Non-Spectrality of Certain Self-Affine Measures on the Generalized Spatial Sierpinski Gasket

Let μM,Dbe a self-affine measure associated with an expanding integer matrix M=[ p1,0,0;p4,p2,0;p5,0,p3]and the digit set D={ 0,e1,e2,e3}in the space R3, where p1,p2,p3∈Z\{ 0,±1 }, p4,p5∈Zand e1,e2,e3are the standard basis of unit column vectors in R3. In this paper, we mainly consider the case p1,p2,p3∈2Z+1, p2≠p3, p4=l(p1−p2), p5=l(p3−p1),where l∈2Z. We prove that μM,Dis a non-spectral measure, and there are at most 4-element μM,D-orthogonal exponentials, and the number 4 is the best. The results here generalize the known results.

AN APPROXIMATION METHOD TO ESTIMATE THE HAUSDORFF MEASURE OF THE SIERPINSKI GASKET

In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/(l + l/2n-3)s≤Hs(S)≤ Pn(S). An algorithm is presented to get Pn(S) for n ≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.

Sierpinski Gasket上的布朗运动与Dirichlet问题

利用布朗运动的性质获得了SierpinskiGasket上Dirichlet问题及Neumann问题的解 ,并使解具有更为简捷的表达式 。

SOME PATH PROPERTIES OF BROWNIAN MOTION AND SUPER-BROWNIAN MOTION ON THE SIERPINSKI GASKET

We study in this paper the path properties of the Brownian motion and super-Brownian motion on the fractal structure-the Sierpinski gasket. At first some results about the limiting behaviour of its increments are obtained and a kind of law of iterated logarithm is proved. Then A Lower bound of the spreading speed of its corresponding super-Brownian motion is obtained.

Sierpinski Gasket上Brownian运动的局部时

本文证明了Sierpinskigasket上Brown运动局部时的联合连续性．

Sierpinski gasket上Brown运动的重点

设｛Ｘ（ｔ）≥０｝为 Ｓｉｅｒｐｉｎｓｋｉｇａｓｋｅｔ上的Ｂｒｏｗｎ运动，Ｅ为［０，＋∞）中闭子集，本文考虑了何时Ｘ（Ｅ）具有ｋ重点，给出了一个充分条件和一个必要条件。

THE BOUNDEDNESS OF BILINEAR SINGULAR INTEGRAL OPERATORS ON SIERPINSKI GASKETS

In the paper we give the boundedness estimate of bilinear singular integral operators on Sierpinski gasket inspired from [ 1 ].

A NEW LOWER BOUND OF THE HAUSDORFF MEASURE OF THE SIERPINSKI GASKET

For the Sierpinski gasket, by using a sort of cover consisting of special regular hexagons, we define a new measure that is equivalent to the Hausdorff measure and obtain a lower bound of this measure. Moreover, the following lower bound of the Hausdroff measure of the Sierpinski gasket has been achieved H^s（S）≥0.670432,where S denotes the Sierpinski gasket, s = dimn（S） = log23, and H^s（S） denotes the s-dimensional Hausdorff measure of S. The above result improves that developed in .

Nonconstant Harmonic Functions on the Level 3 Sierpinski Gasket

We give a detailed description of nonconstant harmonic functions on the level 3 Sierpinski gasket. Then we extend the method onβ-set with 1/3〈β〈1/2.

The Mean Time to Absorption on Horizontal Partitioned Sierpinski Gasket Networks

The random walk is one of the most basic dynamic properties of complex networks,which has gradually become a research hotspot in recent years due to its many applications in actual networks.An important characteristic of the random walk is the mean time to absorption,which plays an extremely important role in the study of topology,dynamics and practical application of complex networks.Analyzing the mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random walks on the network.The existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network,but they have failed to prove whether these effects are related to the scale of these self-similar structures.In this article,we construct and study a class of Horizontal Par-titioned Sierpinski Gasket network model based on the classic Sierpinski gasket net-work,which is composed of local self-similar structures,and the scale of these structures will be controlled by the partition coefficient k.Then,the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained,which prove that the size of the self-similar structure in the network will directly restrict the influence of the self-similar structure on the properties of random walks on the network.Finally,we also analyzed the mean time to absorption of different absorption nodes on the network tofind the location of the node with the highest absorption efficiency.

Martin boundary and exit space on the Sierpinski gasket

We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.

Sierpinski Gasket格点图上的渗流模型

本文研究了Sierpinskigasket上的边渗流模型．首先证明了该模型没有临界现象，进一步给出其一个指数衰减律，进而证明了临界值的唯一性．

SOME GEOMETRIC PROPERLIES OFBROWNIAN MOTION ON SIERPINSKI GASKET

Let {X(t), ≥ 0} be Brownian motion on Sierpinski gasket.The Hausdorff and packingdimensions of the image of a compact set are studied. The uniform Hausdorff and packingdimensions of the inverse image are also discussed.

Sierpinski Gasket上布朗运动的维数性质

本文研究Sierpinski Gasket上的布朗运动X的象集、图集的Hausdorff维数性质,证明了存在零概率集Ⅳ,若ω∈Nc,则对任意紧集F(?)[0,∞),有(i)dimX(F+t)=min(α-1dimF,df),a.e.t>0,(ii)dimGrX|F+t=min(α-1dimF,(1-α)df+dimF),a.e.t>0,其中ds=log3/log2,α=log2/log5.

The Non-Degeneracy of Harmonic Structures on Planar Sierpinski Gaskets

We present a direct and short proof of the non-degeneracy of the harmonic structures on the level-n Sierpinski gaskets for any n≥2,which was conjectured by Hino in[1,2]and confirmed to be true by Tsougkas[8]very recently using Tutte’s spring theorem.

Dimensional Results for the Moran-Sierpinski Gasket

In this paper, the dimensional results of Moran-Sierpinski gasket are considered. Moran-Sierpinski gasket has the Moran structure, which is an extension of the Sierpinski gasket by the method of Moran set. By the technique of Moran set, the Hausdorff, packing, and upper box dimensions of the Moran-Sierpinski gasket are given. The dimensional results of the Sierpinski gasket can be seen as a special case of this paper.

Hausdorff measure of Sierpinski gasket

By farming a sequence of coverings of the Sierpinski gasket,a descending sequence of the upper limits of Hausdorff measure is obtained.The limit of the sequence is the best upper limit of the Hausdorff measure known so far.

Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.

上凸密度函数与Hausdorff测度──Sierpinski垫片

主要讨论了Ｓｉｅｒｐｉｎｓｋｉ垫片的上凸密度函数在其端点处的计算问题，并通过具体的数值计算，得出了在端点处的上凸密度函数不等于１的结论．