Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page

Levi-Civita cylinders with fractional angular deficit

The angular deficit factor in the Levi-Civita vacuum metric has been parametrized using a Riemann-Liouville fractional integral. This introduces a new parameter into the general relativistic cylinder description, the fractional index {\alpha}. When the fractional index is continued into the negative {\alpha} region, new behavior is found in the Gott-Hiscock cylinder and in an Israel shell.Comment: 5 figure

Ramanujan sums analysis of long-period sequences and 1/f noise

Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the Fourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over 13 years and of the coronal index of solar activity over 69 years are taken as illustrative examples. Distinct long periods may be discriminated in place of the 1/f^{\alpha} spectra of the Fourier transform.Comment: 10 page

Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).Comment: 11 pages, LaTe

Characterizations and simulations of a class of stochastic processes to model anomalous diffusion

In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defined through the explicit construction of the underline probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the $H$-{\bf sssi} nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differenital equation of fractional type.Comment: 25 pages, 9 figure

A method for dynamic subtraction MR imaging of the liver

BACKGROUND: Subtraction of Dynamic Contrast-Enhanced 3D Magnetic Resonance (DCE-MR) volumes can result in images that depict and accurately characterize a variety of liver lesions. However, the diagnostic utility of subtraction images depends on the extent of co-registration between non-enhanced and enhanced volumes. Movement of liver structures during acquisition must be corrected prior to subtraction. Currently available methods are computer intensive. We report a new method for the dynamic subtraction of MR liver images that does not require excessive computer time. METHODS: Nineteen consecutive patients (median age 45 years; range 37–67) were evaluated by VIBE T1-weighted sequences (TR 5.2 ms, TE 2.6 ms, flip angle 20°, slice thickness 1.5 mm) acquired before and 45s after contrast injection. Acquisition parameters were optimized for best portal system enhancement. Pre and post-contrast liver volumes were realigned using our 3D registration method which combines: (a) rigid 3D translation using maximization of normalized mutual information (NMI), and (b) fast 2D non-rigid registration which employs a complex discrete wavelet transform algorithm to maximize pixel phase correlation and perform multiresolution analysis. Registration performance was assessed quantitatively by NMI. RESULTS: The new registration procedure was able to realign liver structures in all 19 patients. NMI increased by about 8% after rigid registration (native vs. rigid registration 0.073 ± 0.031 vs. 0.078 ± 0.031, n.s., paired t-test) and by a further 23% (0.096 ± 0.035 vs. 0.078 ± 0.031, p < 0.001, paired t-test) after non-rigid realignment. The overall average NMI increase was 31%. CONCLUSION: This new method for realigning dynamic contrast-enhanced 3D MR volumes of liver leads to subtraction images that enhance diagnostic possibilities for liver lesions

Common Scaling Patterns in Intertrade Times of U. S. Stocks

We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U. S. economy over a period of four years. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution; (ii) when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form; (iii) the intertrade times exhibit power-law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales, in agreement with the correlation behavior of the absolute price returns for the corresponding company, and (iv) the magnitude series of intertrade time increments is characterized by long-range power-law correlations suggesting the presence of nonlinear features in the trading dynamics, while the sign series is anti-correlated at small scales. Our results suggest that independent of industry sector, market capitalization and average level of trading activity, the series of intertrade times exhibit possibly universal scaling patterns, which may relate to a common mechanism underlying the trading dynamics of diverse companies. Further, our observation of long-range power-law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock, support the hypothesis that the dynamics of transaction times may play a role in the process of price formation.Comment: 8 pages, 5 figures. Presented at The Second Nikkei Econophysics Workshop, Tokyo, 11-14 Nov. 2002. A subset appears in "The Application of Econophysics: Proceedings of the Second Nikkei Econophysics Symposium", editor H. Takayasu (Springer-Verlag, Tokyo, 2003) pp.51-57. Submitted to Phys. Rev. E on 25 June 200