In the forth chapter , we present the subsequent development of the renormalization group methods at the beginning . then based upon the electron - phonon model , we explain how to apply these methods to obtain the flow equations of the model 在第四章中首先介绍了重整化群方法的最新发展，并结合电子?声子模型详细介绍了如何用重整化群流方程方法导出系统的流方程。
Though we know the forms of partition function before and after transformation , we will not study directly the property of partition function itself but the transformation which makes it unchanged with the idea of renormalization group 我们虽然利用p - v路方法写出了重整化变换前后配分函数的形式，但并没有直接研究配分函数本身的性质，而采用了重整化群的思想，研究使配分函数保持不变的变换。
Here are some references for nrqcd & nrqed : renormalization group scaling in nonrelativistic qcd on the multipole expansion , the qcd heavy quark potential to order v * * 2 : one loop matching conditions on matching calculations 11下面是非相对论性量子色动力学和非相对论性量子电动力学的一些参考资料：关于多极展开的非相对论性量子色动力学中的重整化群定标，关于匹配计算的v * * 2阶的量子色动力学重夸克势：单圈匹配条件。
Yoccoz introduced a powerful puzzle technique and obtained the result that if the quadratic polynomial pc ( z ) with , which has no indifferent cycle , is not infinitely renormalizable , then the julia set j ( pc ) of pc is locally connected while dm is also locally connected at c 利用这个技巧，他得到了对c m ，如果二次多项式pc ( z )不是无穷可重整化的，并且没有抛物周期点，那么julia集j ( pc )是局部连通的，并且m在c也是局部连通的。
If fc has no parabolic periodic points and is not renormalizable , then j ( fc ] is locally connected . here fc is renormalizable if there are simply connected domains u and u " and a positive integer n such that contains the critical point c or - c1 and is a quadratic - like mapping with connected julia set 这里， it称为可重整化的定义为：存在单连通区域l ’ 、 u ’及正整数n ；使得i ’ u ，仁’包含临界点c或一c ， ” ： r一u是一个二层分支覆盖，且拟共形共轭于一个有连通julia集的二次多项式。
It is just the difficulty that the strongly correlated electron systems can not be studied by analytical methods , a variety of simulative numerical methods come out sequentially . there are a few typical methods such as the exact diagonalization ( ed ) , the qutum monte carlo ( qmc ) , the variation monte carlo ( qmc ) , the renormalization group ( rg ) , and the density matrix renormalization group ( dmrg ) and so on 正是强关联系统遇到了解析研究的困难，各种数值模拟方法才相继出现，最典型的数值方法有：严格对角化( ed ) ，量子蒙特卡洛( qmc )模拟、变分蒙特卡洛模拟( vmc ) 、数值重整化群( rg )以及密度矩阵重整化群( dmrg )等。
Then the structure unit of hexangular lattice is found according to the principle of invariable symmetry . renormalization transformation is processed when we regard the structure unit and the growth model as graphs before and after transformation respectively . after choosing the 11 thermodynamic function fugacity as parameter , we can write out the partition functions before and after transformation and the formula of renormalization transformation 然后根据对称性不变的原则，从整体晶格中选取结构单元，把结构单元和生长模型分别作为重整化变换前、后的图形来进行重整化变换，选取热力学函数易逸度为参量，写出了重整化变换前后的配分函数和重整化变换关系式，求出了这一变换的不动点。
Whereas string theory cannot be tested directly ? it deals with energies that cannot be created in the laboratory ? connes points out that noncommutative geometry makes testable predictions , such as the higgs mass ( 160 billion electron volts ) , and he argues that even renormalization can be verified 例如，弦论无法以实验来检验，因为弦论所涉及的能量太高，物理学家在实验室中造不出那麽高的能量，但是巩讷指出非可换几何所做的预测，譬如希格斯质量（ 1600亿电子伏特） ，却是可以检验的，他说甚至重整化也可以证实。
A generalization , n > 4 , is given in this paper . in result the critical points of these koch curve with different values of tv by decimation based on ising model are same , but the critical exponents are different . the four ones , , , are same , and the other two v , are different 本文对其进行一种推广，推广后n 4 ，对于取不同n值的科赫曲线应用重整化群的格点消元法，仍采用ising模型，结果得到同样的相变点，但是临界指数不完全相同，其中， ， ，相同，而，不同。
The exact expression is n = 4 " , in which n is the stage of koch curve . since we only consider one generator in computation process , so we simplify it as n = 4 ) . the critical point of this kind of koch curve is zero , also called zero temperature phase transition , and this is the character of all the limited branching systems 无分支科赫曲线是一种典型的分形，前人的研究都局限于n = 4 ( n是用线元，面元，或体元覆盖分形系统所需要的覆盖次数，确切的应该写成n = 4 ~ n ，其中n为科赫曲线的级，但我们在重整化群计算时只考虑一个生成元，所以简化为n = 4 )情况，这种科赫曲线的相变点为零，是一种零温相变，这也是有限分岔系统的相变特征。
