Rhythmic precipitate patterns and fractal structure

Liesegang patterns of parallel precipitate bands are obtained when solutions containing co-precipitate ions interdiffuse in a 1D gel matrix.The sparingly soluble salt formed,displays a beautiful stratification of discs of precipitate perpendicular to the 1D tube axis.The Liesegang structures are analyzed from the viewpoint of their fractal nature.Geometric Liesegang patterns are constructed in conformity with the well-known empirical laws such as the time,band spacing and band width laws.The dependence of the band spacing on the initial concentrations of diffusing（outer）and immobile（inner）electrolytes（A0 and B0,respectively）is taken to follow the Matalon-Packter law.Both mathematical fractal dimensions and box-count dimensions are calculated.The fractal dimension is found to increase with increasing A0 and decreasing B0.We also analyze mosaic patterns with random distribution of crystallites,grown under different conditions than the classical Liesegang gel method,and report on their fractal properties.Finally,complex Liesegang patterns wherein the bands are grouped in multiplets are studied,and it is shown that the fractal nature increases with the multiplicity.

Visitor flow pattern of Expo 2010

Expo 2010 Shanghai China was a successful, splendid, and unforgettable event, leaving us with valuable experi- ences. The visitor flow pattern of the Expo is investigated in this paper. The Hurst exponent, the mean value, and the standard deviation of visitor volume indicate that the visitor flow is fractal with long-term stability and correlation as well as obvious fluctuation in a short period. Then the time series of visitor volume is converted into a complex network by using the visibility algorithm. It can be inferred from the topological properties of the visibility graph that the network is scale-free, small-world, and hierarchically constructed, confirming that the time series are fractal and a close relationship exists among the visitor volumes on different days. Furthermore, it is inevitable that will be some extreme visitor volumes in the original visitor flow, and these extreme points may appear in a group to a great extent. All these properties are closely related to the feature of the complex network. Finally, the revised linear regression is performed to forecast the next-day visitor volume based on the previous 10-day data.

Fractals and Spatial Statistics of Point Patterns

The relationship between fractal point pattern modeling and statistical methods of pa- rameter estimation in point-process modeling is reviewed. Statistical estimation of the cluster fractal dimension by using Ripley＇s K-function has advantages in comparison with the more commonly used methods of box-counting and cluster fractal dimension estimation because it corrects for edge effects, not only for rectangular study areas but also for study areas with curved boundaries determined by re- gional geology. Application of box-counting to estimate the fractal dimension of point patterns has the disadvantage that, in general, it is subject to relatively strong ＂roll-off＂ effects for smaller boxes. Point patterns used for example in this paper are mainly for gold deposits in the Abitibi volcanic belt on the Canadian Shield. Additionally, it is proposed that, worldwide, the local point patterns of podiform Cr, volcanogenic massive sulphide and porphyry copper deposits, which are spatially distributed within irregularly shaped favorable tracts, satisfy the fractal clustering model with similar fractal dimensions. The problem of deposit size （metal tonnage） is also considered. Several examples are provided of cases in which the Pareto distribution provides good results for the largest deposits in metal size-frequency distribution modeling.