Scale-invariant Structure Research Articles

Scale-invariant cluster structure of the chemical network of epoxy-phenol polymer

The concentration dependences of the relative viscosity of a mixture of precondensates of epoxy-4,4′-isopropylidenediphenol and phenol-formaldehyde resins in a Cellosolve solution were examined. The surface tension curves at the solution-air interface at concentrations c = 0.5–45 % were obtained. The optimal concentration modes of preparation of coatings from solutions of precondensates of the resin mixture were calculated on the basis of the cluster model. Using electron microscopy with silver chloride surface decoration, the cluster structure parameters of the chemical network of the epoxy-phenol polymer were determined for coatings with high protective performance.

Scale-invariant structure of strongly conserved sequence in genomic intersections and alignments

A power-law distribution of the length of perfectly conserved sequence from mouse/human whole-genome intersection and alignment is exhibited. Spatial correlations of these elements within the mouse genome are studied. It is argued that these power-law distributions and correlations are comprised in part by functional noncoding sequence and ought to be accounted for in estimating the statistical significance of apparent sequence conservation. These inter-genomic correlations of conservation are placed in the context of previously observed intra-genomic correlations, and their possible origins and consequences are discussed.

Structure of Magnetic Noise in Dynamic Fracture

We present an experimental study of magnetic emission spectra recorded during impact fracture of steel. Novel features of dynamic fracture are revealed, i.e., the distribution of the voltage signals of the spectra; furthermore, the areas and energies of voltage peaks exhibit a power law behavior. The value of the exponents of the distributions proved to be characteristic for the failure mode: ductile failure gives rise to exponents significantly higher than brittle failure. The results imply that magnetic crackling noise accompanying impact fracture has a scale invariant structure which reveals new aspects of the dynamics of the fracture process.

On network form and function

Moving from the observation that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution, we review theoretical and observational evidence on river patterns and their scale-invariant structure. Exact results complemented by numerical annealing of the basic equation in the presence of additive noise suggest that configurations at (or very close to) the global minimum of energy dissipation differ from dynamically accessible states, which have rather different scaling properties and conform much better to natural forms. Thus we argue that, at least in the fluvial landscape, Nature works through imperfect searches for dynamically accessible optimal configurations. We also show that optimal networks are spanning loopless configurations only under precise physical requirements. This is stated in a form applicable to generic networks, suggesting that other branching structures occurring in Nature (e.g. scale-free and looping) may possibly arise through optimality to selective pressures. Indeed, we show that this is the case.

Equations for effective stress and permeability of soft mud–sand mixtures

Equations for the Terzaghi effective stress and permeability of clay–silt–sand mixtures are derived on the basis of a scale-invariant structure of the soil to enhance physical interpretation of the empirical coefficients. The behaviour of soft sediment is assumed to be dominated by the clay fraction, but the sediment may also contain silt and sand. A theoretical model is presented of the structure of the sediment, based on the concept of a scale-invariant structure of the clay–silt matrix. Using this scale-invariant model, equations for effective stress and permeability are derived. The equations take effects due to partial segregation into account. The equations for effective stress and permeability are validated against data (effective stress, permeability and particle size distributions) obtained from a consolidation experiment with a sandy soft mud. It is shown that the equation for effective stress relates the effective stress to the clay and sand volume fraction accurately for the present data set. The correspondence between modelled and measured permeabilities is only fair, which may be caused either by a shortcoming of the model or by the poor quality of the data.

Equations for effective stress and permeability of soft mud–sand mixtures

Equations for the Terzaghi effective stress and permeability of clay–silt–sand mixtures are derived on the basis of a scale-invariant structure of the soil to enhance physical interpretation of the empirical coefficients. The behaviour of soft sediment is assumed to be dominated by the clay fraction, but the sediment may also contain silt and sand. A theoretical model is presented of the structure of the sediment, based on the concept of a scale-invariant structure of the clay–silt matrix. Using this scale-invariant model, equations for effective stress and permeability are derived. The equations take effects due to partial segregation into account. The equations for effective stress and permeability are validated against data (effective stress, permeability and particle size distributions) obtained from a consolidation experiment with a sandy soft mud. It is shown that the equation for effective stress relates the effective stress to the clay and sand volume fraction accurately for the present data set. The correspondence between modelled and measured permeabilities is only fair, which may be caused either by a shortcoming of the model or by the poor quality of the data.

Precursory behavior of fractal characteristics of the ULF electromagnetic fields in seismic active zones before strong earthquakes

It is now recognized that the earthquake hazard systems evolve naturally to the self-organized critical (SOC) state, in which the system is very sensitive to any external perturbations, and when the large avalanches (strong earthquakes) are probable events. Since the principal feature of the SOC state is fractal organization of the output parameters both in space (scale-invariant structure) and in time (flicker noise or 1/ f noise) we can use fractal methods to investigate the evolutionary processes in the earthquake hazard system at different stages of the catastrophic event preparation. Here we apply fractal methods for extraction of earthquake precursory signatures from ULF ( f=0.001–1 Hz) geomagnetic data obtained in seismic active regions before strong earthquake events. We focus our attention on the massive Guam earthquake of August 8, 1993 ( M=8, depth=60 km), and on a swarm of Japanese earthquakes of June–August 2000 occurred near Izu Peninsula ( M>6, depth ≈ 10 km). We analyze scaling (fractal) characteristics of ULF geomagnetic fields registered in the area less than 100 km from the earthquake epicenters, study their dynamics as approaching the earthquake dates, and compare the results obtained in both regions. Three methods have been applied to calculate scaling parameters (spectral exponents and fractal dimensions of the ULF geomagnetic time series): FFT procedure, Burlaga–Klein approach and Higuchi method. It is found that fractal characteristics of the ULF emissions manifest specific precursory behavior with some common and individual peculiarities in both seismic active areas. As the common feature, we have revealed the same increase of the ULF time series fractal dimensions (and the corresponding decrease of spectral exponents) before the both events. As the distinctive peculiarity, we have found different character of such dynamics: gradual increase (decrease) in the case of the Guam earthquakes and relatively sharp alteration in the case of the Izu earthquake swarm. For the case of Japanese earthquakes, it turned out to be possible to reveal the most effective frequency range (around f=0.01 Hz), in which precursory behavior of fractal characteristics is more pronounced and manifested earlier than in the other frequency ranges. We give a physical interpretation of the peculiarities revealed taking into account possible specifics of the SOC processes at different depths of the earthquake focuses.

Occurrence of finite-time singularities in epidemic models of rupture, earthquakes, and starquakes.

We present a new kind of critical stochastic finite-time singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star, as well as of earthquake sequences.

Scale invariance in the nonstationarity of human heart rate.

We introduce a segmentation algorithm to probe the temporal organization of heterogeneities in human heartbeat interval time series. We find that the lengths of segments with different local mean heart rates follow a power-law distribution and show that this scale-invariant structure is not a simple consequence of the long-range correlations present in the data. The differences in mean heart rates between consecutive segments display a common functional form, but with different parameters for healthy individuals and for heart-failure patients. These findings suggest that there is relevant physiological information hidden in the heterogeneities of the heartbeat time series.

Incorporating spatial distribution into stochastic modelling of fractures: multifractals and Levy-stable statistics

A new approach to the stochastic modelling of fractures is developed using data acquired in horizontal wells and applied to one-dimensional simulations. It differs from previous studies in the use of both Levy-stable statistics to describe the fracture attributes and of a (multifractal) strain-based model to create the spatial distribution. The structure of the multifractal strain is generated with a multiplicative cascade and used as a template or guide in the simulation. Resulting simulations have scale-invariant structure. Spacing distribution functions depend on the spatial partitioning of strain. For the case where strain distribution is homogeneous an approximate negative exponential spacing distribution results. Heterogeneous strain, exhibiting intermittency, leads to power-law (fractal) spacing distributions and spatial clustering. When plotting spacing as log-log cumulative frequency, the slope (fractal dimension) quantifies the degree of clustering. Small fractal dimensions are indicative of more clustering than large ones, the degree of clustering decreasing with an increase in fractal dimension.

Multifractal intermittency of Eulerian and Lagrangian turbulence of ocean temperature and plankton fields

Abstract. In this paper, we present evidence that intermittency of Eulerian and Lagrangian turbulence of ocean temperature and plankton fields is multifractal and furthermore can be analysed with the help of universal multifractals. We analyse time series of temperature and in vivo fluorescence taken from a drifter in the mixed coastal waters of the eastern English Channel. Two analysis techniques are used to compute the fundamental universal multifiractal parameters, which describe all the statistics of the turbulent fluctuations: the analysis of the scale invariant structure function exponent ζ(q) and the Double Trace Moment technique. At small scales, we do not detect any significant difference between the universal multifiractal behavior of temperature and fluorescence in an Eulerian framework. This supports the hypothesis that the latter is passively advected with the flow as the former. On the one hand, we show that large scale measurements are Lagrangian and indeed we obtain for temperature fluctuations a ω2 power spectrum corresponding to the theoretical scaling of a Lagrangian passive scalar. Furthermore, we show that Lagrangian temperature fluctuations are multiscaling and intermittent. On the other hand, the flatter slope at large scales of the fluorescence power spectrum points out that the plankton is at these scales a "biologically active" scalar.

Multifractal analysis of Vesuvius volcano eruptions

The standard statistical analysis of volcanic eruptions could be not very useful tools to describe temporal variations of the volcanic events clustering due to the shortness of the catalogues. A more appropriate approach is the multifractal analysis. If interevent times between eruptions are fractally distributed, they have a scale invariant structure and the behaviour of the catalogue can be described in terms of Poissonian or stationary clustered distribution. On the contrary, if a multifractal distribution is observed, the clustering cannot be considered constant. We analyse the catalogue of occurrence times of Vesuvius eruptions from 1631 to 1944 that can be considered complete. We find a multifractal distribution. Such a result implies a clustering degree that varies in time. At small scales a weak clustering is observed, while at large scale the catalogue can be considered Poissonian. This implies that the correlation between the events of the volcanic activity cycles is very weak and that the dynamics evolves at the same time scales within a given cycle and between cycles.

Initial-state colour dipole emission associated with QCD Pomeron exchange

The initial-state radiation of soft colour dipoles produced together with a single QCD Pomeron exchange (BFKL) in onium-onium scattering is calculated in the framework of Mueller's approach. The resulting dipole production grows with increasing energy and reveals an unexpected feature of a power-law tail at appreciably large transverse distances from the collision axis, this phenomenon being related to the scale-invariant structure of dipole-dipole correlations.

Estimating the cost of a telecommunications network using the fractal structure of the human population distribution

Results are presented which exploit the strong scale-invariant structure present in the spatial distributions of the populations of Great Britain and the United States to estimate the cost of star networks to serve these populations. The costs of the network components are modelled as affine functions of the capacities of the components. The main application for this work is to assist in making decisions in the design and choice of network components on a national scale.

Is there any scaling in the cluster distribution?

view Abstract Citations (13) References (55) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Is There Any Scaling in the Cluster Distribution? Borgani, Stefano ; Martinez, Vicent J. ; Perez, Miguel A. ; Valdarnini, Riccardo Abstract We apply fractal analysis methods to investigate the scaling properties in the Abell and ACO catalogs of rich galaxy clusters. The methods are adapted to account for the incompleteness of the samples by appropriately using the known selection functions in the partition sums. We also discuss different technical aspects of the method when applied to data sets with small number of points as the cluster catalogs. Results are compared with simulations based on the Zel'dovich approximation. We limit our analysis to scales less than 100 h^-1^ Mpc. The cluster distribution show a scale-invariant multifractal behavior in a limited scale range. For the Abell catalog this range is 15-60 h^-1^ Mpc, while for the ACO sample it extends to smaller scales. Despite this difference in the extension of the scale-range where scale-invariant clustering takes place, both samples are characterized by remarkably similar multifractal spectra in the corresponding scaling regime. In particular, the correlation dimension turns out to be D_2_ ~ 2.2 for both Abell and ACO clusters. Although it is difficult to point out the scale at which homogeneity is reached with the present size of these redshift surveys, our results indicate that the cluster distribution shows a tendency to homogeneity at large scales, disproving the picture of a pure scale invariant fractal structure extending to arbitrarily large distances. Publication: The Astrophysical Journal Pub Date: November 1994 DOI: 10.1086/174791 arXiv: arXiv:astro-ph/9403010 Bibcode: 1994ApJ...435...37B Keywords: Astronomical Catalogs; Astronomical Models; Cluster Analysis; Cosmology; Fractals; Galactic Clusters; Scaling Laws; Computerized Simulation; Error Analysis; Mathematical Models; Statistical Correlation; Astrophysics; GALAXIES: CLUSTERING; METHODS: ANALYTICAL; Astrophysics E-Print: LaTeX, 26 pages + 11 figures available on request, DFUPG-94-10 full text sources arXiv | ADS |

Scale-invariance of time hierarchies in viscoelastic media relaxation processes

It is shown that the time hierarchy governing shear and bulk relaxation in viscoelastic media has a fractal (scale-invariant) structure. It is noted that temporal fractality eases the modelling of viscoelastic media and leads to simple universal relaxation functions. In particular, in media with scale-invariant distributions of relaxational characteristics an algebraic relaxation law may occur, which leads to rheological models and equations of state containing fractional derivatives.

On self-organized criticality in nonconserving systems.

Two models with nonconserving dynamics and slow continuous deterministic driving, a stick-slip model (SSM) of earthquake dynamics and a toy forest-fire model (FFM), have recently been argued to show numerical evidence of self-organized criticality (generic, scale-invariant steady states). To determine whether the observed criticality is indeed generic, we study these models as a function of a parameter \ensuremath{\gamma} which was implicitly tuned to a special value, \ensuremath{\gamma}=1, in their original definitions. In both cases, the maximum Lyapunov exponent vanishes at \ensuremath{\gamma}=1. We find that the FFM does not exhibit self-organized criticality for any \ensuremath{\gamma}, including \ensuremath{\gamma}=1; nor does the SSM with periodic boundary conditions. Both models show evidence of macroscopic periodic oscillations in time for some range of \ensuremath{\gamma} values. We suggest that such oscillations may provide a mechanism for the generation of scale-invariant structure in nonconserving systems, and, in particular, that they underlie the criticality previously observed in the SSM with open boundary conditions.

Intermittency and jets

We show that intermittency, i.e. unlimited growth of multiplicity fluctuations with increasing resolution, is obtained in hadron cascade models with scale invariant branching structure. Strong effects are expected if in a multiparticle event not only the rapidity but simultaneously the azimuthal angle around the beam axis is analysed. The occurrence of narrow jets (“pencil jets”) is expected. The relevance of this type of study for models of hadronization is pointed out.

Self-organized criticality.

We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal ``fingerprint'' of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.

On the ballistic model of aggregation

A simple model of aggregation is discussed. It is argued on the example of this model that the aggregates may exhibit interesting scale invariant structure while having a “trivial” fractal dimension (i.e. equal to the dimensionality of space).