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There are numerous existing works on investigating the dynamics of particle production process in ultrarelativistic nuclear collision. In the past, fluctuation of spatial pattern has been analyzed in terms of the scaling behavior of voids. But analysis of the scaling behavior of the void in fractal scenario has not been explored yet. In this work, we have analyzed the fractality of void probability distribution with a completely different and rigorous method called visibility graph analysis, analyzing the void-data produced out of fluctuation of pions in ${}^{32}$S–AgBr interaction at 200 GeV in pseudo-rapidity $(\eta )$ and azimuthal angle $(\varphi )$ space. The power of scale-freeness of visibility graph denoted by PSVG is a measure of fractality, which can be used as a quantitative parameter for the assessment of the state of chaotic system. As the behavior of particle production process depends on the target excitation, we can dwell down the void probability distribution in the event-wise fluctuation resulted out of the high energy interaction for different degree of target excitation, with respect to the fractal scenario and analyze the scaling behavior of the voids. From the analysis of the PSVG parameter, we have observed that scaling behavior of void probability distribution in multipion production changes with increasing target excitation. Since visibility graph method is a classic method of complex network analysis, has been applied over fractional Brownian motion (fBm) and fractional Gaussian noises (fGn) to measure the fractality and long-range dependence of a time series successfully, we can quantitatively confirm that fractal behavior of the void probability distribution in particle production process depends on the target excitation.

A modified Diffusion Limited Aggregation (DLA) model has been established for single and multi-center fractal growth. Number of particles $N$, size of one step $N_S$, deposition probability $p$, growth direction, and interaction effect are had been take into consideration for fractal analysis. In addition, the effect of internal interaction in multi-center growth have been taken into consideration. Fractal growth morphology shows strong boundary and interaction effects.

The electronic packaging and systems are very important topics as the limitation of miniaturization approaches in semiconductor industry. Regarding the optimal materials microstructure for these applications, we studied different alloys such as Sn-3.0Ag-0.5Cu (wt.%)/organic solderability preservative (SAC305/OSP) Cu and SAC305–0.05Ni/OSP Cu solder joints. We implemented the fractal dimension characterization and microstructure morphology reconstruction. This is the first time that we applied fractals on such alloys. The morphology reconstruction is important for predicting and designing the optimal microstructure for the advanced desirable properties these alloys. These analyzed parameters are important for the hand-held devices and systems especially for the exploitation. The fractal reconstruction was applied on the prepared microstructures with five different magnifications. The results confirmed successful application of fractals in this area of materials science considering the grains and shapes reconstructions.

In this paper, Cu thin films were successfully deposited on glass substrates using DC magnetron sputtering at varying deposition times. The deposition time was varied as 5, 9, 11 and 17min. The obtained Cu thin films were analyzed for morphology and topography using atomic force microscopy (AFM). The size of the surface structures/grains was seen to evolve with deposition time. The conventional/statistical, fractal and multifractal analyses were carried out on AFM images using existing imaging algorithms. The arithmetic roughness and interface width parameters were seen to evolve with the sputtering time. The autocorrelation and height–height correlation functions revealed that the surfaces of all the Cu thin films exhibited self-affine character, but were not mounded properties. The fractal dimensions computed using box counting and power spectral density functions revealed that larger dimensions were associated with larger surface features. The lacunarity coefficients were too small indicating that the surfaces were generally deficient in porosity and other defects. The multifractal analyses revealed that spatial roughness does not exhibit linear relationship with the deposition time. The study reveals that surface evolution and nanoscale behavior is significantly influenced by the deposition time although a linear relationship is not established.

In this paper, in order to figure out the variations before an earthquake and extract abnormal signals related to it, the geomagnetic three component Z, H, F minute values of 15 geomagnetic stations within 600km of the epicentral distance before the MS 6.6 Minxian–Zhangxian earthquake in Gansu were analyzed. The following are the results. (1) After the fractal analysis was used directly, only three geomagnetic stations in 15 geomagnetic stations showed synchronous anomalous signals; (2) After the method of this paper was used, 9 of the 15 geomagnetic stations (including the three stations in the first point) extracted two synchronous anomalous signals, and six of the nine geomagnetic stations presented additional synchronous anomalous signals. (3) Of the three abnormal signals extracted, one had a medium-term effect and two had short-term effects. (4) The anomalous duration of the Z component of nine geomagnetic stations was longer than that of H and F. And as the epicentral distance increased, duration decreased. While the proposed method could not clearly indicate the exact relationship between the anomalous signals and the earthquake, it was proved that the signals extracted are effective and well-correlative to the earthquake.

This paper contains the application of fractal concept in analyzing heartbeat (RR interval) fluctuations measured under controlled physical activity for subjects with stable angina pectoris (SAP). Results that illustrate the separation ability of the nonlinear methods, such as the Hurst R/S method, the detrended fluctuation analysis, DFA, and the method of G-moments, in distinguishing healthy from SAP subjects in scaling parameter space are presented.

Much evidence suggests complexity in cognitive and motor task performances [Gilden, 2001]. The present study builds upon this work, treating reading of text as a kind of complex coordination or coupling between reader and reading conditions. Three self-paced reading conditions presented connect text in units of different sizes: word, phrase, or sentence units, and repeatedly measure times between spacebar presses to advance the text. The three conditions reveal different patterns across the data. These patterns were evaluated using fractal analyses and Recurrent Quantification Analyses to distinguish highly fluent readers, PhD candidates in English literature, from competent but less fluent undergraduate readers.

The prevalence of a Patent Foramen Ovale is described in merely 30% of the asymptomatic population. This patency has been shown to be an increasing risk factor for paradoxical cerebral embolization. Some desaturation or decompression situations in human activities such as scuba diving or altitude flight are prone to provoke embolisations. The association with the presence of a patent Foramen Ovale and the onset of cerebral decompression sickness seems to be presenting an odds ratio value of about 5.1.¹ The presence of asymptomatic brain lesion-like "spots" has been investigated in a randomized population of diving individuals (n=42 randomized out of 200). The inclusion criteria were drastic and included: age (less than 41 years of age); diving experience (more than 200 logged dives); no decompression sickness episodes; no contraindications for the MRI examination; and no known central nervous system conditions. Data of the magnetic resonance investigation of the brain has been performed in 42 (diving) volunteers fully informed on the experimental procedures. The statistical comparison (Anova test after Kolmogorov-Smirnov compatibility testing and Neuman–Keuls discriminant post-test) of the fractal dimension obtained by means of the box counting method with the slope analysis (Harfa fractal analysis program). The comparison was performed with known pathological images such as multiple sclerosis (a pathology not emerging from vascular problems), ischemic thrombotic lesions (vascular problem), diver's asymptomatic brain spots, and the arteriography of the internal carotid in non-pathological humans (clearly vascular). The statistical difference (p<0.001) between the vascular related images, as well as the absence of statistical difference (p>0.05) with the non-vascular spots images advocates with a non-vascular origin of the diver's asymptomatic spots and thus the link between the patency of the cardiac Foramen Ovale and the brain "spots" seems not to be as clear as it was believed.

This paper describes fractal behaviors in a soccer game according to the player's position. It is quite important for us to characterize the fractal motion behaviors of the objects during the game. We obtained two-dimensional coordinates of the objects using standard video processing techniques from a computer soccer game. We calculated values of regularization dimensions of the time series to characterize their fractal behaviors. To see positional dependence, we averaged individual player's values over the same position in the same team. When a team is one-sidedly experiencing a severe attack, its defenders have higher fractal dimensions than those of the opponent's corresponding players. We propose a new measure of relative dominance in attack against the opponent team.

Through analysis of the morphology of dendritic arborisation of neurons from the substantia gelatinosa of dorsal horns from four different species, we have established that two types of cells (stalked and islet) are always present. The aim of the study was to perform the intra- and/or inter-species comparison of these two neuronal populations by fractal analysis, as well as to clarify the importance of the fractal dimension as an objective and usable morphological parameter. Fractal analysis was carried out adopting the box-counting method. We have shown that the mean fractal dimensions for the stalked cells are significantly different between species. The same is true for the mean fractal dimensions of the islet cells. Still, no significant differences were found for the fractal dimensions of the stalked and islet cells within a particular species. The human species has shown as the only exception where fractal dimensions of these two types of cells differ significantly. This study shows once more that the fractal dimension is a useful and sensitive morphological descriptor of neuronal structures and differences between them.

In pathological brain, a variety of morphological forms exist that reflect differences in functional requirements. To better understand microglia function in neurological disease, it is important to identify and quantify microglia morphology associated with specific neuropathologies. Traditional feature parameters such as area or cell diameter are not sufficient. In this study microglia were quantified by the box-counting fractal dimension (D_B). One hundred and four cells from post-mortem tissue were analyzed comprising cells of controls, Alzheimer's disease, schizophrenia and affective disorder. The D_B was significantly different from the control (1.36) compared to schizophrenia (1.41), Alzheimer's disease (1.41) and affective disorder (1.43) with p < 0.01. Thus fractal analysis provides a useful quantitative and objective measure of microglial form associated with normal function and diverse neuropathology. The distribution of fractal dimensions associated with microglia structure and activation with disease progression also differs, suggesting a different etiology for these diseases.

The prevalence of Patent Foramen Ovale (PFO) is an increasing risk factor for paradoxical cerebral embolism. Decompression Sickness (DCS) results from gas coming out of solution in the bodily fluids and tissues when a diver ascends too quickly. This occurs because decreasing pressure lowers the solubility of gas in liquid. Also, the expansion of gas in the lungs may lead to alveolar rupture which may, in turn, result in Arterial Gas Embolism (AGE). The amount of nitrogen that diffuses into the bodily tissues during a dive depends on the depth of the dive and the duration of exposure. PFO acquires significance in various congenital heart diseases leading to right to left shunt and thus to paradoxical embolism. The presence of asymptomatic brain lesion like "spots" has been investigated in a randomized population of diving individuals. The partial pressure of the gas can be calculated by Henry's Law and the pressure of the blood stream can be calculated by Poiseuille's Law. The statistical comparison of the fractal dimension was obtained by means of Box Counting Method with slope analysis. We observe that PFO plays a significant role in the occurrence of unexplained cerebral DCS and white spots in the brain.

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

Soil structure plays an important role in flow and transport phenomena, and a quantitative characterization of the spatial heterogeneity of the pore space geometry is beneficial for prediction of soil physical properties. Morphological features such as pore-size distribution, pore space volume or pore–solid surface can be altered by different soil management practices. Irregularity of these features and their changes can be described using fractal geometry. In this study, we focus primarily on the characterization of soil pore space as a 3D geometrical shape by fractal analysis and on the ability of fractal dimensions to differentiate between two a priori different soil structures. We analyze X-ray computed tomography (CT) images of soils samples from two nearby areas with contrasting management practices. Within these two different soil systems, samples were collected from three depths. Fractal dimensions of the pore-size distributions were different depending on soil use and averaged values also differed at each depth. Fractal dimensions of the volume and surface of the pore space were lower in the tilled soil than in the natural soil but their standard deviations were higher in the former as compared to the latter. Also, it was observed that soil use was a factor that had a statistically significant effect on fractal parameters. Fractal parameters provide useful complementary information about changes in soil structure due to changes in soil management.

Chaotic dynamical systems are pervasive in nature and can be shown to be deterministic through fractal analysis. There are numerous methods that can be used to estimate the fractal dimension. Among the usual fractal estimation methods, variance fractal dimension (VFD) is one of the most significant fractal analysis methods that can be implemented for real-time systems. The basic concept and theory of VFD are presented. Recent research and the development of several applications based on VFD are reviewed and explained in detail, such as biomedical signal processing and pattern recognition, speech communication, geophysical signal analysis, power systems and communication systems. The important parameters that need to be considered in computing the VFD are discussed, including the window size and the window increment of the feature, and the step size of the VFD. Directions for future research of VFD are also briefly outlined.

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.

Because of the low permeability, multi-stage hydraulic fractured horizontal wells (MHFHWs) occupy a dominant position among production wells in tight gas reservoir. However, net present value (NPV) estimation method for MHFHW in tight gas reservoirs often ignores the effect of heterogeneity in microscopic pore structure. Apart from that, a new fractal model is presented for NPV of MHFHW, based on the fractal expressions of formation parameters. First, with the aid of apparent permeability model, a pseudo pressure expression considering both reservoir fractal features and slippage effect is derived, contributing to establish the productivity model. Secondly, economic assessment method is built based on the fractal productivity model, in order to obtain the NPV of MHFHW. Thirdly, the type curves are illustrated and the influences of different fractal parameters are discussed. The pore fractal dimensions $D_f$ and the capillary tortuosity fractal dimensions $D_T$ have significant effects on the NPV of an MHFHW. Finally, the proposed model in this paper provides a new methodology for analyzing and predicting the NPV of an MHFHW and may be conducive to a better understanding of the optimal design of MHFHW.

The diagonal effect of orders is well documented in different markets, which states that the orders are more likely to be followed by the orders of the same aggressiveness and implies the presence of short-term correlations in order flows. Based on the order flow data of 43 Chinese stocks, we investigate if there are long-range correlations in the time series of order aggressiveness. The detrending moving average analysis shows that there are crossovers in the scaling behaviors of overall fluctuations and order aggressiveness exhibits linear long-term correlations. We design an objective procedure to determine the two Hurst indexes delimited by the crossover scale. We find no correlations in the short term and strong correlations in the long term for all stocks except for an outlier stock. The long-term correlation is found to depend on several firm specific characteristics. We also find that there are nonlinear long-term correlations in the order aggressiveness when we perform the multifractal detrending moving average analysis.

Analysis of human movements is an important category of research in biomedical engineering, especially for the rehabilitation purpose. The movement of limbs is investigated usually by analyzing the movement signals. Less efforts have been made to investigate how neural that correlate to the movements, are represented in the human brain. In this research, for the first time we decode the limb movements by fractal analysis of Electroencephalogram (EEG) signals. We investigated how the complexity of EEG signal changes in different limb movements in motor execution (ME), and motor imagination (MI) sessions. The result of our analysis showed that the EEG signal experiences greatest level of complexity in elbow flexion and hand-close movements in ME, and MI sessions respectively. On the other hand, the lowest level of complexity of EEG signal belongs to hand-open and rest condition in ME, and MI sessions, respectively. Employing fractal theory in analysis of bio signals is not limited to EEG signal, and can be further investigated in other types of human’s bio signals in different conditions. The result of these investigations can vastly been employed for the rehabilitation purpose.

Analysis of eye movement due to different visual stimuli always has been one of the major research areas in vision science. An important category of works belongs to decoding of eye movement due to variations of color of visual stimuli. In this research, for the first time, we employ fractal analysis in order to investigate the variations of complex structure of eye movement time series in response to variations of color of visual stimuli. For this purpose, we applied two different images in three different colors (red, green, blue) to subjects. The result of our analysis showed that eye movement has the greatest complexity in case of green visual stimulus. On the other hand, the lowest complexity of eye movement was observed in case of red stimulus. In addition, the results showed that except for red visual stimulus, applying the visual stimulus with greater complexity causes the lower complexity in eye movements. The employed methodology in this research can be further applied to analyze the influence of other variations of visual stimuli on human eye movement.