Fractal Behavior Research Articles

Image analysis of the pore structures: An intensive study for Middle Bakken

Pores with sizes ranging from nanometers to micrometers are widely distributed in shale gas and shale oil formations. These pores are the sites for hydrocarbon accumulation and provide the flow paths for hydrocarbons during production. The Middle Bakken member is the main production zone of the Bakken Formation in North Dakota. In order to reveal the pore structures of the Middle Bakken, we employed field emission scanning electron microscopy (FE-SEM). After segmentation of the SEM images, we calculated the surface area and shape of the pores using image analysis and then quantified the complexity and heterogeneity of the pore structures by applying both fractal and multifractal analyses. Finally, we employed the fractal permeability model to estimate the permeability of the samples. The results showed that different pore types, such as interparticle and intraparticle pores exist in the Middle Bakken samples. Even under the same scale of the same sample, the pore parameters could be different. Sample 2 has the largest average porosity, followed by Sample 1 and Sample 3. The mean pore size of these samples is less than 31 nm indicating that the pores in Middle Bakken samples are very small. The pore structures in the Middle Bakken exhibited fractal and multifractal behavior. The fractal dimension from the entire size range of pores is the largest compared with the fractal dimension of the subdivided groups. The pore size distribution in Sample 2 is the most heterogeneous.

Fractal Behavior of a Ternary 4-Point Rational Interpolation Subdivision Scheme

In this paper, a ternary 4-point rational interpolation subdivision scheme is presented, and the necessary and sufficient conditions of the continuity are analyzed. The generalization incorporates existing schemes as special cases: Hassan–Ivrissimtzis’s scheme, Siddiqi–Rehan’s scheme, and Siddiqi–Ahmad’s scheme. Furthermore, the fractal behavior of the scheme is investigated and analyzed, and the range of the parameter of the fractal curve is the neighborhood of the singular point of the rational scheme. When the fractal curve and surface are reconstructed, it is convenient for the selection of parameter values.

Snow Depth Structure, Fractal Behavior, and Interannual Consistency Over Haut Glacier d'Arolla, Switzerland

AbstractSnow depth patterns over glaciers are controlled by precipitation, snow redistribution due to wind and avalanches, and the exchange of energy with the atmosphere that determines snow ablation. While many studies have advanced the understanding of ablation processes, less is known about winter snow patterns and their variability over glaciers. We analyze snow depth on Haut Glacier d'Arolla, Switzerland, in the two winter seasons 2006–2007 and 2010–2011 to (1) understand whether snow depth over an alpine glacier at the end of the accumulation season exhibits a behavior similar to the one observed on single slopes and vegetated areas; and (2) investigate the snow pattern consistency over the two accumulation seasons. We perform this analysis on a data set of high‐resolution lidar‐derived snow depth using variograms and fractal parameters. Our first main result is that snow depth patterns on the glacier exhibit a multiscale behavior, with a scale break around 20 m after which the fractal dimension increases, indicating more autocorrelated structure before the scale break than after. Second, this behavior is consistent over the two years, with fractal parameters and their spatial variability almost constant in the two seasons. We also show that snow depth patterns exhibit a distinct behavior in the glacier tongue and the upper catchment, with longer correlation distances on the tongue in the direction of the main winds, suggesting spatial distinctions that are likely induced by different processes and that should be taken into account when extrapolating snow depth from limited samples.

Parameters of a Microstructure of Erosion Regions during the Action of a Spark Discharge on a Copper Electrode Surface in Air

The parameters of a microstructure of regions of action of a spark discharge on a copper plane electrode surface in air at atmospheric pressure in the spike–plane gap both for positive and negative polarity of a spike have been studied using scanning electron microscopy. It has been found that regions of action have the appearance of rounded spots or groups of spots with an internal structure in the form of agglomeration of a large number of microcraters. It has been shown that the statistical distributions of microcrater diameters and intermicrocrater distances are characterized by several local maximums that determine the presence of several predominant values of these parameters in a spot. It has been revealed that the microstructure of a spot is characterized by the fractal behavior and the fractal dimensionalities of the microstructure have been determined.

Statistical Characteristics of Long-Term High-Resolution Precipitable Water Vapor Data at Darwin

This paper studies the statistical characteristics of a unique long-term high-resolution precipitable water vapor (PWV) data set at Darwin, Australia, from 12 March 2002 to 28 February 2011. To understand the convective precipitation processes for climate model development, the U.S. Department of Energy’s Atmospheric Radiation Measurement (ARM) program made high-frequency radar observations of PWV at the Darwin ARM site and released the best estimates from the radar data retrievals for this time period. Based on the best estimates, we produced a PWV data set on a uniform 20-s time grid. The gridded data were sufficient to show the fractal behavior of precipitable water with Hausdorff dimension equal to 1.9. Fourier power spectral analysis revealed modulation instability due to two sideband frequencies near the diurnal cycle, which manifests as nonlinearity of an atmospheric system. The statistics of PWV extreme values and daily rainfall data show that Darwin’s PWV has El Nino Southern Oscillation (ENSO) signatures and has potential to be a predictor for weather forecasting. The right skewness of the PWV data was identified, which implies an important property of tropical atmosphere: ample capacity to hold water vapor. The statistical characteristics of this long-term high-resolution PWV data will facilitate the development and validation of climate models, particularly stochastic models.

Eigenfunction distribution for the Rosenzweig-Porter model

The statistical distribution of eigenfunctions for the Rosenzweig-Porter model is derived for the region where eigenfunctions have fractal behaviour. The result is based on simple physical ideas and leads to transparent explicit formulas which agree very well with numerical calculations. It constitutes a rare case where a non-trivial eigenfunction distribution is obtained in a closed form.

Fractal analysis applied to light curves of δ Scuti stars★

Fractal behaviour, i.e. scale invariance in spatio-temporal dynamics, have been found to describe and model many systems in nature, in particular fluid mechanics and geophysical related geometrical objects, like the convective boundary layer of cumulus cloud fields, topographic landscapes, solar granulation patterns, and observational astrophysical time series, like light curves of pulsating stars. The main interest in the study of fractal properties in such physical phenomena lies in the close relationships they have with chaotic and turbulent dynamic. In this work we introduce some statistical tools for fractal analysis of light curves: Rescaled Range Analysis (R/S), Multifractal Spectra Analysis, and Coarse Graining Spectral Analysis (CGSA), an FFT based algorithm, which can discriminate in a time series the stochastic fractal power spectra from the harmonic one. An interesting application of fractal analysis in asteroseismology concerns the joint use of all these tools in order to develop classification criteria and algorithms for {\delta}-Scuti pulsating stars. In fact from the fractal and multi-fractal fingerprints in background noise of light curves we could infer on different mechanism of stellar dynamic, among them rotation, modes excitation and magnetic activity.

Multifractal topography of several planetary bodies in the solar system

Topography is the expression of both internal and external processes of a planetary body. Thus hypsometry (the study of topography) is a way to decipher the dynamics of a planet. For that purpose, the statistics of height and slopes may be described by different tools, at local and global scale. We propose here to use the multifractalapproach to describe fields of topography. This theory encompasses height and slopes and other statistical moments of the field, taking into account the scale invariance. Contrary to the widely used fractalformalism, multifractals are able to describe the intermittency of the topography field. As we commonly observe a juxtaposition of rough and smooth topographies at a given scale, the multifractal framework seems to be appropriate for hypsometric studies. Here we analyze the data at global scale of the Earth, Mars, Mercury and the Moon and find that the statistics are in good agreement with the multifractaltheory for scale larger than ∼ 10 km. Surprisingly, the analysis shows that all bodies have the same fractal behavior for scale smaller than ∼ 10 km. We hypothesize that dynamic topography of the mantle may be the explanation at large scale, whereas the smaller scale behavior may be related to elastic thickness.

Fractal properties and morphological investigation of nano-amiodarone using image processing

Many people hear their heartbeats when they are in certain emotional conditions such as joy, fear, or anxiety or are exercising. But if the heart rate constantly decreases or increases, it shows that the heart has an arrhythmia that, if ignored, may turn off the vital sound forever. Heart arrhythmia is one of the heart diseases, in which case the heart rhythm is abnormal. Amiodarone is one of the drugs used to maintain normal heartbeats in people with heart rhythm. In this study, we will explain a scrutiny on the effectiveness of amiodarone with the help of the “fractal dimension.” To better investigate the fractal dimensions, we converted this drug to nano-size, and then using a scanning electron microscope (SEM), performing of nano-amiodarone was identified. We calculated the fractal dimensions of nano-amiodarone using image processing values on a computer and histogram plot with SEM image using MATLAB software. We obtained new results from fractal dimensions and histogram diagrams of this nano-drug based on the SEM image. The primary objective of this study is to prove the relationship between fractal behavior and therapeutic effects of drugs in the body. This study could open up a new perspective between chemistry and mathematics sciences as well as pharmacy sciences in the development of treatment and release of medications in medical science in the near future.

A fractional perspective to the modelling of Lisbon’s public transportation network

Urban growth originates multiscale spatial patterns, such as those of transportation networks. Here, the public transportation network (PTN) of the city of Lisbon is analysed from 1901 to 2015, employing several mathematical tools. In a first stage, the fractal dimension and the fractional entropy are used to quantify the evolution of the structure of the PTN in space and time. In a second stage, the PTN is analysed adopting additional information, namely considering different levels of the network based on transportation schedule and passenger capacity, and studying the significance of the distance between stops. Both the fractal dimension and the fractional entropy reveal time patterns compatible with known historical events, showing them to be appropriate for quantifying the growth of the PTN. When the routes’ schedules are used to stratify the PTN, not only the fractal behaviour is observed at different levels, but also the evolution of the network in respect to the homogenization of the capacity of different routes. Finally, when the distance between consecutive stops is analysed, a power law behaviour is revealed, as expected from the fractal geometry of the network. This result is then confirmed using the ht-index.

Fractal and Multifractal Characteristics of Pore Throats in the Bakken Shale

To evaluate pore structures of the Bakken Shale, which is one of the most important factors that affect petrophysical properties, high-pressure mercury intrusion was employed in this study. Pore structures such as pore-throat size, pore-throat ratio, and fractal attributes are investigated in this major shale play. Pore-throat size from 3.6 to 200 um is widely distributed in these shale samples. Accordingly, pore-throat size distributions demonstrate the multimodal behavior within the samples. The whole pore-throat network can be divided into four clusters: one set of large pores, two transitional/intermediate pore groups, and one set of smaller pores. The fractal analysis revealed that fractal dimensions decrease as the pore-throat size decreases. The multifractal analysis demonstrated that as the maturity of the shale samples increases, pore-throat size distributions would become more uniform and pore structures tend to become more homogeneous. The results are compared to our previous results obtained from nitrogen gas adsorption for further verifications of fractal behavior. Finally, although fractal analysis of mercury intrusion and nitrogen gas adsorption were comparable, the results of multifractal analysis from these two methods were not identical.

Cycle-dependent and cycle-independent surface tracers of solar magnetic activity

AbstractWe consider several tracers of magnetic activity that separate cycle-dependent contributions to the background solar magnetic field from those that are independent of the cycle. The main message is that background fields include two relative separate populations. The background fields with a strength up to 100 Mx cm−2 are very poorly correlated with the sunspot numbers and vary little with the phase of the cycle. In contrast, stronger magnetic fields demonstrate pronounced cyclic behaviour. Small-scale solar magnetic fields demonstrate features of fractal intermittent behaviour, which requires quantification. We investigate how the observational estimate of the solar magnetic flux density B depends on resolution D in order to obtain the scaling In BD = −k In D + a in a reasonably wide range. The quantity k demonstrates cyclic variations typical of a solar activity cycle. k depends on the magnetic flux density, i.e. the ratio of the magnetic flux to the area over which the flux is calculated, at a given instant. The quantity a demonstrates some cyclic variation, but it is much weaker than in the case of k. The scaling is typical of fractal structures. The results obtained trace small-scale action in the solar convective zone and its coexistence with the conventional large-scale solar dynamo based on differential rotation and mirror-asymmetric convection. Here we discuss the message for solar dynamo studies hidden in the above results.

Aggregate shape determination via light scattering by aligned and randomly oriented polydisperse aggregates

We present calculations of light scattered by simulated aligned and randomly oriented fractal aggregates of touching spheres. The alignment by an applied electric field is in the direction yielding the minimum energy associated with the polarizability of the aggregate. This direction is also within a few degrees of the direction associated with the smallest inertial eigenvalue. Aggregates were generated to have a fractal dimension of 1.78 (characteristic of diffusion–limited cluster-cluster aggregation) and to simulate the range of aggregate sizes (30 to 2000) and size distributions for post-flame generated soot by a range of hydrocarbon fuels. More nearly monodisperse aggregates were also generated to simulate the size distributions obtained via mobility or mass classification. We show that a ratio of slopes computed from small angle light scattering intensities (Guinier plots) is related to the shape parameter A31, which is the ratio of the largest to the smallest principle radii of gyration of the inertia tensor. A geometric interpretation of the overall shape is presented based on the semi-axes of an ellipsoid. Results on the correlation between the maximum structure factor ratio {S(q)}a/ {S(q)}r for the monodisperse and polydisperse clusters and the average value of A31 indicate that light scattering measurements in the q range of fractal behavior would also be feasible for shape characterization. Our calculations indicate that light scattering measurements have a good potential for characterizing aggregate shape with an advantage of being more sensitive to shape than mobility measurements.

Fractal behavior of GNSS time offsets and fractal interpolation forecasting method for multi-GNSS time synchronization

Synchronization between system times of Global Navigation Satellite Systems (GNSSs), or GNSS time interoperability, is one of the prerequisites of GNSS interoperability. We explored the fractal behavior of time offsets between Coordinated Universal Time (UTC) and GNSS time scales (GNSST). A fractal-based prediction method for forecasting UTC-GNSST time offsets is developed. Through data analysis, the self-similarity behavior in UTC-GNSST time offsets is revealed, which is proven by calculating the Hurst exponent. The values of the fractal dimension and correlation metric indicated that the time series of UTC-GNSST time offsets have long memory, of which the memory span is 1247 days and could be obtained from V-statistics. Subsequently, the fractal interpolation-based mathematic model was built, and the prediction accuracy was tested through numerical experiments. Compared with the polynomial and gray prediction models, the fractal interpolation forecast model was found to have higher prediction accuracy. The prediction accuracies of UTC-GPST and UTC-GLOT using fractal interpolation forecast model in short term are less than 1.3 and 5.5 ns, and in long term are less than 2.3 and 8.3 ns, with an improvement by at least 50 and 30% compared to polynomial and gray model.

RETRACTED: Fractal Behavior of Particle Size Distribution in the Rare Earth Tailings Crushing Process under High Stress Condition

To disclose the fractal transformation mechanism of the size distribution of rare earth tailing particles, this paper utilizes the side confined uniaxial compression test to study the evolution rule of the particle size distribution of the rare earth tailing and the particle breakage features under the high stress condition. The fractal behavior of the particle size distribution in the crushing process is studied based on the fractal model and the particle size distribution. The experimental results show that, under the high stress condition, the particle size distribution tends towards fractal distribution due to the particle breakage of the rare earth tailing. The crushing process is closely related to the breakage amount of the particles and can be described with the increased fractal dimension. Although the initial distribution and particle size of the rare earth tailings are different, the measured data of the particle size distribution with the fractal dimension greater than 2.2 shows fairly strict self-similarity, and this value can be taken as the lower limit value of the fractal dimension when the particle size distribution tends towards fractal distribution. When the particle size distribution becomes a fractal distribution due to the particle breakage, the ratio between the volumetric strain and the relative breakage rate remains constant and is slightly affected by the initial distribution uniformity and particle size.

Fractal properties of biophysical models of pericellular brushes can be used to differentiate between cancerous and normal cervical epithelial cells

Fractal behavior is found on the topographies of pericellular brushes on the surfaces of model healthy and cancerous cells, using dissipative particle dynamics models and simulations. The influence of brush composition, chain stiffness and solvent quality on the fractal dimension is studied in detail. Since fractal dimension alone cannot guarantee that the brushes possess fractal properties, their lacunarity was obtained also, which is a measure of the space filling capability of fractal objects. Soft polydisperse brushes are found to have larger fractal dimension than soft monodisperse ones, under poor solvent conditions, in agreement with recent experiments on dried cancerous and healthy human cervical epithelial cells. Additionally, we find that image resolution is critical for the accurate assessment of differences between images from different cells. The images of the brushes on healthy model cells are found to be more textured than those of brushes on model cancerous cells, as indicated by the larger lacunarity of the former. These findings are helpful to distinguish monofractal behavior from multifractality, which has been found to be useful to discriminate between immortal, cancerous and normal cells in recent experiments.

A lower bound of the hyperbolic dimension for meromorphic functions having a logarithmic Hölder tract

We improve existing lower bounds of the hyperbolic dimension for meromorphic functions that have a logarithmic tract Ω \Omega which is a Hölder domain. These bounds are given in terms of the fractal behavior, measured with integral means, of the boundary of Ω \Omega at infinity.

Evidence of centrality dependent fractal behavior in high energy heavy ion interactions: Hint of two different sources

Studies on large density fluctuations in produced particle spectra in high energy interactions provide enough information regarding the dynamics of the process of particle production. Various analyses have revealed self-similarity in particle production process. The concept of self-similarity is indicative of fractal geometry. The present analysis reports an exhaustive study on centrality dependence of event-by-event fluctuation of pions produced in16O-AgBr interactions at 60 A GeV using a non conventional tool based on complex network analysis, viz. visibility graph method. The analysis reveals different fractal behaviour as well as different clustering property for different centrality events and the amount of fractality and average clustering coefficient decrease with the increase of centrality. Estimation of Hurst exponent hints towards two different sources of fluctuation, fractional Brownian motion (fBm) and fractional Gaussian noises (fGn) for two different centrality classes.

Fractal characteristics of acoustic emissions from coal under multi-stage true-triaxial compression

The fractal characteristics of acoustic emissions (AEs) are widely used for monitoring and warning of coal and rock dynamic disasters such as coal and gas outburst and rock bursts. However, most research has focused on fractal behavior during coal and rock failure under uniaxial and conventional triaxial compression. For this study, we conducted true-triaxial multi-stage loading tests on coal under different stress conditions. Based on the AE count rate data collected from these experiments, we calculated the fractal dimension of every loading step by the Grassberger and Procaccia algorithm using phase space reconstruction theory. The results show that the AE count rate represents the load during the failure well. In the initial loading stage, AEs are very active; in the initial portion of the load maintenance stage, AE counts decrease gradually until they almost cease later in that stage. AE count rates during failure for all loading steps exhibit fractal features and show a generally consistent pattern. The fractal dimension fluctuates in the early loading stages and then begins to decline, reaching a minimum when failure occurs. The critical fluctuation coefficient where the fractal dimension begins to enter the continuous decline stage is about 55.5%–66.5% of the stress level. The relationship between the change of fractal dimension and the ‘quiet period’ of AE before coal failure is discussed and the proposal that the combination of two monitoring methods can more effectively predict coal and rock dynamic disasters is put forward.

A re-evaluation of thermal expansion measurements of metallic liquids and glasses from x-ray scattering experiments.

Previous studies reported a number of anomalies when estimates of linear thermal expansion coefficients of metallic liquids and glasses from x-ray scattering experiments were compared with direct measurements of volume/length changes with temperature. In most cases, the first peak of the pair correlation function showed a contraction, while the structure factor showed an expansion, but both at rates much different from those expected from the direct volume measurements. In addition, the relationship between atomic volume and the characteristic lengths obtained from the structure factor from scattering experiments was found to have a fractional exponent instead of one equal to three, as expected from the Ehrenfest relation. This has led to the speculation that the atomic packing in liquids and glasses follow a fractal behavior. These issues are revisited in this study using more in-depth analysis of recent higher resolution data and some new ideas suggested in the literature. The main conclusion is that for metallic alloys, at least to a large extent, most of these anomalies arise from complicated interplays of the temperature dependences of the various partial structure factors, which contribute to the total intensities of the scattering peaks.