Multifractal analysis of the branch structure of diffusion-limited aggregates

We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered

Multifractal analysis of selected rare-earth elements.

The multifractal formalism is applied to the energy eigenvalues of Ce I, CeII, Nd II, SmI, SmII, and Tb I. The R´enyi dimensionsDq , mass exponents τ(q) and f (α) spectra are calculated and used to characterize the eigenvalue spectra. It is found that these elements show multi-scaling behaviour that can be accurately modelled by simple multifractal recursive Cantor sets. The effect of unfolding the spectra is also investigated

Multifractal analysis of the branch structure of diffusion-limited aggregates

We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered

Multifractal analysis of the branch structure of diffusion-limited aggregates

We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered

Left-sided multifractality in a binary random multiplicative cascade.

In this paper we study a binary random multiplicative cascade. Specifically, the cascade is used to produce and study left-sided multifractal random measures. Extensive numerical simulations of the random cascade process were undertaken and f (a) spectra obtained and compared with the analytical results. We believe that this model and approach can serve as a simple and fundamental tool in the analysis and understanding of physical systems possessing an underlying multiplicative structure

Multifractal analysis of selected rare-earth elements.

The multifractal formalism is applied to the energy eigenvalues of Ce I, CeII, Nd II, SmI, SmII, and Tb I. The R´enyi dimensionsDq , mass exponents τ(q) and f (α) spectra are calculated and used to characterize the eigenvalue spectra. It is found that these elements show multi-scaling behaviour that can be accurately modelled by simple multifractal recursive Cantor sets. The effect of unfolding the spectra is also investigated

Multifractal analysis of selected rare-earth elements.

The multifractal formalism is applied to the energy eigenvalues of Ce I, CeII, Nd II, SmI, SmII, and Tb I. The R´enyi dimensionsDq , mass exponents τ(q) and f (α) spectra are calculated and used to characterize the eigenvalue spectra. It is found that these elements show multi-scaling behaviour that can be accurately modelled by simple multifractal recursive Cantor sets. The effect of unfolding the spectra is also investigated

Multifractal analysis of selected rare-earth elements.

The multifractal formalism is applied to the energy eigenvalues of Ce I, CeII, Nd II, SmI, SmII, and Tb I. The R´enyi dimensionsDq , mass exponents τ(q) and f (α) spectra are calculated and used to characterize the eigenvalue spectra. It is found that these elements show multi-scaling behaviour that can be accurately modelled by simple multifractal recursive Cantor sets. The effect of unfolding the spectra is also investigated

Higher-order phase transitions on financial markets

Statistical and thermodynamic properties of the anomalous multifractal structure of random interevent (or intertransaction) times were thoroughly studied by using the extended continuous-time random walk (CTRW) formalism of Montroll, Weiss, Scher, and Lax. Although this formalism is quite general (and can be applied to any interhuman communication with nontrivial priority), we consider it in the context of a financial market where heterogeneous agent activities can occur within a wide spectrum of time scales. As the main general consequence, we found (by additionally using the Saddle-Point Approximation) the scaling or power-dependent form of the partition function, Z(q'). It diverges for any negative scaling powers q' (which justifies the name anomalous) while for positive ones it shows the scaling with the general exponent τ(q'). This exponent is the nonanalytic (singular) or noninteger power of q', which is one of the pilar of higher-order phase transitions. In definition of the partition function we used the pausing-time distribution (PTD) as the central one, which takes the form of convolution (or superstatistics used, e.g. for describing turbulence as well as the financial market). Its integral kernel is given by the stretched exponential distribution (often used in disordered systems). This kernel extends both the exponential distribution assumed in the original version of the CTRW formalism (for description of the transient photocurrent measured in amorphous glassy material) as well as the Gaussian one sometimes used in this context (e.g. for diffusion of hydrogen in amorphous metals or for aging effects in glasses). Our most important finding is the third- and higher-order phase transitions, which can be roughly interpreted as transitions between the phase where high frequency trading is most visible and the phase defined by low frequency trading. The specific order of the phase transition directly depends upon the shape exponent $\alpha$ defining the stretched exponential integral kernel. On this basis a simple practical hint for investors was formulated