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We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν↔1/ν, for integer-value ν of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as . The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.

Three aspects of time series are uncertainty (dispersion at a given time scale), scaling (time-scale dependence) and intermittency (inclination to change dynamics). Simple measures of dispersion are the mean absolute deviation and the standard deviation; scaling exponents describe how dispersions change with the time scale. Intermittency has been defined as a difference between two scaling exponents. After taking a moving average, these measures give descriptive information, even for short heart rate records. For this data, dispersion and intermittency perform better than scaling exponents.

A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.