格点巢分形上的渗流模型

该文研究了一类格点分形图 (格点巢分形 )上的渗流模型 ,证明了该模型没有临界现象 ,进一步给出一个指数衰减律 .同时 ,指出一般有限分岔图上的渗流模型没有临界现象     

Sierpinski Carpet格点图上边渗流及其临界现象

本文讨论了一种非平移不变格点图──格点分形图上的边渗流及其临界现象。     

高密度情形格点Sierpinski地毯上渗流模型无穷开串的唯一性

本文证明高密度情形格点Ｓｉｅｒｐｉｎｓｋｉ地毯上边渗流模型无穷开串的唯一性,同时给出本模型相变存在性的一个新的证明．一种再标度技巧被发展并用作我们证明的主要工具．     

分形Sierpinski Gasket上单点介质作用下的超过程

本文考虑分形ＳｉｅｒｐｉｎｓｋｉＧａｓｋｅｔ上的分枝粒子系统，当其受单点介质作用而导出了一类超过程．在理论上对过程的存在性给出了证明，同时研究了这类超过程作为随机测度值过程的轨道特征即证明了其轨道的连续性和联合连续的密度场的存在性．     

一阶线性和拟线性双曲型方程的格点模型

一阶线性和拟线性双曲型方程的格点模型李元香（武汉大学软件工程国家重点实验室）黄樟灿（武汉工学院）ＬＡＴＴＩＣＥＭＯＤＥＬＳＦＯＲＦＩＲＳＴＯＲＤＥＲＬＩＮＥＡＲＡＮＤＱＵＡＳＩ－ＬＩＮＥＡＲＨＹＰＥＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＬｉＹｕａｎ－ｘｉａｎ...     

Zakharov格点动力系统的指数吸引子

利用一般格点动力系统存在指数吸引子的充分条件,证明自治Zakharov格点动力系统的指数吸引子的存在性.     

外场中分形晶格上Gauss模型的临界现象

在分形晶格上把Gauss模型加以推广 ,认为Gauss分布常数和重整化的外磁场都依赖于晶格格点的配位数 ,且格点i和j上的Gauss分布常数b _qi 和bb_qj 满足关系b_qi b_qj = q_i q_j (q_i 和q_j分别是格点i和j的配位数 ) .利用实空间重整化群变换的方法 ,在Koch型曲线和一族钻石型等级 (DH)晶格上计算了外场中Gauss模型的临界点和临界指数 .结果表明 :对于这些晶格 ,在临界点 ,格点近邻相互作用参量和外磁场都可表示为K =b_qi q_i 和h q_i =0的形式 ,h_qi 是格点i上的简化磁场 ,而临界指数则决定于分形系统的分形维数d_f;另外 ,对于DH晶格 ,临界指数与平移对称晶格上的结果完全相同 ,且在d_f=4时和平均场理论的结果完全一致     

格点形心问题的若干结果

设n(k)为满足如下条件的最小整数，给定平面上任意n个格点，其中必存在k个点的形心也是格点，文献[4]提出关于确定n(4)的未解问题，本文给出解答n(4)=13，并进一步给出相关的一些问题的结果。     

格上Ising模型的临界失真估计

本文是我们先前工作「１」，「２」的继续，对格上Ｉｓｉｎｇ模型的临界失真ｄｃ的估计，Ｎｅｗｍａｎ和Ｂａｋｅｒ「６」证明了ｄｃ和Ｉｓｉｎｇ模型的Ｍａｙｅｒ级数之收敛半径Ｒ有以下关系：ｄｃ＝Ｒ／（１＋Ｒ），在「１」中匀提出了估计Ｒ及ｄｃ的新方法，并它计算了二维矩形格Ｚ＾２上Ｉｓｉｎｇ模型的临界失真ｄｃ，此文中我们继续应用此方法首次计算了定义在其它二维和三维格上Ｉｓｉｎｇ模型的临界失真ｄｃ，数值计算的结果     

关于Sierpinski垫片的一个注记

指出本刊2001年发表的“关于S ierp insk i垫片的H ausdorff测度”一文的主要结论是错误的,并给出有关讨论.     

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

Uniqueness of the Infinite Open Cluster for High-density Percolation on Lattice Sierpinski Carpet

We prove the uniqueness of infinite open cluster for high-density bond percolation on lattice Sierpinski Carpet; forthermore, an alternative proof of the existence of phase transition of the model is given. A rescaling technique is developed and used as the main tool of our proofs. Research supported by the National Natural Science Foundation of China (Grant number 19771008) and Doctoral Program Foundation of Institution of Higher Education (Grant number 96002704)     

Isoperimetric Estimates on Sierpinski Gasket Type Fractals

For a compact Hausdorff space that is pathwise connected, we can define the connectivity dimension to be the infimum of all such that all points in can be connected by a path of Hausdorff dimension at most . We show how to compute the connectivity dimension for a class of self-similar sets in that we call point connected, meaning roughly that is generated by an iterated function system acting on a polytope such that the images of intersect at single vertices. This class includes the polygaskets, which are obtained from a regular -gon in the plane by contracting equally to all vertices, provided is not divisible by 4. (The Sierpinski gasket corresponds to .) We also provide a separate computation for the octogasket (), which is not point connected. We also show, in these examples, that , where the infimum is taken over all paths connecting and , and denotes Hausdorff measure, is equivalent to the original metric on . Given a compact subset of the plane of Hausdorff dimension and connectivity dimension , we can define the isoperimetric profile function to be the supremum of , where is a region in the plane bounded by a Jordan curve (or union of Jordan curves) entirely contained in , with . The analog of the standard isperimetric estimate is . We are particularly interested in finding the best constant and identifying the extremal domains where we have equality. We solve this problem for polygaskets with . In addition, for we find an entirely different estimate for as , since the boundary of has infinite measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.

     

关于Sierpinski垫片的Hausdorff测度

本文给出了 Sierpinski垫片的另一构造方法 ,并给出了它的 Hausdorff测度的精确值     

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.     

Entanglement in Percolation

We study finite and infinite entangled graphs in the bond percolationprocess in three dimensions with density p. After a discussionof the relevant definitions, the entanglement critical probabilitiesare defined. The size of the maximal entangled graph at theorigin is studied for small p, and it is shown that this graphhas radius whose tail decays at least as fast as exp(–n/logn); indeed, the logarithm may be replaced by any iterate oflogarithm for an appropriate positive constant . We explorethe question of almost sure uniqueness of the infinite maximalopen entangled graph when p is large, and we establish two relevanttheorems. We make several conjectures concerning the propertiesof entangled graphs in percolation. http://www.statslab.cam.ac.uk/\simgrg/1991 Mathematics Subject Classification: primary 60K35; secondary05C10, 57M25, 82B41, 82B43, 82D60.