Multifractals

A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, and C. Vaillant, Wavelet based multifractal formalism Application to DNA sequences, satellite images of the cloud structure and stock market data, in The Science of Disasters Climate Disruptions, Heart Attacks, and Market Crashes, Springer-Verlag, Berlin, 2002, pp. 26-102. 

Extracting Information ABOUT Surface Heterogeneity Effect on Heterogeneous Reactions Using Multifractal Scaling... 

We have applied the multifractal scaling [55] which relates the analysis of the distribution of the reaction probabilities over the length of the rough surfaee. The steps in the multifractal scaling analysis are given below. 

There are three basic equations in the multifractal scaling analysis. 

Figure 5. q-x(q) and a-F(a) multifractal plots for the three surfaces of RDWD. 

Figure 6. Comparision between the multifractal plots for the rough surface of 50x50 for RD and RDWD.

Figure 7. Effect of sticking probability on the multifractal plots.

For each case above, the reaction probabilities of different surface sites are recorded and analyzed using multifractal analysis. 

Figure 10. q-x(q) multifractal plots for different Pini, m, surface roughness andNE. 

The q-T(q) curves are nonlinear for different values of m, Pini, NE and surface roughness as ean be seen from Figure 10 whieh indicates multifractality From Figure 10(a), for q >0, as m deereases, the curvature of q-x(q) curves also decreases gradually, indieating relatively homogeneous reaetion probability distribution. This also indicates that, the number of aetive... 

Baveye, P., Boast, C. W., Gaspard, S., and Tarquis, A. M. (2008). Introduction to fractal geometry, fragmentation processes and multifractal measures Theory and operational aspects of their application to natural systems In Bio-Physical Chemistry of Fractal Structures and Process in Environmental Systems, Senesi, N., and Wilkinson, K., eds. IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Vol. 11, John Wiley Sons, Chichester, pp. 11-67. 

Kantelhardt J.W. Zschiegner S.A. Koscielny-Bunde E. Havlin S. Bunde A. and Stanley H.E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica, A316(14), 87-114. 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

The application of the loop-type interface for LC-GC for multifraction introduction has been introduced [134]. The use of microbore LC columns have been used as a means to reduce the injection volumes of solvent [135,136]. 

Figure 7 demonstrates on a logarithmic scale the dependence of perimeter P on area A of the pores obtained from the binary TEM image of CAS30 in Figure 6b. The (log P - log A) plots obtained from the carbon specimen displayed two straight lines with different slopes that can be divided into region I and II, indicating multifractal geometiy of the carbon specimen. The individual surface fractal dimensions in regions I and II were determined from Eqs. (26) and (27) to be 2.08 + 0.018 and 2.72 + 0.046, respectively. The transition area Ab from region I to II were determined to be 108 nm2, which corresponds to the pore diameter of 12 nm based upon spherical pore shape.

These features are then passed on to machine learning algorithms to classify the signals. In particular, multifractal properties are a clear indicator of arrhythmia in ECG signals. The onset of deformity in the ECG signal has a signature pattern that can be used to predict the onset of a heart attack. Scinova proposes to exploit this feature of Rx as an early warning system for intensive care units. 

Multifractal Interpolation and Fractal Concentration-Area (C-A) Method 137... 

In this chapter, the application of multifractal inverse distance weighted (MIDW) interpolation method and a fractal filtering technique, named spatial and spectral analysis or simply (S-A) method (Cheng, 2003), will be illustrated to evaluate geochemical background at the regional and local scale. For this purpose, two case studies will be discussed for two different Italian areas ... 

Agterberg, F. P. (2001). Multifractal simulation of geochemical map patterns. In Geologic Modeling and Simulation Computer Applications in the Earth Sciences (D. F. Merriam and J. C. Davis, eds.), pp. 31—39. Plenum Press, New York. 

Lima, A., De Vivo, B., Cicchella, D., Cortini, M., and Albanese, S. (2003). Multifractal IDW interpolation and fractal filtering method in environmental studies An application on regional stream sediments of Campania Region (Italy). Appl. Geochem. 18, 1853—1865. 

Cheng, Q. (1999a). Multifractality and spatial statistics. Comput. Geos a. 25(10), 946—961. 

---