Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性.     

Growth of Self‐Similar Graphs

Locally finite self‐similar graphs with bounded geometry and without bounded geometry as well as non‐locally finite self‐similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor μ can be defined similarly to the corresponding parameters of continuous self‐similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self‐similar graphs, it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self‐similar fractals: . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 224–239, 2004     

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, normal, and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small H cells. Our results are of the formN=AL ^D _SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD _SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.     

Weak separation properties for self-similar sets

We develop a theory for self-similar sets in that fulfil the weak separation property of Lau and Ngai, which is weaker than the open set condition of Hutchinson.

     

Fractals in physics: Squig clusters,diffusions, fractal measures,and the unicity of fractal dimensionality

The three topics discussed in this paper are largely independent. Part 1: Fractal squig clusters are introduced, and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality. Physics on these new geometric shapes should prove tractable. As background, the author's theories of squig intervals and squig trees are reviewed, and restated in more versatile form. Part 2: The notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality. Scaling noises are touched upon. A common formalism is presented for three forms of anomalous diffusion: the ant in the fractal labyrinth, fractional Brownian motion, and Lévy stable motion. The fractal dimensionalities common to diverse shapes generated by diffusion are given, in Table I, as functions of the latent dimensionalities of the support of the motion and of the diffusion itself. Part 3: It is argued that every fractal point set has a unique fractal dimensionality, but it is pointed out that many fractals involve diverse combinations of many fractal point sets. Such is, in particular, the case for fractal measures and for fractal graphs, often called hierarchical lattices. The fractal measures that the author had introduced in the early 1970s are described, including new developments.     

Self-avoiding walks on random fractal environments

Self-avoiding random walks (SAWs) are studied on several hierarchical lattices in a randomly disordered environment. An analytical method to determine whether their fractal dimensionD _saw is affected by disorder is introduced. Using this method, it is found that for some lattices,D _saw is unaffected by weak disorder; while for othersD _saw changes even for infinitestimal disorder. A weak disorder exponent is defined and calculated analytically [ measures the dependence of the variance in the partition function (or in the effective fugacity per step)vL on the end-to-end distance of the SAW,L]. For lattices which are stable against weak disorder (<0) a phase transition exists at a critical valuev=v ^* which separates weak- and strong-disorder phases. The geometrical properties which contribute to the value of are discussed.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Elastic–plastic transition in three-dimensional random materials: massively parallel simulations,fractal morphogenesis and scaling functions

Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscopically uniform loadings. The materials are cubic-shaped domains (of up to 100?×?100?×?100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J ₂ flow rule. The spatial assignment of the grains’ elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The randomness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the ‘reduced von-Mises stress’, ‘reduced plastic volume fraction’ and ‘reduced fractal dimension’), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitutive properties and their random noise levels.