S. C. Lim

Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates

In this paper, we investigate the fractal scaling behaviors of foreign currency exchange rates with respect to Malaysian currency, Ringgit Malaysia. These time series are examined piecewise before and after the currency control imposed in 1st September 1998 using the monofractal model based on fractional Brownian motion. The global Hurst exponents are determined using the R/S analysis, the detrended fluctuation analysis and the method of second moment using the correlation coefficients. The limitation of these monofractal analyses is discussed. The usual multifractal analysis reveals that there exists a wide range of Hurst exponents in each of the time series. A new method of modelling the multifractal time series based on multifractional Brownian motion with time-varying Hurst exponents is studied.

Generalized Ornstein–Uhlenbeck processes and associated self-similar processes

We consider three types of generalized Ornstein–Uhlenbeck processes: the stationary process obtained from the Lamperti transformation of fractional Brownian motion, the process with stretched exponential covariance and the process obtained from the solution of the fractional Langevin equation. These stationary Gaussian processes have many common properties, such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency limit. In addition, the generalized Ornstein–Uhlenbeck processes can be shown to be local stationary representations of fractional Brownian motion. Two new self-similar Gaussian processes, in addition to fractional Brownian motion, are obtained by applying the (inverse) Lamperti transformation to the generalized Ornstein–Uhlenbeck processes. We study some of the properties of these self-similar processes such as the long-range dependence. We give a simulation of their sample paths based on numerical Karhunan–Loeve expansion.

On some possible generalizations of fractional Brownian motion

Abstract Fractional Brownian motion (fBm) can be generalized to multifractional Brownian motion (mBm) if the Hurst exponent H is replaced by a deterministic function H ( t ). The two possible generalizations of mBm based on the moving average representation and the harmonizable representation are first shown to be equivalent up to a multiplicative deterministic function of time by Cohen [S. Cohen, in: M. Dekking et al. (Eds.), Fractals: Theory and Applications in Engineering, Springer, Berlin, 1999, p. 3.] using the Fourier transform method. In this Letter, we give an alternative verification of such an equivalence based on the direct computation of the covariances of these two Gaussian processes. There also exists another equivalent representation of mBm, which is a variant version of the harmonizable representation. Finally, we consider a generalization based on the Riemann–Liouville fractional integral, and study the large time asymptotic properties of this version of mBm.

Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type

The relationship between standard fractional Brownian motion (FBM) and FBM based on the Riemann-Liouville fractional integral (or RL-FBM) is clarified. The absence of stationary property in the increment process of RL-FBM is compensated by a weaker property of local stationarity, and the stationary property for the increments of the large-time asymptotic RL-FBM. Generalization of RL-FBM to the RL-multifractional Brownian motion (RL-MBM) can be carried out by replacing the constant Holder exponent by a time-dependent function. RL-MBM is shown to satisfy a weaker scaling property known as the local asymptotic self-similarity. This local scaling property can be translated into the small-scale behaviour of the associated scalogram by using the wavelet transform.

Fractal analysis of lyotropic lamellar liquid crystal textures

We apply fractal analysis to study the birefringence textures of lyotropic lamellar liquid crystal system (water/cethylpyridinium chloride/decanol). Birefringence texture morphologies are important as they provide information on the molecular ordering as well as defect structures and therefore has been adopted as a standard method in characterizing different phases of liquid crystals. The system under consideration shows a gradual morphological transition from mosaic to oily streak structures and then to maltese cross texture when the water content is increased. Since these textures are the characteristic fingerprints for the lyotropic lamellar phases, it is necessary to have robust techniques to obtain image quantifiers that can characterize the morphological structure of the textures. For this purpose we employ three different approaches namely the Fourier power spectrum for monofractal analysis, the generalized box-counting method for multifractal analysis and multifractal segmentation technique for estimating the space-varying local Hurst exponents. The relationships between estimated image parameters such as the spectral exponent, the Hurst exponent and the fractal dimension with respect to patterns observed in the birefringence textures are discussed.

ON THE SPECTRA OF RIEMANN-LIOUVILLE FRACTIONAL BROWNIAN MOTION

We study the spectrum of the fractional Brownian motion of Riemann-Liouville type using two approaches, namely the double frequency spectral density and the Wigner-Ville spectrum. The bifrequency representation gives a complex-valued function which contains two distinctive terms. These terms can be identified as the diagonal and off-diagonal distribution of the spectrum in a frequency-frequency plane. The physical interpretation of these two terms is briefly discussed. A calculation of Wigner-Ville spectrum gives an alternative way of representing the spectrum of this nonstationary process in the time-frequency plane. Asymptotic approximation of the Wigner-Ville spectrum is obtained. We show that the large-time average spectrum of Riemann-Liouville fractional Brownian motion exhibits a power law in the high-frequency range.

Free electromagnetic potentials in Minkowsku and Euclidean regions

Free electromagnetic vector potentials in Coulomb and Gupta-Bleuler gauges are shown to be unitary equivalent in both Minkowski and Euclidean regions. For covariant gauges, the Euclidean electromagnetic potential is Markovian but non-reflective, whereas for the Coulomb gauge it is reflective but only satisfies the Markov property with respect to special half-spaces. The Feynman-Kac-Nelson formula can be established for the case of the Coulomb gauge.

Fractional dynamics of single file diffusion in dusty plasma ring

Single file diffusion (SFD) refers to the constrained motion of particles in quasi‐one‐dimensional channel such that the particles are unable to pass each other. Possible SFD of charged dust confined in biharmonic annular potential well with screened Coulomb interaction is investigated. Transition from normal diffusion to anomalous sub‐diffusion behaviors is observed. Deviation from SFDs mean square displacement scaling behavior of 1/2‐exponent may occur in strongly interacting systems. A phenomenological model based on fractional Langevin equation is proposed to account for the anomalous SFD behavior in dusty plasma ring.

Fractional brownian motion: Theory and application to DNA walk

This paper briefly reviews the theory of fractional Brownian motion (FBM) and its generalization to multifractional Brownian motion (MBM). FBM and MBM are applied to a biological system namely the DNA sequence. By considering a DNA sequence as a fractal random walk, it is possible to model the noncoding sequence of human retinoblastoma DNA as a discrete version of FBM. The average scaling exponent or Hurst exponent of the DNA walk is estimated to be H = 0.60 ± 0.05 using the monofractal R/S analysis. This implies that the mean square fluctuation of DNA walk belongs to anomalous superdiffusion type. We also show that the DNA landscape is not monofractal, instead one has multifractal DNA landscape. The empirical estimates of the Hurst exponent falls approximately within the range H ~ 0.62 - 0.72. We propose two multifractal models, namely the MBM and multiscale FBM to describe the existence of different Hurst exponents in DNA walk.

Local Asymptotic Properties of Multifractional Brownian Motion

One direct way to generalize fractional Brownian motion to multifractional Brownian motion is to replace the constant Holder exponent by a certain deterministic function of time. In this paper local asymptotic properties of two types of multifractional Brownian motion will be considered.