Fractal Exponent Research Articles

Fractal characterization of an after-sales spare parts supply chain in telecom industry

Data series generated by complex systems exhibit fluctuations on a wide range of time scales, which often follow a scaling relation over several orders of magnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal scaling exponents, which can serve as characteristic fingerprints of the systems in comparisons with other systems and models. In this article was developed a fractal characterization of the data series generated by the closed loop supply chain that supports the availability of spare parts in Telecom industry since this complex system displays fluctuations in its processes caused by endogenous and exogenous variables that create a difficulty for matching the recovery process with the demand process. Key words: Spare parts, closed loop supply chain, complex systems, fluctuations, data series, fractals.

Shrinkage and Deformation of Agave atrovirens Karw Tissue during Convective Drying: Influence of Structural Arrangements

The influence of structural arrangements of Agave discs cut transversally (ADCT) and longitudinally (ADCL) on moisture loss, shrinkage, and shape was evaluated during convective drying by physical and image parameters. ADCT showed lower drying and shrinkage rates than ADCL. The fractal exponent (d) relating volume and thickness of samples rose from 1.552 ± 0.126 to 2.394 ± 0.128 and from 1.662 ± 0.111 to 1.848 ± 0.070 for ADCT and ADCL, respectively, which indicates that shrinkage was nonisotropic for both cases. Parameters considered for evaluating the changes of size and shape of Agave discs at macro- and microscopic levels during drying were projected area (PA), major length (ML), shape factor (SF), and fractal dimension of contour (FDC). The values obtained for these parameters demonstrated that changes in shrinkage and shape of both samples were dependent on structural arrangements of the samples. During the drying of ADCT samples, which have short and rigid structures, pronounced creasing was observed compared to ADCL materials, which are mainly formed by long and rigid structures. Also, it was observed that orientation of fibers influenced shrinkage and deformation.

Contour lines of the discrete scale-invariant rough surfaces

We study the fractal properties of the two-dimensional (2D) discrete scale-invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale-invariant rough surfaces. The fractal properties of the 2D DSI rough surfaces apart from possessing the discrete scale-invariance property follow the properties of the contour lines of the corresponding scale-invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents, some other new fractal exponents are also calculated.

Breakdown of the Intermediate-Term Fractal Scaling Exponent in Sinus Node Dysfunction - New Method for Non-Invasive Evaluation of Sinus Node Function -

The aim of the present study was to characterize the heart rate dynamics of sinus bradycardia (SB) from sinus node dysfunction (SND) using non-linear dynamical system analysis. No data are yet available on how the dynamics change in the presence of SND. Conventional time and frequency domain analysis, the short- (DFAα(1)) and intermediate-term fractal scaling exponent (DFAα(2)), approximate entropy (ApEn) and sample entropy (SampEn) were calculated in 60-min sinus RR interval data of SB from 24-h ambulatory electrocardiograms of 110 patients: 44 SND patients, 44 age-matched controls, and 22 younger controls. All of the time and frequency domain parameters, ApEn and SampEn, were significantly reduced in the age-matched control group, compared with the young control group. DFAα(1) and DFAα(2) increased with aging. Both the DFAα(1) and DFAα(2) of SND patients were paradoxically reduced, which was not appropriate for their age. Only the percentage of consecutive RR intervals with absolute differences >50ms (pNN(50)), low-frequency power, and DFAα(2) made a significant contribution to prediction of SND on logistic regression analysis. Among them, DFAα(2) was the most significant variable for prediction of SND (odds ratio, 0.927; 95% confidence interval: 0.888-0.969, P=0.001). DFAα(2) remained as a significant variable for prediction of SND, when compared with overall control patients, combining the 2 control groups. Inappropriate reduction of DFAα(2) is a robust measure and could be an adjunctive tool for improvement of diagnostic performance in detection of SND.

Fractal correlation property of heart rate variability in chronic obstructive pulmonary disease

BackgroundIt was reported that autonomic nervous system function is altered in subjects with chronic obstructive pulmonary disease (COPD). We evaluated short- and long-term fractal exponents of heart rate variability (HRV) in COPD subjects.Patients and methodsWe analyzed data from 30 volunteers, who were divided into two groups according to spirometric values: COPD (n = 15) and control (n = 15). For analysis of HRV indices, HRV was recorded beat by beat with the volunteers in the supine position for 30 minutes. We analyzed the linear indices in the time (SDNN [standard deviation of normal to normal] and RMSSD [root-mean square of differences]) and frequency domains (low frequency [LF], high frequency [HF], and LF/HF), and the short- and long-term fractal exponents were obtained by detrended fluctuation analysis. We considered P < 0.05 to be a significant difference.ResultsCOPD patients presented reduced levels of all linear exponents and decreased short-term fractal exponent (alpha-1: 0.899 ± 0.18 versus 1.025 ± 0.09, P = 0.026). There was no significant difference between COPD and control groups in alpha-2 and alpha-1/alpha-2 ratio.ConclusionCOPD subjects present reduced short-term fractal correlation properties of HRV, which indicates that this index can be used for risk stratification, assessment of systemic disease manifestations, and therapeutic procedures to monitor those patients.

Detrended fluctuation analysis of membrane flickering in discocyte and spherocyte red blood cells using quantitative phase microscopy

Dynamic analyses of vibrational motion in cell membranes provide a lot of information on the complex dynamic motilities of a red blood cell (RBC). Here, we present the correlation properties of membrane fluctuation in discocyte and spherocyte RBCs by using quantitative phase microscopy (QPM). Since QPM can provide nanometer sensitivity in thickness measurement within a millisecond time scale, we were able to observe the membrane flicking of an RBC in nanometer resolution up to the bandwidth of 50 Hz. The correlation properties of the vibrational motion were analyzed with the detrended fluctuation analysis (DFA) method. Fractal scaling exponent α in the DFA method was calculated for the vibrational motion of a cell surface at various surface points for normal discocyte and abnormal spherocyte RBCs. Measured α values for normal RBCs are distributed between 0.7 and 1.0, whereas those for abnormal spherocyte RBCs are within a range from 0.85 to 1.2. We have also verified that the vibrational motion of background fluid outside of a cell has an α value close to 0.5, which is a typical property of an uncorrelated white noise.

Fractal analysis of the ore-forming process in a skarn deposit: a case study in the Shizishan area, China

Abstract This paper presents a tool for analysing the element distribution and mineralization intensity. The Hurst exponents and a - f ( a ) multifractal spectrum are utilized to analyse the irregular element distribution in Shizishan skarn orefield, China. The Hurst exponents reveal the Cu, Ag, Au and Zn distributions in the skarn-dominated drill cores are persistent and those in marble-dominated drill core are nearly random; the persistence indicates the mineralized segments are repeatedly developed, with accordance to multi-layer structure of the ore-controlling bedding faults and orebodies. The small a min (minimum multifractal singularity) of the Cu, Ag and Au in M 1 reflect bare mineralization. The a min also displays that the mineralization intensities are varied for distinct elements and for different locations, yet the similarity of the distinct ore-forming processes is manifested by constant a min / f ( a min ) ratio. The constant ratio indicates the wider mineralization range denotes a more compact concentration distribution. The compact distributions represent the wide Cu, Ag and Au mineralization in skarns, and the loose distributions reflect the bare Cu, Ag and Au mineralization in marbles. Moreover a min shows a positive correlation with Hurst exponents in the Shizishan skarn orefield. Using fractal analysis the author's show that although the mineralization intensities for different elements and different locations along the Shizishan skarn orefield is not consistent, similar mineralization processes can be correlated to similar fractal exponents.

Strong Anticipation: Complexity Matching in Interpersonal Coordination

Anticipation in sensori-motor synchronization tasks is often attributed to the presence of internal model based on situation regularities. However this interpretation seems unable to explain the synchronization of an organism with a complex environment. Instead the notion of “strong anticipation” provides a new point of view about synchronization between complex systems. It implies a global coordination on non-local time scales between an organism and its environment. The matching between the long-range correlations of the organism and the long-range correlations of the environment could attest for the presence of such a strong anticipatory process. We propose here to test the presence of strong anticipation in interpersonal coordination task. Results show a close matching of fractal exponents of participants within a dyad. Moreover this matching cannot be attributed to only short-term adaptations as revealed by the low percentage of local cross-correlations.

Fractal Weyl law for Linux Kernel architecture

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be $\nu \approx 0.63$ that corresponds to the fractal dimension of the network $d \approx 1.2$. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension $d<2$.

P122 The use of impulse oscillometry (IOS) to study fractal scaling and sample entropy in airway resistance time series in severe asthma

IntroductionSevere asthma affects airway calibre and can be monitored using IOS. Sample entropy (SampEn) is a measure of complexity and is defined as the probability that sequences of patterns (template...

Fractal kinetic analysis of polymers/nonionic surfactants to eliminate lignin inhibition in enzymatic saccharification of cellulose

The profile of enzymatic saccharification of Avicel in the presence and absence of lignin has been described with a fractal kinetic model (Wang and Feng, 2010), in which the retarded hydrolysis rate of enzymatic saccharification of cellulose has been represented with a fractal exponent. The lignin inhibition in the enzymatic saccharification of cellulose is indexed by the increase of fractal exponent, which can not be fully counterbalanced by high cellulase loading due to the high fractal exponent at high cellulase loading. On the contrary, fractal kinetic analysis indicates that an addition of some nonionic surfactant/polymers decrease the fractal exponent to the original values of enzymatic saccharification of Avicel without lignin and the corresponding toxicity of nonionic surfactants/polymers on the consecutive ethanol fermentation strain Saccharomyces cerevisiae is also examined.

Monogenetic vent self-similar clustering in extending continental crust: Examples from the East African Rift System

The spatial clustering of fracture networks and vents in basaltic volcanic fields has been analyzed in three sectors of the East African Rift System, the classical example of an active continental rift. Fracture trace maps and monogenetic basaltic vents have been thus collected in the Afar Depression, in the Main Ethiopian Rift, and in the Virunga Belt (Western Rift). The mapped vents are generally younger than 2 Ma, and most are of Holocene age. All the analyzed fracture networks have self-similar clustering with fractal exponents ( D f ) varying in the 1.54−1.85 range. Also, vents show a self-similar clustering with fractal exponents ( D v ) in the 1.17−1.50 range. For all the studied sectors, the relationship D f > D v has been observed. The fractal exponents for vents ( D v ) of power-law distributions are computed in a range of lengths with a lower and an upper cutoff. The upper cutoff ( Uco ) for the fractal clustering of vents in the studied sectors of the East African Rift System are compared with the respective crust thickness derived by independent geophysical data. The computed Uco s for the studied sectors well match the crust thickness in the volcanic fields. A preliminary conceptual model to explain the relationships between the upper cutoffs of the fractal distribution of vents and the thickness of the crust in the volcanic fields is thus proposed in the light of the percolation theory.

Do Arnold tongues really constitute a fractal set?

We review and analyze the main features of fractal sets, as well as argue that the geometric property called fractality is not a well defined concept in the literature. As an example, we consider the mode locking phenomenon exhibited by the Sine Circle Map, concerning those sets in parameter space called Arnold Tongues and Devil's Staircase (more exactly, their complements). It is shown that well-known results for the fat fractal exponent of the ergodic region turn out to be valid only in a tiny region of parameter space where ρ ∊ [0,a], a → 0, being ρ the winding number. A careful geometric analysis shows that a misleading simplification has led to a loss of information that hindered relevant conclusions. We propose an alternative, broader and more rigorous approach in that it selects a generic interval from the whole domain which excludes the mentioned restricted region. Our results reveal that the measure shows a different dependence on tongue widths, so we argue that such a discrepancy is only possible if we assume that the set in question does not satisfy a scale-free property. Since there is no invariant exponent for the complementary set of Arnold Tongues (and consequently for that of Devil's Staircase), we conclude that the exponent found by Ecke et al.[4] does not characterize those sets as true fractals. We also consider those sets as an example that statistical criteria are insufficient to characterize any set as being a fractal; they should always follow an analysis of the topological process responsible for generating the set.

A model for wet aggregation of ash particles in volcanic plumes and clouds: 2. Model application

The occurrence of particle aggregation has a dramatic effect on the transport and sedimentation of volcanic ash. The aggregation process is complex and can occur under different conditions and in multiple regions of the plume and in the ash cloud. In the companion paper, Costa et al. develop an aggregation model based on a fractal relationship to describe the rate particles are incorporated into ash aggregates. The model includes the effects of both magmatic and atmospheric water present in the volcanic cloud and demonstrates that the rate of aggregation depends on the characteristics of the initial particle size distribution. The aggregation model includes two parameters, the fractal exponent Df, which describes the efficiency of the aggregation process, and the aggregate settling velocity correction factor ψe, which influences the distance at which distal mass deposition maxima form. Both parameters are adjusted using features of the observed deposits. Here this aggregation model is implemented in the FALL3D volcanic ash transport model and applied to the 18 May 1980 Mount St. Helens and the 17–18 September 1992 Crater Peak eruptions. For both eruptions, the optimized values for Df (2.96–3.00) and ψe (0.27–0.33) indicate that the ash aggregates had a bulk density of 700–800 kg m−3. The model provides a higher degree of agreement than previous fully empirical aggregation models and successfully reproduces the depositional characteristics of the deposits investigated over a large range of scales, including the position and thickness of the secondary maxima.

Self-Similar and Self-Affine Properties of Two-Dimensional Fracture Patterns in Rocks

In this paper, we investigate the fractal properties of binary maps of rock fractures at different scales and different geological types, as well as different families of fracture patterns obtained from a two-dimensional Laplacian growth model (LGM). From these analyses we figure out which families of the LGM patterns match the structural properties of the fracture binary maps. The LGM is defined in terms of a nonlinear map that depends on two parameters, λ and \(\mathfrak{a}\), that respectively define the area and shape of the elements of the aggregate that conforms the patterns. The fractal dimension and roughness exponent of the LGM patterns are found to depend on \(\mathfrak{a}\), with \(0<\mathfrak {a}<1\). From a detailed statistical analysis of these patterns we found that the fractal dimensions of capacity, correlation and information decrease monotonically as \(\mathfrak{a}\) increases. We also found that the values of these three fractal dimensions tend to collapse on top of each other as \(\mathfrak{a}\lessapprox1\). Remarkably, the fractal properties of rock fractures in the scales from millimeters up to a few meters appear to be well represented by the fractal structure of the LGM families of patterns with \(\mathfrak{a}=0.15\) and 0.30, while the fractal properties of rock fractures in the scale of kilometers seems to be well represented by the LGM family with \(\mathfrak{a}=0.90\). In addition, the three fractal dimension values of fracture binary maps in the scales from millimeters up to meters were found to be different between them. Nonetheless, for fractures in the scale of kilometers, the values of the three fractal dimensions are very close to each other as an indication of self-similar behavior. Analysis of the corrections to the scaling of the roughness exponent, ζ, suggests that they are negligible for the LGM family of fracture patterns with \(\mathfrak{a}=0.9\). This finding points to a self-affine structure for this family of patterns. In fact, the calculated roughness exponent results are in the range of values characteristic of rock fractures.

Experimental study of regional fractal speckle produced in large angle scattering

Using the interference of speckle and the spherical reference wave, we extract the real and the imaginary parts of the complex amplitudes of the speckle in large angle scattering. By calculating the spatial correlation functions of intensity, we find that the speckle has a property of regional fractality, i.e. the fractal exponent equals 1 in a small spatial region and it becomes less than 1 in a bigger region. The empirical analytic expression of the intensity correlation function is gained. The probability density distributions of the intensities and the complex amplitudes show that the regional fractal speckle still obeys zero-mean circular complex Gaussian statistics.

Fractal kinetic analysis of the enzymatic saccharification of cellulose under different conditions

Fractal kinetic analysis has been applied for the kinetics of enzymatic saccharification of cellulose. Based on the first order cellulose degradation kinetic, a fractal kinetic model with two parameters (rate coefficient and fractal exponent) has been developed. The model fits very well with the experimental data of enzymatic saccharification of cellulose under different conditions, including cellulase loading, nature of substrate, hydrolysis temperature, and addition of nonionic surfactant or polymer in the reaction medium. It indicates that the complex kinetics of enzymatic saccharification of cellulose can be described with the fractal kinetic model. The model exhibits that an addition of nonionic surfactant or polymer increases the rate coefficient and decreases the fractal exponent at the same time. A nonionic surfactant or polymer aqueous solution may be a potential medium engineering method for enzymatic saccharification of cellulose in biofuel industry.

Fractal correlation of heart rate variability in obese children

Background The loss of variability and fractal properties in the heart rate is related to greater morbidity and mortality rates. Objective The aim of the present study was to analyze heart rate dynamics in obese children by measuring short and long-term fractal exponents and heart rate variability (HRV) indices. Materials and methods 112 male and female children aged 8 to 12 years were divided into two groups based on body mass index: obese ( n = 51) and normal weight range ( n = 61). Heart rate was monitored beat-to-beat in the dorsal decubitus position for 20 min, analyzing the low frequency (LF) and high frequency (HF) spectral indices in normalized units and ms² as well as short and long-term fractal exponents (alpha-1 and alpha-2) of the detrended fluctuation analysis (DFA). The non-paired Student's t-test and Mann–Whitney test were used for the statistical analysis, with the level of significance set at P ≤ 0.05. Results The following were lower in the obese children: alpha-1 exponent value (0.930 ± 0.14 vs. 0.992 ± 0.11; P = 0.012), LF index (219.0 vs. 361.0; P = 0.001) and HF index (175.0 vs. 227.0; P = 0.019) in ms 2. No significant differences between groups were found for the alpha-2 exponent, LF and HF indices in normalized units or LF/HF ratio. Conclusion The obese children exhibited a reduction in short-term fractal correlation properties in heart frequency dynamics associated to a reduction in both sympathetic and parasympathetic activities, thereby demonstrating a need for early treatment for these children in order to avoid future complications.

Heart Rate Nonlinear Dynamics During Sudden Hypoxia at 8230 m Simulated Altitude

Acute hypobaric hypoxia is associated with autonomic changes that bring a global reduction of linear heart rate variability (HRV). Although changes in nonlinear HRV can be associated with physiologic stress and are relevant predictors of fatal arrhythmias in ischemic heart disease, to what extent these components vary in sudden hypobaric hypoxia is not known. Twelve military pilots were supplemented with increasing concentrations of oxygen during decompression to 8230 m in a hypobaric chamber. Linear and nonlinear HRV was evaluated at 8230 m altitude before, during, and after oxygen flow deprivation. Linear HRV was assessed through traditional time-domain and frequency-domain analysis. Nonlinear HRV was quantified through the short-term fractal correlation exponent alpha (alphas) and the Sample Entropy index (SampEn). Hypoxia was related to a decrease in linear HRV indexes at all frequency levels. A non-significant decrease in alphas (basal, 1.39 +/- 0.07; hypoxia, 1.11 +/- 0.13; recovery, 1.41 +/- 0.05; P = .054) and a significant increase in SampEn (basal, 1.07 +/- 0.11; hypoxia, 1.45 +/- 0.12; recovery, 1.43 +/- 0.09; P = .018) were detected. The observed pattern of diminished linear HRV and increased nonlinear HRV is similar to that seen in subjects undergoing heavy exercise or in patients with ischemic heart disease at high risk for ventricular fibrillation.

Identification of Mineral Intensity along Drifts in the Dayingezhuang Deposit, Jiaodong Gold Province, China

AbstractGreat mineral intensity in drifts or drills is characterized by both the high proportion and spatial cluster of the high grades. Great mineral intensity indicates the high ore quality and low dilution in the exploitation. The fractal dimension, the fluctuation exponent proposed in this paper and the lacunarity are utilized to analyze the mineral intensity along drifts in the Dayingezhuang gold deposit in Jiaodong gold province, China. It is proven that the combination of the parameters can identify the mineral intensity. The fractal dimension and fluctuation exponent are negatively correlated. In the places where the fractal dimension is small and fluctuation exponent is great, the mineralization is more pronounced. While in the case that the fractal dimensions are similar, smaller fluctuation exponent means more clustered structure of high grades and greater mineral intensity. In the diagram of fractal dimension versus lacunarity, the drifts with great mineral intensity can also be identified. The methods used in this paper provide a relatively comprehensive description for local mineral intensity, providing information for both the ore‐forming process and the exploitation.