the homogeneous models of fractal weighted Koch networks and non-homogeneous models of fractal weighted Koch networks cases,the sum of the shortest path of each vertex of the network to a target node is studied.The whole text is divided into six chapters:In Chapter 1,we briefly introduced some of the background reviews and motivations of this dissertation,as well as the main results of this dissertation.In Chapter 2,we have mentioned some basic properties,metric spaces;contraction map-ping;attractors;iterated function systems(IFS)and many of the conceptual lemmas and theo-rems involved in our dissertation.In Chapter 3,we discussed the problem of fractal interpolation surfaces.As a fixed point of a certain type of Read-B ajraktarevic operator,we prove that the image of the fractal interpo-lation surface is an attractor of the compressed IFS under a generalized metric df.In Chapters 4,5,and 6,we discuss the homogeneous and non-homogeneous weighted Koch network models using the recursive division method.For homogeneous models,it relies on a scale factor t ?(0,1),for non-homogeneous models,we usually take different scale factors r,s?(0,1)or t,r,s?(0,1).Finally,the average homogeneous weighted receiving time(AHWRT)by taking the longest paths depending on unvaried scaling factor t ?(0,1)and the average non-homogeneous weight-ed receiving time(AN-HWRT)by taking the longest paths depending on two different scaling factors r,s?(0,1)or three different scaling factors t,r,s ?(0,1)was obtained.