Spanning Trees on the Sierpinski Gasket

We present the numbers of spanning trees on the Sierpinski gasket SG _d (n) at stage n with dimension d equal to two, three and four. The general expression for the number of spanning trees on SG _d (n) with arbitrary d is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket SG _d,b (n) with d = 2 and b = 3,4 are also obtained.     

Dimer Coverings on the Sierpinski Gasket

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.     

Emergence of the Sierpinski Gasket in Coin-Dividing Problems

The present paper proposes a generation mechanism of a fractal pattern related to a coin system. The problem is formulated in terms of a situation of dividing coins among people. Remarkably, a fractal pattern like the Sierpinski gasket is obtained, by marking all the possible division of coins as a point set. The mechanism for this fractal structure is reduced to nested relations, owing to a hierarchical property of coin denominations. Relevance to dynamical systems is also discussed.     

Hydrodynamic Limit for a Zero-Range Process in the Sierpinski Gasket

We consider a system of random walks on graph approximations of the Sierpinski gasket, coupled with a zero-range interaction. We prove that the hydrodynamic limit of this system is given by a nonlinear heat equation on the Sierpinski gasket.     

基于Sierpinski地毯结构的类分形光子晶体特性研究

将谢尔宾斯基(Sierpinski)地毯结构引入二维光子晶体,设计了一种具有规则分形结构特征的光子晶体-Sierpinski类分形光子晶体.采用时域有限差分法仿真分析了空气背景介质柱和介质背景空气孔结构Sierpinski类分形光子晶体的透射谱.结果表明,只有当TM波入射空气背景介质柱结构时,Sierpinski类分形...     

Two-Point Resistance of a Cobweb Network with a 2r Boundary

We consider the problem of the two-point resistance on an m × n cobweb network with a 2r boundary,which has never been solved before. Up to now researchers just only solved the cases with free boundary or null resistor boundary. This paper gives the general formulae of the resistance between any two nodes in both finite and infinite cases using a method of direct summation pioneered by Tan [Z. Z. Tan, et al., J. Phys. A 46(2013) 195202], which is simpler and can be easier to use in practice. This method contrasts the Green's function technique and the Laplacian matrix approach, which is difficult to apply to the geometry of a cobweb with a 2r boundary. We deduce several interesting results according to our general formula. In the end we compare and illuminate our formulae with two examples. Our analysis gives the result directly as a single summation, and the result is mainly composed of the characteristic roots.     

Chaos for the Sierpinski Carpet

We study the chaotic behavior of the Sierpinski carpet. It is proved that this dynamical system has a chaotic set whose Hausdorff dimension equals that of the Sierpinski carpet.     

States on the Sierpinski Triangle

States on a Sierpinski triangle are described using a formally exact and general Hamiltonian renormalization. The spectra of new (as well as previously examined) models are characterized. Numerical studies based on the renormalization suggest that the only models which exhibit absolutely continuous specta are effectively one-dimensional in the limit of large distances.     

Two-Point Resistance on the Centered-Triangular Lattice

The resistance between any two lattice points in an infinite,centered-triangular lattice of equal resistors is determined using the lattice Green function method.It is shown that the two-point resistance on the centeredtriangular lattice is expressed in terms of the resistance of a triangular lattice.Some exact values for the resistance near the origin of the lattice are presented.For large separation between lattice points the asymptotic forms of the resistance are calculated.     

Dimer-monomer model on the Sierpinski gasket

We present the numbers of dimer-monomers M_d(n) on the Sierpinski gasket SG_d(n) at stage n with dimension d equal to two, three and four. The upper and lower bounds for the asymptotic growth constant, defined as z_SGd=lim_v→∞lnM_d(n)/v where v is the number of vertices on SG_d(n), are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of z_SGd can be evaluated with more than a hundred significant figures accurate. From the results for d=2,3,4, we conjecture the upper and lower bounds of z_SGd for general dimension. The corresponding results on the generalized Sierpinski gasket SG_d,b(n) with d=2 and b=3,4 are also obtained.     

Ising thermodynamics on the Sierpinski gasket

We report a calculation of the thermodynamic properties of an Ising system on a fractal lattice, the Sierpinski gasket (SG). The scale-invariant geometry of SG leads to a wider critical region compared to that in translationally invariant systems. We calculate exactly the near-neighbor correlation function and specific heat and discuss their critical behaviour.     

Ising models on the lattice Sierpinski gasket

Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.     

Two-Point Resistance of a Non-Regular Cylindrical Network with a Zero Resistor Axis and Two Arbitrary Boundaries

We study a problem of two-point resistance in a non-regular m × n cylindrical network with a zero resistor axis and two arbitrary boundaries by means of the Recursion-Transform method. This is a new problem never solved before, the Green's function technique and the Laplacian matrix approach are invalid in this case. A disordered network with arbitrary boundaries is a basic model in many physical systems or real world systems, however looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of the arbitrary boundaries, the boundary is like a wall or trap which affects the behavior of finite network. In this paper we obtain a general resistance formula of a non-regular m × n cylindrical network, which is composed of a single summation. Further,the current distribution is given explicitly as a byproduct of the method. As applications, several interesting results are derived by making special cases from the general formula.     

Recurrence and return times of the Sierpinski carpet

We compute the limit distribution of the recurrence and of the normalizedk th return times to small sets of the Sierpinski carpet with respect to a natural measure defined on it. It is proved that this dynamical system follows the Poisson law, as one could have expected for such schemes. We study different sequences which converge in finite distribution to the Poisson point process. This limit in law is very interesting in ergodic theory, and it seems to appear for chaotic dynamical systems such as the one we study.     

Generalized Rayleigh Criterion of Two-Point Resolution

We propose a criterion of resolution of one-type incoherent signals with respect to two scalar parameters and estimate the corresponding resolution limits. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 5, pp. 441–445, May 2005.     

Monte Carlo study of the XY-model on Sierpinski gasket

We have performed a Monte Carlo study of the classical XY-model on two-dimensional Sierpinski gaskets (SGs) of several cluster sizes. From the dependence of the helicity modulus on the cluster size we conclude that there is no phase transition in this system at a finite temperature. This is in agreement with previous findings for the harmonic approximation to the XY-model on SG and is analogous to the absence of finite-temperature phase transition for the Ising model on fractals with a finite order of ramification.     

Thresholds and Universality of the Site Percolation on the Sierpinski Carpets

Following the methods proposed by Yonezawa, Sakamoto and Hori, we have calculated the percolation thresholds P_c, their error bars ΔP_c, and the correlation length exponents v of a family of the Sierpinski carpets for the site percolation problems by making use of MonteCarlo simulations and finite size scaling. We have found the dependence of P_c and v on the fractal dimensionality D_f and the lacunarity. We ascertain that the site percolation problems on a family of Sierpinski carpets with central cutouts and different D belong to different universal classes, and those on Sierpinski carpets with same D_f but of different lacunarities belong to different universal classes.     

Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

We prove \({|x|^{-2}}\) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^{d}}\), in the upper critical dimension d = 4. This is a statement that the critical exponent \({\eta}\) exists and is equal to zero. Results of this nature have been proved previously for dimensions \({d \ge 5}\) using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.     

Simultaneous studies of the elasticity and the conductivity exponents of Sierpinski carpets

Simultaneous studies of the conductivity and the clasticity exponents on Sierpinski carpet made of metal and voids are reported. The elasticity exponent is =0.29±0.01, the conductivity exponent =0.22±0.01. It is obvious that they belong to different universality classes.     

Coupled maps and pattern formation on the Sierpinski gasket

The bifurcation structure of coupled maps on the Sierpinski gasket is investigated. The fractal character of the underlying lattice gives rise to stability boundaries for the periodic synchronized states with unusual features and spatially inhomogeneous states with a complex structure. The results are illustrated by calculations on coupled quadratic and cubic maps. For the coupled cubic map lattice bistability and domain growth processes are studied.