STUDY OF DIFFERNTIAL EQUATIONS BY RENORMALIZATION GROUP METHODS

1.伊藤和田村(金泽大学)用重整化群(块自旋变换)方法研究了经典的O(N)对称自旋模型。他们首先证明了关联函数可以用自回避行走来描述，这使他们能够在临界温度下获得几乎最优的界限。在第二阶段，他们论证了函数行列式Det^&lt；N/2&gt；(1+2iGpsi/Roo&lt；N&gt；)相对于Psi的可积性，其中Psi是真正引入的傅里叶变换域。利用聚合物(团簇)膨胀技术，他们证明了逆临界温度β_C在二维上服从约束的β_C&GT；N log N，这意味着存在强偏差。(大于或等于3的维度为Beta_C-N。)这种方法有望为猜想BETA_C=*建立一个完整的证明，目前他们正朝着这个方向工作。2.Teramoto利用柱坐标法研究了不可压缩粘性流体绕圆柱的流动。3.Teramoto和Ito研究了湍流的性质，其中包括关于能量耗散和偏离能量的Kolmogorov定律。他们试图从Navier-Stokes方程推导出偏差，但今年未能得到具体结果。4.池边和岛田(Setuanan University.)证明了具有位势V(x)=muf(x)delta(x&lt；@D12@&gt；D1-a&lt；@D12@&gt；D1)的薛定谔算子在球面上的凸性(在预解收敛意义下)为u*(]sy.+-.[)*。这与原子的阿尔法衰变有关。

1.ITO and TAMURA (Kanazawa Univ.) studied classical O(N) symmetric spin model by renormalization group (block spin transformation) method. They first showed that the correlation functions can be described by self-avoiding walks which enabled them to obtain almost optimal bounds for the critical temperatures. In the second stage, they argued the integrability of the functional determinent det^<N/2>(1+2iGpsi/ROO<N>) with respect to psi, where psi is the auxially field introcuced for Fourier Transformation. Using the technique called the polymer (cluster) expansion, they showed that the inverse critical temperature beta_C obeys the bound beta_C>N log N in two dimensions, which implies the existence of strong deviation. (beta_C-N for the dimension more than or equal to 3.) This method is expected to establish a complete proof of the conjecture beta_C=* and they are currently working in this direction.2.Teramoto investigated flow of non-compressible viscous fluid around a cyclinder by using cylindrical coordinate. He established that the equation exhibits global solution in time if the initial condition is sufficiently close to the stational current.3.Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.4.Ikebe and Shimada (Setuanan Univ.) showed that the Schroedinger Operator with the potential V(x)=muf(x)delta(x<@D12@>D1-a<@D12@>D1) distributed on the sphere conveges (in the sense of resolvent convergence) as mu*(]SY.+-。[)*. This is related the the alpha decay of atoms.