Graph Concatenations to Derive Weighted Fractal Networks

Given an initial weighted graph G0, an integer m>1, and m scaling factors f1,…,fm∈0,1, we define a sequence of weighted graphs Gkk=0∞ iteratively. Provided that Gk−1 is given for k≥1, we let Gk−11,…,Gk−1m be m copies of Gk−1, whose weighted edges have been scaled by f1,…,fm, respectively. Then, Gk is constructed by concatenating G0 with all the m copies. The proposed framework shares several properties with fractal sets, and the similarity dimension dfract has a great impact on the topology of the graphs Gk (e.g., node strength distribution). Moreover, the average geodesic distance of Gk increases logarithmically with the system size; thus, this framework also generates the small-world property.

Research and Proof of Decidability on Self-Similar Fractals

Some undecidability on self-affine fractals have been supported. In this paper, we research on the decidability for self-similar fractal of Dubes type. In fact, we prove that the following problems are decidable to test if the Hausdorff dimension of a given Dubes self-similar set is equal to its similarity dimension, and to test if a given Dubes self-similar set satisfies the strong separation condition.

BOUNDS FOR DIMENSION OF THE ATTRACTOR OF A SCALED IFS

A hyperbolic iterated function system (IFS) consists of a complete metric space X together with a finite set of contraction mappings on X. In this paper, the notion of scaled IFS is defined and its existence conditions are examined. The relation between the similarity dimension of the attractors of a given homogeneous IFS and a scaled IFS and its dependency on the scaling factor are studied. A lower and upper bounds for the Hausdorff dimension of the attractor of a scaled IFS is obtained.

SEPARATION PROPERTIES FOR ITERATED FUNCTION SYSTEMS OF BOUNDED DISTORTION

In this paper we study a general separation property for subsystems G, whose attractor KG is a sub-self-similar set. This is a generalization of the Lau-Ngai weak separation property for the bounded distortion case. For subsystems with positive Hausdorff measure in its similarity dimension, we characterize the subsets of KG with positive measure where the separation property may fail. We exhibit two examples of fractal sets, one not satisfying the weak separation property and whose existence was questioned by Zerner, the other having positive Hausdorff measure in its dimension and with the separation property failing on a subset of positive measure.

The Calculation Method of Similarity Degree for City Logistics Facilities

For the multi-attribute characteristics of scale, quantity, service radius and target of service in city logistics facilities，this paper considered the similar phenomena between city logistics facilities caused by the interaction of different social and economic attributes. Based on the analysis of the similarity degree calculation methods in computer science and mechanical engineering, it proposed two calculation methods of similarity degree in city logistics facilities. The qualitative and quantitative attributes were considered separately in the first method, the quantitative attributes were disposed by triangular fuzzy number.The similarity dimension was introduced as the basis of the similarity degree calculation in the second method. A merge processing method was used to incorporate all similar characteristics of every similarity dimension and a similarity calculation formula was deduced from the theory of similarity.

Exceptional sets for self-similar fractals in Carnot groups

We consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

EXPERIMENTAL VERIFICATION FOR FRACTAL TRANSITION USING A FORCED DAMPED OSCILLATOR

A damped oscillator stochastically driven by temporal forces is experimentally investigated. The dynamics is characterized by a set Γ(C) of trajectories in a cylindrical space, where C is a set of initial states on the Poincaré section. Two sets, Γ(C) and C, are attractive and unique invariant fractal sets that approximately satisfy specific equations derived previously by the authors. The correlation dimension of the set C is in good agreement with the similarity dimension obtained for a strictly self-similar set constructed by contraction mappings while C is a self-affine set constructed by non-contraction mappings.