3-D in vivo brain tumor geometry study by scaling analysis

A new method, based on scaling analysis, is used to calculate fractal dimension and local roughness exponents to characterize in vivo 3-D tumor growth in the brain. Image acquisition was made according to the standard protocol used for brain radiotherapy and radiosurgery, i.e., axial, coronal and sagittal magnetic resonance T₁-weighted images, and comprising the brain volume for image registration. Image segmentation was performed by the application of the k-means procedure upon contrasted images. We analyzed glioblastomas, astrocytomas, metastases and benign brain tumors. The results show significant variations of the parameters depending on the tumor stage and histological origin.     

Experimental Study and Fractal Analysis of Heterogeneity in Naturally Fractured Rocks

Capillary pressure curves of six low porosity and low permeability core samples from The Geysers geothermal field were measured using the mercury-intrusion approach to characterize the heterogeneity of rock. One high permeability Berea sandstone core sample was analyzed similarly, for comparison. The maximum pressure of mercury intruded into the rock was about 200 MPa to reach the extremely small pores. Experimental data showed that the capillary pressure curves of The Geysers rock are very different from that of the Berea sandstone. It was found that the frequently used capillary pressure models could not be used to represent the data from The Geysers rock samples. This might be because of the fractures in the rock. To this end, a fractal technique was proposed to model the features of the capillary pressure curves and to characterize the difference in heterogeneity between The Geysers rock and Berea sandstone. The results demonstrated that the rock from The Geysers geothermal field was fractal over a scaling range of about five orders of magnitude. The values of the fractal dimension of all the core samples (six from The Geysers and one Berea sandstone) calculated using the proposed approach were in the range from 2 to 3. The results showed that The Geysers rock with a high density of fractures had a greater fractal dimension than Berea sandstone which is almost without fractures. This shows that The Geysers rock has greater heterogeneity, as expected.     

河北平原地区断层系的分形特征分析

断层形成机理的复杂性和时空展布的不规则性,传统的评价方法往往很难取得令人满意的定量化结果,应用分形理论可以对断裂构造系进行定量评价,断层分维是断层数量、规模、组合方式及动力学机制的综合体现,因此分维值的大小可以作为断层构造复杂程度一个定量化指标。运用分形理论,对河北平原区主要断裂构造带进行了定量评价。得出河北平原区NNE向分维值为1.358,纬向断层系分维值为1.183,河北平原区活断层系空间分维值较高,这和研究区地震活动频繁、地质灾害多发一致。最后对河北平原区、渭河盆地和川滇断层系分形值和地震活动性进行了分析对比,发现地震空间分布和分形值的大小有对应关系,且渭河盆地、川滇的分形值大于河北平原地区,河北平原所发生的地震多沿NNE向呈带状分布。     

泥石流类型与地貌分形特征的关系研究

泥石流是一种频繁发生且危害性很大的自然灾害。作为中国泥石流灾害重灾区之一的陇南地区,泥石流不仅严重阻碍G212线的正常运营,也严重影响沿线地区的经济发展和人民生活。通过对G212线陇南段232条泥石流沟的调查资料分不同属性、不同发展阶段和不同危害程度进行统计,结合分形理论,分析了泥石流沟道数与地貌要素的定量关系。结果表明:地貌形态要素对泥石流的属性、发展阶段和危害程度等不同类型泥石流的影响程度不同; 综合分维值、形成区形态参数和地形坡度等参数都在一定程度上定量反映了不同类型泥石流的特征。     

The Finite Element Method on the Sierpinski Gasket

For certain classes of fractal differential equations on the Sierpinski gasket, built using the Kigami Laplacian, we describe how to approximate solutions using the finite element method based on piecewise harmonic or piecewise biharmonic splines. We give theoretical error estimates, and compare these with experimental data obtained using a computer implementation of the method (available at the web site http://mathlab.cit.cornell.edu/\sim gibbons). We also explain some interesting structure concerning the spectrum of the Laplacian that became apparent from the experimental data. March 29, 2000. Date revised: March 6, 2001. Date accepted: March 21, 2001.     

小波分形插值医学图像处理

为了得到高分辨率图像,本文利用小波变换后各方面上的高频图像具有相似性的特点,提出了小波分形插值处理医学图像的方法。仿真实验表明,这种处理方法比常用的方法（如双线性插值等）具有更好的效果。     

高磺化度聚苯胺体系中的分形结构研究

通过透射电镜的观察研究发现磺化聚苯胺的胶体聚集体和胶粒内部结构都具有分形体的特征 ,从而将分形的概念及其数学模型引入共轭导电聚合物体系之中 .磺化聚苯胺胶体的聚集体为很不均匀的分支状开放结构 ,其形成过程可用扩散控制集团聚集模型 (DLCA)进行模拟 ,计算机模拟生成的图形及其分形维数都与实验观测结果相当吻合 .胶粒由于是在分散介质所形成的平均化场中生成 ,屏蔽效应减弱 ,是比由它组成的聚集体要致密的球形结构 ,该结构的生成可用随机雨点模型模拟且结果相近 .     

自由基引发的凝胶化反应的Monte Carlo模拟

基于自由基凝胶化反应的基本原理,在一个具有周期性边界条件的二维网络上,利用动态ＭｏｎｔｅＣａｒｌｏ 方法模拟了聚合物凝胶的自由基凝胶化反应,得到了凝胶的具体结构,研究了总单体浓度（ 单体浓度和交联剂浓度之和） 对凝胶的分形结构和孔径分布的影响．模拟中首次考虑了聚合后单体的运动对凝胶结构的影响．结果表明：考虑聚合后单体的运动,可使所得凝胶网络的分形维数和凝胶化所需的最低浓度均显著小于动力学凝胶化模型和ＤＬＡ 模型的相应值．用移动气泡法得到了凝胶网络的孔径分布,发现凝胶网络中大孔所占百分率明显多于随机纤维网络模型．