Fractal Behavior Research Articles

Complexity of information processing in obsessive-compulsive disorder based on fractal analysis of EEG signal.

The human brain is considered as a self-organizing system with self-similarities at various temporal and spatial scales called “fractals”. In this scale-free system, it is possible to decode the complexity of information processing using fractal behavior. For instance, the complexity of information processing in the brain can be evaluated by fractal dimensions (FDs). However, it is unclear how over-elaboration of information processing impacts the dimensionality of its fractal behavior. In this study, we hypothesized that FDs of electroencephalogram (EEG) in obsessive-compulsive disorder (OCD) should be higher than healthy controls (HCs) because of exaggeration of information processing mainly in the frontal regions. Therefore, a group of 39 OCDs (age: 34.76±8.22, 25 female, 3 left-handed) and 19 HCs (age: 31.94±8.22, 11 female, 1 left-handed) were recruited and their brain activities were recorded using a 19-channel EEG recorder in the eyes-open resting-state condition. Subsequently, fractal dimensions of the cleaned EEG data were calculated using Katz's method in a frequency band-specific manner. After the test of normality, significant changes in the OCDs as compared to the HCs were calculated using a two-sample t-test. OCDs showed higher FDs in the frontal regions in all frequency bands as compared to HCs. Although, significant increases were only observed in the beta and lower gamma bands, mainly at the high beta. Interestingly, neurophysiological findings also show association with severity of obsessive behaviors. The results demonstrate that complexity of information processing in the brain follows an intimate nature of structural and functional impairments of the brain in OCD.

Micro-Mechanical and 3D Fractal Analysis, Durability, and Thermal Behaviour of Nano-Modified Cementitious Lightweight Composites for Building Facades

There are increasing research endeavours on the application of nanotechnology in the construction industry and lightweight composites. In this study, the influence of different percentage (1%, 2%, and 3% by weight of cement) colloidal nano-silica particles on the mechanical, thermal, and durability properties of lightweight cementitious composites was studied through measurement of compressive strength, flexural response, micro-hardness measurement, pore structure analysis, thermal conductivity, water permeability, and chloride penetration. Moreover, 3D X-ray Compute Tomography together with digital image analysis and 3D fractal analysis was used to characterize the nano-silica, micro-structures, and the fracture surfaces. The experimental results show that incorporating nano-silica particles resulted in a mechanical strength increase up to 45.4 % and a water permeability and chloride migration decrease up to 51.2% and 48.2%, respectively. The micro-structural and 3D fractal analysis also indicated that dense, flaw-free, and thus more resistant, interfaces to micro-cracks were formed and greater fractal dimensions were obtained with the increase of the nano-silica content. Finally, the 3D views confirmed that the nano-silica clusters were well interconnected which further increase the carrying capacity and reducing the heat flow.

Some stochastic chaotic attractors with global derivative and stochastic fractal mapping: Existence, uniqueness and applications

One of the great ability of humans is to use mathematical formulas to replicate what they see in their daily activities, they start with simple models, when these models are unable to achieve their goals and they modify or replace them with more complex mathematical formulas. Chaos has been modelled using classical differential operators, classical with fractional orders. On the other hand, many mathematical mapping have been used to simulate some fractals behaviours. In this paper, in order to capture more fractal and chaos behaviours, we introduce the notion of chaos stochastic models and fractal stochastic mapping. We additionally present the conditions of existence and uniqueness of chaos‐stochastic models. Some numerical simulations are presented to help see the effect of stochastic functions.

Fractal Analysis of Fragmentation Distribution of Rockbursts Induced by Low-Frequency Seismic Disturbances

Low-frequency seismic disturbances frequently induce violent rockburst hazards, seriously threatening the safety of deep excavation and mining engineering. To investigate the characteristics and mechanisms of rockbursts induced by seismic disturbances, in this study a series of true triaxial experiments, including the moderate seismically induced, the weak seismically induced, and the self-initiated rockburst experiments under different conditions were conducted. The fractal geometry theory was applied to study rockbursts and the fractal dimensions of fragmentation distribution of different types of rockbursts were calculated. The results show that the fragmentation distributions of both the seismically induced and self-initiated rockbursts exhibit fractal behaviors. For the moderate seismically induced rockbursts, as the static stresses (i.e., the maximum and minimum static stresses) and disturbance amplitude increase, the fractal dimension increases, whereas, as the disturbance frequency increases, the fractal dimension decreases first and then increases. Under similar static loading conditions, the moderate seismically induced rockbursts have the largest fractal dimension, followed by the self-initiated rockbursts, and the weak seismically induced rockbursts have the smallest fractal dimension. There is a linear relationship between the average fractal dimension and kinetic energy of these rockbursts, implying that the fractal dimension can serve as an indicator for estimating rockburst intensity. Furthermore, from a fractal point of view, the energy input, dissipation, and release of these rockbursts are all linear processes.

Influence of weak inclusions on the fracturing and fractal behavior of a jointed rock mass containing an opening: Experimental and numerical studies

To gain a better understanding of the interaction between weak inclusions and jointed rock masses, a conceptual model containing a joint set and an opening is prepared, and cases involving unfilled and filled openings are considered. The influence of weak inclusions on the fracturing and fractal behavior of these models is investigated by using laboratory experiments, the rock failure process analysis (RFPA) code and the fractal geometry. An overhanging beam model is proposed to explain the initiation mechanism of tensile cracks around the unfilled and filled openings. The RFPA simulations provide deep and quantitative insight into the fracturing behavior. The coalescence patterns of the surrounding rock mass obtained from physical and numerical tests are in reasonable agreement, and can be classified into two categories: shearing along the joint set and cutting through the rock bridges. The inclusion with a low strength exerts little influence on the failure pattern but has an appreciable reduction in the stress values and an appreciable increment in the mechanical properties. The crack coalescence exhibits fractal properties, and there are good correlations between the fractal dimension and mechanical properties of these jointed rock specimens containing unfilled and filled openings.

Polypropylene foams under CO2 batch conditions: From formulation and rheological modeling to cell-growth simulation

The objective of this work was to study the development of polypropylene (PP) foams with low density (<100 kg m−3) using a CO2 batch process. To carry out this study, we used a PP-g-MA that was chemically modified by reactive extrusion to obtain polymer branched structures. Two reagent systems were selected on the basis of chemical reactions with maleic anhydride—the first one is based on the reaction between maleic anhydride and primary amines (MA/NH2, triamine, Jeffamine T-403), and the second one is based on ionic interactions from Zn neutralization. Both of these systems have shown that it is possible to obtain branched PP structures favorable to CO2 foaming and were compared to a commercial, high-melt-strength PP.We have shown that the branched structure of polymer chains can be defined on the basis of a rheological criterion. It defines the notion of fractal behavior (coupling of relaxation modes), which polymer chains must have for nonlinear behavior (strain hardening). This criterion has been defined by analogy with the sol-gel transition, tan δ = 1.Finally, in terms of modeling and simulation, the plasticizing effect of CO2 was modeled from the experimental rheological behavior of the PP plasticized with synthetic oil. A theoretical power law was derived and was then introduced in the mass balance equations to predict the cell growth during depressurization.

Fractional Charge of Quarks and Fractal Properties of Hadrons

The origin of fractional charge of a quark is investigated considering the fractal structure of a hadron. Hadron is suggested to be a fractal object with fractal dimension 9/2. Describing quark as a quasiparticle in an analogy with quasi particle in Fractional Quantum Hall Effect, the filling factors are extracted which show large fractional plateau. It is suggested that quarks behave like quasi particles and the fractional charges of quarks can be attributed to the fractal behavior of a hadron.

NEW PERSPECTIVE AIMED AT LOCAL FRACTIONAL ORDER MEMRISTOR MODEL ON CANTOR SETS

In this paper, we structure the idea of the fractal memristor to simulate the nonlinear self-similarity dopant drift effect for the first time. We investigate the fractal memristor device with the use of the local fractional Laplace transform, and the dynamic fractal behavior of the fractal HP TiO2 memristor model is simulated by changing the integer order of traditional memristor model to [Formula: see text]. Compared with the traditional model, the fractal memristor can be well-presented in describing the non-differentiable dopant drift effect.

Defining a no-reference image quality assessment by means of the self-affine analysis.

In this paper we propose a novel Blind Image Quality Assessment via Self-Affine Analysis (BIQSAA) method by considering the wavelet transform as a linear operation that decomposes a complex signal into elementary blocks at different scales or resolutions. BIQSAA decomposes a distorted image into a set of wavelet planes ωλ, ϕ of different spatial frequencies λ and spatial orientations ϕ, and it transforms these wavelet planes into one-dimension vector Ω using a Hilbert scanning. From the vector Ω there were obtained their wavelet coefficient fluctuations estimated by the inverse of the Hurst exponent in decibels, whose scaling-law or fractal behavior was obtained by applying Fractal Geometry or Self-Affine Analysis. The scaling exponents calculated for the coefficient fluctuation behavior of Image Lena at 24bpp, at 1.375bpp, and at 0.50bpp were H24bpp = 0.0395, H1.375bpp = 0.0551, and H0.50bpp = 0.0612, respectively. Our experiments show that BIQSAA algorithm improves in 14.36% the Human Visual System correlation, respect to the four state-of-the-art No-Reference Image Quality Assessments.

Morphologic evaluation of some promising A3B porphyrinic type compounds designed for theranostic applications in cancer

• New A 3 B porphyrinic structure with pharmacological potential were investigated. • Morphologic and textural properties of A 3 B porphyrins were assessed by AFM. • A 3 B porphyrinic compounds shown fractal (self-similar) behaviour. • HT-29 colon carcinoma and U-87 glioblastoma membrane integrity not altered by A 3 B. • P and Zn(II)P similarly incorporated into tumor cells as proved by flow cytometry. The use of porphyrinic compounds for photodynamic therapy (PDT) in cancer is well documented, but the aggregation tendency in physiologically-relevant media impedes on their medical use. The present study argues for the use of atomic force microscopy (AFM) for the evaluation of the aggregation tendency in terms of topography, self-similarity (fractal behavior) and textural properties of some promising A3B-type porphyrinic compounds, previously described by our group. It was found that the samples are isotropic and rough enough to favor cell adhesion. Despite having different aggregation tendencies, the P and Zn(II)P compounds (5 µM) were similarly incorporated into tumour cells (HT-29 human colon carcinoma and U-87 MG human glioblastoma), as demonstrated by flow cytometry data. Results suggest that there might be an aggregation, “window” that does not impede on the interaction of the investigated porphyrins with tumor cells.

Assessing the contribution of groundwater to catchment travel time distributions through integrating conceptual flux tracking with explicit Lagrangian particle tracking

Travel time distributions (TTDs) provide an effective way to describe the transport and mixing processes of water parcels in a subsurface hydrological system. A major challenge in characterizing catchment TTD is quantifying the travel times in deep groundwater and its contribution to the streamflow TTD. Here, we develop and test a novel modeling framework for an integrated assessment of catchment scale TTDs through explicit representation of 3D-groundwater dynamics. The proposed framework is based on the linkage between a flux tracking scheme with the surface hydrologic model (mHM) for the soil-water compartment and a particle tracking scheme with the 3D-groundwater model OpenGeoSys (OGS) for the groundwater compartment. This linkage provides us with the ability to simulate the spatial and temporal dynamics of TTDs in these different hydrological compartments from grid scale to regional scale. We apply this framework in the Nägelstedt catchment in central Germany. Simulation results reveal that both shape and scale of grid-scale groundwater TTDs are spatially heterogeneous, which are strongly dependent on the topography and aquifer structure. The component-wise analysis of catchment TTD shows a time-dependent sensitivity of transport processes in soil zone and groundwater to driving meteorological forcing. Catchment TTD exhibits a power-law shape and fractal behavior. The predictive uncertainty in catchment mean travel time is dominated by the uncertainty in the deep groundwater rather than that in the soil zone. Catchment mean travel time is severely biased by a marginal error in groundwater characterization. Accordingly, we recommend to use multiple summary statistics to minimize the predictive uncertainty introduced by the tailing behavior of catchment TTD.

Limited fractal behavior in cement paste upon mercury intrusion porosimetry test: Analysis and models

The fractal nature of cement paste has been discussed a lot in recent decades. Most of studies are concentrated on ordinary fractal dimension of a complete fractality, while an incomplete or limited fractality is exhibited in fact. This paper presents some results towards a comprehensive understanding about the limited fractal behavior in cement paste upon the mercury intrusion porosimetry test. To that end, two modules of analysis are adopted, i.e., the ordinary fractal and the multiscale fractal. Moreover, ordinary and multiscale fractal models are introduced to simulate pore structure, and pore size distributions are compared among ordinary fractal model, multiscale fractal model and test. Results indicate that multiscale fractal provides a much richer and preciser way than ordinary fractal to describe the limited fractal behavior in cement paste even it may deviate far from perfect fractaltiy.

Fractal characteristics of pore networks and sealing capacity of Ordovician carbonate cap rocks: A case study based on outcrop analogues from the Tarim Basin, China

Understanding and predicting the main controls on the sealing capacity of carbonate cap rocks is of great significance for ultradeep carbonate reservoir exploration and production. This study focuses on revealing the pore networks and sealing capacity of the Ordovician carbonate cap rocks in the Tarim Basin by analyzing samples from outcrop analogues using optical and scanning electron microscopy and a combination of mercury intrusion capillary pressure and nitrogen gas adsorption. Three classes of cap rocks are defined here according to their pore-throat structure, fractal dimension, and sealing capacity. These carbonate cap rocks are dominated by limestones and dolomitic limestones. Four pore types are identified: microfractures, intragranular, intercrystalline, and intracrystalline. Six pore structure types show multiscale variability from macropores to micropores. The pore structures present multiple fractal behaviors, with fractal dimensions showing an increasing trend as the pore diameter decreases. The cover coefficient, a parameter that allows characterization of the caprock sealing performance, shows an increasing trend along with increasing the fractal dimension of pore structure. The average cover coefficients of six pore structure types not only show good correlations (either exponential or linear) with certain fractal dimensions, but they also demonstrate a strong positive correlation with the average fractal dimension. These results suggest that the sealing capacity of the studied rocks increases with increasing fractal dimension. The sealing performance of cap rocks significantly decreases with an increasing amount of macropores. This work provides a relevant case study for further understanding of pore structures and sealing capacity of carbonate cap rocks.

Research on Synchronization of Fractal Behaviors in 2-D Logistic Map

In this paper, the fractal behaviors in 2-D Logistic map are discussed. First, the definition and some properties of fractal set in 2-D Logistic map are given. Moreover, several synchronous definitions between different 2-D Logistic maps are introduced in this paper, such as complete synchronization, coupled synchronization, and adaptive synchronization. The synchronization of the fractal set between different Logistic maps is accomplished by synchronizing their iterative tracks. The simulations illustrate the effectiveness and correctness of these methods.

Fractal distribution of mineral species among the crystallographic point groups

Abstract Crystallographic data from 5289 IMA-approved mineral species in the RRUFF database were used to examine the distribution of species among the 32 crystallographic point groups. It is found that within each crystal system, minerals strongly prefer point groups with higher group orders. Within a crystal system, the abundance of minerals belonging to each point group approximately obeys a power law with respect to group order, the same mathematical formalism that describes objects with fractal geometry. In this framework, each crystal system has its own fractal dimension; crystal systems possessing threefold (or sixfold) symmetry elements (i.e., trigonal, hexagonal, isometric) have significantly lower fractal dimension (&amp;lt;2), while those with only one-, two-, or fourfold symmetry elements (triclinic, monoclinic, orthorhombic, tetragonal) have higher fractal dimension (&amp;gt;2). While higher symmetry is preferred within a crystal system, the opposite trend is observed when comparing between crystal systems, with more species preferring crystals systems with lower order symmetry elements than those with higher order symmetry elements at constant group order. The combination of these two competing trends leads to a complex distribution of minerals among the crystal systems, and to the monoclinic group 2/m, the orthorhombic group 2/m2/m2/m, and the triclinic group 1 being the three most popular point groups, respectively. The fractal behavior of symmetry distribution among minerals points toward universal scaling patterns not just in physical, geometric objects but also in the way that symmetry is incorporated into natural periodic structures.

Some New Iterative Algorithms for Solving One-Dimensional Non-Linear Equations and Their Graphical Representation

Solving non-linear equation is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton's method. However, it is very difficult to select reasonable initial guess of the solution for most systems of non-linear equations. Besides, the computational efficiency is not high enough. Taking this into account, based on variational iteration technique, we develop some new iterative algorithms for solving one-dimensional non-linear equations. The convergence criteria of these iterative algorithms has also been discussed. The superiority of the proposed iterative algorithms is illustrated by solving some test examples and comparing them with other well-known existing iterative algorithms in literature. In the end, the graphical comparison of the proposed iterative algorithms with other well-known iterative algorithms have been made by means of polynomiographs of different complex polynomials which reflect the fractal behavior and dynamical aspects of the proposed iterative algorithms.

Analysis and FPGA of semi-fractal shapes based on complex Gaussian map

Abstract This paper studies the fractal-like behavior exhibited by the complex form of Gaussian chaotic map and the capability of digital architectures to mimic that behavior. Digitally realized chaotic attractors had many applications; hopefully, a digital realization of fractals may achieve the same eventually. The Gauss map is viewed concerning its bifurcation behavior, time waveform plots, Lyapunov exponent, and attractor performance through parameter variation. The Fractal-like entities emerging from the perceived complex map are examined versus different map coefficients for the highest chaotic periods to extract an interpretation for the fractal behavior. FPGA implementation of the fractal behavior is discussed viewing an optimized hardware architecture that eventually displays a fractal entity experimentally.

Fractal and re-breakage behavior of the saturated rock granular material under compaction

A compaction experiment of the saturated rock granular material is conducted to investigate its fractal and re-breakage behavior under compaction. The samples after compaction have obvious fraction characteristics, the variation of grain size distribution (GSD) and the fractal characteristics of samples with different Talbot power exponent (TPE) are analyzed. It presents that samples after compaction still obey to the modified Talbot continuous gradation equation, and its coefficient increases with the compaction degree going on; however, the samples with larger TPE are much easier to far away from the modified equation than samples with smaller TPEs. Sample with different TPEs reach their most obvious re-breakage at the different compaction levels. It can be inferred that 10 mm is an appropriate compression for the sample with the TPE of 0.5, and 30 mm for 0.6. The suitable compaction for the samples with TPEs of 0.7, 0.8, and 0.9 could be found as the curves are converged. During the compaction process, the fractal dimension (FD) and re-breakage degree of samples with the different TPEs increase with the compression going on, which result in the new GSD. These results could be used to guide engineering practice to obtain the optimal compaction for samples with different TPE.

Networked Microgrid Operations: Supporting a Resilient Electric Power Infrastructure

The fundamental requirements for electric power-system operations are the same regardless of the system size. While specific operational implementations will change based on the measurements-regardless of whether the power system spans a continent or a single building, it is necessary to regulate the frequency and voltage, adjust generating unit outputs, and to protect equipment from electrical faults. In this way, the operational requirements of electric power systems exhibit fractal behaviors. Because of this operational characteristic, it is technically feasible to separate or segment any portion of an electric power system and operate it independently, if sufficient generating assets and control systems are present. However, the generating assets and controls necessary to implement this type of operation have not traditionally been deployed; centralized generation and control is currently the predominant operational paradigm. For example, the frequency is typically regulated at a regional level by balancing the generation at large central units with the consumption of power by end-use customers and system losses. Small changes in the frequency are used to coordinate the sharing of the load between generators on a regional level.

Higher-Order Root-Finding Algorithms and Their Basins of Attraction

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.