Scaling in Non-stationary time series I

Most data processing techniques, applied to biomedical and sociological time series, are only valid for random fluctuations that are stationary in time. Unfortunately, these data are often non stationary and the use of techniques of analysis resting on the stationary assumption can produce a wrong information on the scaling, and so on the complexity of the process under study. Herein, we test and compare two techniques for removing the non-stationary influences from computer generated time series, consisting of the superposition of a slow signal and a random fluctuation. The former is based on the method of wavelet decomposition, and the latter is a proposal of this paper, denoted by us as step detrending technique. We focus our attention on two cases, when the slow signal is a periodic function mimicking the influence of seasons, and when it is an aperiodic signal mimicking the influence of a population change (increase or decrease). For the purpose of computational simplicity the random fluctuation is taken to be uncorrelated. However, the detrending techniques here illustrated work also in the case when the random component is correlated. This expectation is fully confirmed by the sociological applications made in the companion paper. We also illustrate a new procedure to assess the existence of a genuine scaling, based on the adoption of diffusion entropy, multiscaling analysis and the direct assessment of scaling. Using artificial sequences, we show that the joint use of all these techniques yield the detection of the real scaling, and that this is independent of the technique used to detrend the original signal.Comment: 39 pages, 13 figure

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators ($Psi_R^+$, $Psi_R^-$) were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives ($Psi_k^+$, $Psi_k^-$, k = {0,1,2,..}). The main part of the work is to demonstrate that for any smooth real-valued function f in the Schwartz space ($S^-(R)$), the successive derivatives of the n-th power of f (n in Z and n not equal to 0) can be decomposed using only $Psi_k^+$ (Lemma) or with $Psi_k^+$, $Psi_k^-$ (k in Z) (Theorem) in a unique way (with more restrictive conditions). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.Comment: The paper was accepted for publication at Acta Applicandae Mathematicae (15/05/2013) based on v3. v4 is very similar to v3 except that we modified slightly Definition 1 to make it more readable when showing the decomposition with the families of energy operator of the derivatives of the n-th power of

Strange kinetics: conflict between density and trajectory description

We study a process of anomalous diffusion, based on intermittent velocity fluctuations, and we show that its scaling depends on whether we observe the motion of many independent trajectories or that of a Liouville-like equation driven density. The reason for this discrepancy seems to be that the Liouville-like equation is unable to reproduce the multi-scaling properties emerging from trajectory dynamics. We argue that this conflict between density and trajectory might help us to define the uncertain border between dynamics and thermodynamics, and that between quantum and classical physics as well.Comment: submitted to Chemical Physic

Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^{\gamma}}{\partial x^{\gamma}}[P(x,t) ]^{\nu}$. Exact time-dependent solutions are found for $\nu = \frac{2-\gamma}{1+ \gamma}$ ($-\infty<\gamma \leq 2$). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely $q=\frac{\gamma+3}{\gamma+1}$ ($0<\gamma \le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for L\'evy-like superdiffusion (i.e., $\nu=1$ and $0<\gamma \le 2$). Finally, for $(\gamma,\nu)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $\gamma=2$ solutions available in the literature ($\nu<1$ porous medium equation), thus exhibiting nonuniform convergence.Comment: 3 figure

Fractional Fokker-Planck Equation and Oscillatory Behavior of Cumulant Moments

The Fokker-Planck equation is considered, which is connected to the birth and death process with immigration by the Poisson transform. The fractional derivative in time variable is introduced into the Fokker-Planck equation. From its solution (the probability density function), the generating function (GF) for the corresponding probability distribution is derived. We consider the case when the GF reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. Formulas of the factorial moment and the $H_j$ moment are derived from the GF. The $H_j$ moment derived from the GF of the NBD decreases monotonously as the rank j increases. However, the $H_j$ moment derived in our approach oscillates, which is contrasted with the case of the NBD. Calculated $H_j$ moments are compared with those given from the data in $p\bar{p}$ collisions and in $e^+e^-$ collisions.Comment: 10 pages, 8 figures, submitted to Phys. Rev.

Truncated Levy Random Walks and Generalized Cauchy Processes

A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.Comment: 7 pages, 3 figure

Anomalous diffusion and the first passage time problem

We study the distribution of first passage time (FPT) in Levy type of anomalous diffusion. Using recently formulated fractional Fokker-Planck equation we obtain three results. (1) We derive an explicit expression for the FPT distribution in terms of Fox or H-functions when the diffusion has zero drift. (2) For the nonzero drift case we obtain an analytical expression for the Laplace transform of the FPT distribution. (3) We express the FPT distribution in terms of a power series for the case of two absorbing barriers. The known results for ordinary diffusion (Brownian motion) are obtained as special cases of our more general results.Comment: 25 pages, 4 figure

On the Nature and Genesis of EUV Waves: A Synthesis of Observations from SOHO, STEREO, SDO, and Hinode

A major, albeit serendipitous, discovery of the SOlar and Heliospheric Observatory mission was the observation by the Extreme Ultraviolet Telescope (EIT) of large-scale Extreme Ultraviolet (EUV) intensity fronts propagating over a significant fraction of the Sun's surface. These so-called EIT or EUV waves are associated with eruptive phenomena and have been studied intensely. However, their wave nature has been challenged by non-wave (or pseudo-wave) interpretations and the subject remains under debate. A string of recent solar missions has provided a wealth of detailed EUV observations of these waves bringing us closer to resolving their nature. With this review, we gather the current state-of-art knowledge in the field and synthesize it into a picture of an EUV wave driven by the lateral expansion of the CME. This picture can account for both wave and pseudo-wave interpretations of the observations, thus resolving the controversy over the nature of EUV waves to a large degree but not completely. We close with a discussion of several remaining open questions in the field of EUV waves research.Comment: Solar Physics, Special Issue "The Sun in 360",2012, accepted for publicatio

Scale-free static and dynamical correlations in melts of monodisperse and Flory-distributed homopolymers: A review of recent bond-fluctuation model studies

It has been assumed until very recently that all long-range correlations are screened in three-dimensional melts of linear homopolymers on distances beyond the correlation length $\xi$ characterizing the decay of the density fluctuations. Summarizing simulation results obtained by means of a variant of the bond-fluctuation model with finite monomer excluded volume interactions and topology violating local and global Monte Carlo moves, we show that due to an interplay of the chain connectivity and the incompressibility constraint, both static and dynamical correlations arise on distances $r \gg \xi$. These correlations are scale-free and, surprisingly, do not depend explicitly on the compressibility of the solution. Both monodisperse and (essentially) Flory-distributed equilibrium polymers are considered.Comment: 60 pages, 49 figure

Solutions of a particle with fractional δ\delta-potential in a fractional dimensional space

A Fourier transformation in a fractional dimensional space of order \la (0<\la\leq 1) is defined to solve the Schr\"odinger equation with Riesz fractional derivatives of order \a. This new method is applied for a particle in a fractional $\delta$-potential well defined by V(x) =- \gamma\delta^{\la}(x), where $\gamma>0$ and \delta^{\la}(x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0< \la<\a. The results for \la= 1 and \a=2 are in exact agreement with those presented in the standard quantum mechanics