Presence Of Long-range Correlations Research Articles

Compositional heterogeneity within and among isochores in mammalian genomes: II. Some general comments

The presence of long-range correlations and/or mosaicism in DNA sequences results in GC fluctuations, even within individual isochores that are much larger than expected correlation-free ‘random’ sequences. Neglecting the presence of such fluctuations can lead to incorrect conclusions regarding relative homogeneity or isochore borders. In this commentary, we address these and other methodological issues raised by the variations in GC level within human isochores. We also discuss some recent misconceptions.

Thin points for Brownian motion

Let Θ( x, r) denote the occupation measure of the ball of radius r centered at x for Brownian motion { W t } 0≤ t≤1 in R d, d≥2 . We prove that for any analytic set E in [0,1], we have inf t∈E lim inf r→0 Θ(W t,r)/(r 2/| logr|)=1/ dim P (E) , where dim P ( E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which lim inf r→0 Θ(x,r)/(r 2/| logr|)=a , is almost surely 2−2/ a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of `limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.

On the fractal properties of natural human standing

We analyzed the temporal evolution of the displacement of the center of pressure (COP) during prolonged unconstrained standing (30 min) in non-impaired human subjects. The COP represents the collective outcome of the postural control system and the force of gravity and is the main parameter used in studies on postural control. Our analysis showed that the COP displacement during human standing displays fractal properties that were quantified by the Hurst exponent obtained from the classical rescaled adjusted range analysis. The average fractal or Hurst exponent ( H) was 0.35±0.06. The presence of long-range correlations from a few seconds to several minutes due to the fractal characteristics of the postural control system has several important implications for the analysis of human balance.

Long-Range Correlations in the Fossil Record and the Fractal Nature of Macroevolution

Recent studies on the fossil record time series has suggested that there is consistent evidence for self-similarity i.e. long-range correlations with power-law behavior. The existence of such fractal strucutes means that, when looking at a given time frame, some basic properties remain the same if a change of scale is performed. In other words, there is no characteristic time scale, as we could expect if some type of periodic or other low-dimensional dynamics were present. A possible explanation for such long-range order is a dynamical process operating at all scales, as it is the case for systems in the neighborhood of critical points. In this paper these results are further explored by extending previous data analysis and examining the relevance of recent theoretical approaches to the statistical features of the fossil record. The presence of long-range correlations is shown through Hurst analysis using non-interpolated data series from the Fossil Record 2 database. As shown in previous studies, such correlations span over hundreds of millions of years and are compared with a simple model of large-scale evolution displaying self-organized criticality.

LONG-RANGE CORRELATIONS IN COMPUTER PROGRAMS

A different view of software complexity based on long-range correlations is presented in this paper. Three different methods are proposed first and then used in the analysis of C , FORTRAN, and Pascal programs. All methods being used indicate the presence of long-range correlations in programs, but every method gives different results. The presence of longrange correlations denotes that we can regard the complexity of computer programs from a different perspective and use it as an advanced software metric.

Software complexity—an alternative view

A different view on software complexity based on long range correlations is presented in this paper. Four different methods are proposed first and then used in the analysis of twenty FORTRAN programs. All methods being used indicate the presence of long range correlations, but every method gives different results. The presence of long range correlations denotes that we can infact regard the complexity of computer programs from a different perspective.

Further evidence of fractal structure in hydraulic conductivity distributions

Past rescaled range analyses of porosity and hydraulic conductivity (K) distributions have indicated the presence of long‐range correlations in the data typical of the related stochastic functions known as fractional Gaussian noise and fractional Brownian motion [Hewett, 1986; Molz and Boman, 1993]. New K data analyzed herein lend further support to this notion. Horizontal processes that mimic fBm will display a power‐law variogram. The Mandelbrot‐Weierstrass random fractal function is introduced as an analytical model for fBm and used to illustrate several concepts. With the exception of the Hurst coefficient value (H), our analysis supports the existence of fractal‐like K fields similar to those visualized by Neuman [1990, 1994]. Past studies which produced H values greater than or less than 0.5 appear to differ mainly because of the underlying model, fractional motion or fractional noise, that was assumed in the respective analyses.

Effect of correlations on the localization properties of electrons and phonons in the long-wavelength limit.

The problem of Anderson localization in a spatially correlated disordered potential is formulated in the framework of the self-consistent theory of Vollhardt and Wolfle in conjunction with the self-consistent Born approximation for both tight-binding electrons and phonons. An approximation which is exact in the weak scattering limit is introduced for the coherent backscattering contribution to the irreducible two-particle vertex. For the case of electrons in three dimensions, the phase diagrams in the near-band-edge region are studied numerically for the binary alloys with short-range correlations and the idea of quasiuniversality is examined. For the case of phonons in one and two dimensions, the complete expressions for the localization length are obtained in the long-wavelength limit for both short-range and long-range correlations. The universality holds when the correlations are short range. In the presence of long-range correlations, different expressions for the localization length are obtained depending on the definition of the mean free path used in the theory. When the single-phonon mean free path is used, our results give the same asymptotic behaviors found previously by using the replica method. In one dimension, different asymptotic behavior appears when the transport mean free path is used. Discussions are given.