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 Sierpin...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.展开更多
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 obta...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.展开更多
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 f...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 .展开更多
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 verti...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.展开更多
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 discus...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.展开更多
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 rece...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.展开更多
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 tech...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.展开更多
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 s...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.展开更多
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)...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.展开更多
文摘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.
文摘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.
基金Project partially supported by the National Natural Science Committee Foundation of Chinathe Natural Science Foundation of Guangdong Provincethe Foundation of the Department of Education of Guangdong Province.
文摘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 .
文摘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.
文摘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.
基金the Nature Science Foundation of China,Grant No.12071213.
文摘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.
基金Supported by the National Natural Science Foundation of China(10771082 and 10871180)
文摘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.
基金Project partially supported by the Foundation of Guangdong Province and the Foundation of Advanced Research Centre, Zhongshan University.
文摘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.
基金This work was supported partially by the National Natural Science Foundation of China(Grant No.10371062).
文摘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.