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11.
Tetsuya Hattori 《Journal of statistical physics》2007,127(3):609-627
Let W(x,y) = ax
3+ bx
4+ f
5
x
5+ f
6
x
6+ (3 ax
2)2
y+ g
5
x
5
y + h
3
x
3
y
2 + h
4
x
4
y
2 + n
3
x
3
y
3+a
24
x
2
y
4+a
05
y
5+a
15
xy
5+a
06
y
6, and X = , , where the coefficients are non-negative constants, with a > 0, such that X
2(x,x
2)−Y(x,x
2) is a polynomial of x with non-negative coefficients.
Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding
paths on the 3 and 4 dimensional pre-gaskets.
We prove that there exists a unique fixed point (x
f
,y
f
) of Φ in the invariant set .
2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41. 相似文献
12.
通过在任意给定的凸四边形和三角形上构造一个不同于通常欧氏度量的度量,证明了如果把构造经典Sierpinski地毯的初始图形正方形换成任意一个凸四边形或者三角形,则得到的广义Sierpinski地毯与经典的Sierpinski地毯具有相同的Haus-dorff维数. 相似文献
13.
We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters. 相似文献
14.
关于自相似集的Hausdorff测度的一个判据及其应用 总被引:6,自引:1,他引:5
讨论了满足开集条件的自相似集。对于此类分形,用自然覆盖类估计它的Hausdorff测度只能得到一个上限,因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用,统一处理了一类自相似集,得到了平面上的一个Cantor集-Cantor尘的Hausdorff测度的准确值,并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。 相似文献
15.
Gang Tang Rongji Wen Kui Han Hui Xia Dapeng Hao Wei Zhou Xiquan Yang Yuling Chen 《Physica A》2010,389(21):4552-4557
The growth of the modified Family model and the Etching model on the Sierpinski carpet is studied by means of numerical simulations. The evolving interface of the aggregates is described by the well-established Family-Vicsek dynamic scaling approach. The results of the modified Family model prove the universality of the fractional Langevin equation introduced by Lee and Kim [S.B. Lee, J.M. Kim, Phys. Rev. E 80 (2009) 021101]. The Etching model also shows good scaling behavior. We conjecture that the systematic deviations of the data found in the ballistic deposition [C.M. Horowitz, F. Romá, E.V. Albano, Phys. Rev. E 78 (2008) 061118] may be due to the finite-size effects of the Ballistic Deposition model. 相似文献
16.
《Mathematische Nachrichten》2018,291(11-12):1743-1756
The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one. 相似文献
17.
In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite. 相似文献
18.
19.
We present the number of dimer coverings N
d
(n) on the Sierpinski gasket SG
d
(n) at stage n with dimension d equal to two, three, four or five. When the number of vertices, denoted as v(n), of the Sierpinski gasket is an even number, N
d
(n) is the number of close-packed dimers. When the number of vertices is an odd number, no close-packed configurations are possible
and we allow one of the outmost vertices uncovered. The entropy of absorption of diatomic molecules per site, defined as
, is calculated to be ln (2)/3 exactly for SG
2. The numbers of dimers on the generalized Sierpinski gasket SG
d,b
(n) with d=2 and b=3,4,5 are also obtained exactly with entropies equal to ln (6)/7, ln (28)/12, ln (200)/18, respectively. The number of dimer
coverings for SG
3 is given by an exact product expression, such that its entropy is given by an exact summation expression. The upper and lower
bounds for the entropy are derived in terms of the results at a certain stage for SG
d
(n) with d=3,4,5. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical
value of
with d=3,4,5 can be evaluated with more than a hundred significant figures accurate.
This paper is written during the Lung-Chi Chen visit to PIMS, University of British Columbia. The author thanks the institute
for the hospitality. 相似文献
20.
Chao Xie Qi-Meng Wu Ruo-Ning Li Gao-Chen Gu Xue Zhang Na Li Richard Berndt J rg Kr ger Zi-Yong Shen Shi-Min Hou Yong-Feng Wang 《中国化学快报》2016,27(6):807-812
To date, supramolecular chemistry is an ever growing research field owing to its crucial role in molecular catalysis, recognition, medicine, data storage and processing as well as artificial photosynthetic devices.Different isolated supramolecules were prepared by molecular self-assembly on surfaces. This review mainly focuses on supramolecular aggregations on noble metal surfaces studied by scanning tunneling microscopy, including dimers, trimers, tetramers, pentamers, wire-like assemblies and Sierpin′ ski triangular fractals. The variety of self-assembled structures reflects the subtle balance between intermolecular and molecule–substrate interactions, which to some extent may be controlled by molecules, substrates and the molecular coverage. The comparative study of different architectures helps identifying the operative mechanisms that lead to the structural motifs. The application of these mechanisms may lead to novel assemblies with tailored physicochemical properties. 相似文献