Fractal Methodology Research Articles

Determination of alteration zones applying fractal modeling and Spectral Feature Fitting (SFF) method in Saryazd porphyry copper system, central Iran

The target of this research is recognition of the alteration zones utilizing concentration-area fractal methodology according to the reflection of the main minerals of each alteration zone that enhanced by Spectral Feature Fitting (SFF) method due to Advanced Space-borne Thermal Emission and Reflection Radiometer (ASTER) satellite images in Saryazd porphyry system, central Iran. The alterations and mineralization developed in Eocene volcanics. Remote sensing results achieved by the SFF method and Concentration -Area (C - A) fractal modeling represent different parts of propylitic, argillic and phyllic alteration zones due to their intensity and pixel values. In addition, the results reveal that there is a ring-shaped structure in the alteration zones, which are correlated with results, derived from X-Ray Diffraction (XRD) analyses and field observations.

Application of an approximate geostatistical simulation algorithm to delineate the gold mineralized zones characterized by fractal methodology

The Zarshuran Carlin-type gold deposit is located in the NW part of the Sanandaj-Sirjan metamorphic zone which hosts Neoproterozoic carbonate and black shale rock units. Concentration-Volume (C–V) fractal model and Turning Band Simulation method have been applied to effectively separate gold mineralized stages in the Zarshuran deposit. Three mineralized stages determined by the means of C–V fractal model and Turning Bands Simulation method were correlated with the mineralized stages considering geological studies. The results revealed the presence of enriched (>37 ppm) and Au high grade (5.5–37 ppm) mineralized zones within the deposit. A log ratio matrix was used to correlate the results obtained by the C–V fractal model and turning band simulation for Au and lithological units which showed that the main mineralized zones occur in the black gouges and Jasperoids.

Early market efficiency testing among hydrogen players

We study the stock price efficiency of companies with exposure to the hydrogen economy. As hydrogen, a pillar of the energy transition required for the global society to achieve the Sustainable Development Goals for 2030, does not trade as a commodity, we use the Solactive Hydrogen Index NTR as a proxy. Efficiency is assessed through a fractal methodology, with data from November 2018 to June 2021. Additionally, we run a time-varying approach that improves the robustness of the efficiency estimates. We find random price behavior consistent with the weak version of the market efficiency hypothesis, with only slight departures from efficiency in some companies with higher hydrogen exposure. There is also evidence of time-varying behavior of randomness during the acute pandemic period. The study validates the Solactive Hydrogen Index as an adequate proxy for the hydrogen economy.

The aplication of the fractal analysis in oncopathology

Fractal analysis is an objective approach that in oncopathology is one of the important fields of application. In this study is present fractal methodologies at histological level that have been successfully applied to characterize pathological features and able to perform differential diagnosis and prognosis in oncopathology. The basic principles and prospects of fractal geometry in pathology are promising. In particular, fractal analysis is emerging as a powerful tool to perform differential diagnosis and prognosis of the patients in cancer and other malignancies as well to improve the effectiveness and safety of patient care. All fractal objects have Fractal Dimension FDs, commonly calculated with box counting. Morphometry, the measure of shapes of the structures, can be added to every imaging technique in order to obtain objective indexes. In this field, fractal analysis has been applied to histopathology, cytopathology, and electron microscopy with great success. Performing fractal analysis of tissue samples, it’s possible to make differential diagnosis between the early stage of tumors and flogosis, or among the different types of Basal Cell Carcinoma, and different grade of Invasive Bladder Carcinoma, well as to investigate the subtle alterations of the nuclear patterns in human breast tumors, or to evaluate other tumors.

Diagnosis of the Cane Sugar Crystallization Process by Multifractal Analysis of Temperature Time Series

AbstractThe monitoring and diagnosis of crystallization processes are difficult due to the interaction of nucleation and crystal growth phenomena. In recent years, image and time series analysis using fractal methodologies showed potential as an alternative for monitoring crystal growth, although the available results are scarce. In this work, the multifractal detrended fluctuation analysis (MF‐DFA) was applied to temperature time series obtained from a laboratory‐scale cane sugar crystallization operated at different operating conditions. MF‐DFA reflects that the crystallization process exhibits multifractal properties associated with the dynamic behavior of the underlying phenomena. Thus, multifractal analysis can identify how operational changes influence the crystal growth and formed crystal mass.

Fractal geometry applied to the analysis of cervix biopsy.

the measurement of the spaces of occupation of irregular objects in the context of fractal geometry has had some applications at a cellular morphometric level, where characterizations of normality and disease have been established. The objective of the present study is to apply a fractal methodology to characterize images from cervical colposcopy. a mathematical and geometrical characterization of 67 cell samples was performed by measuring cellular fractal characteristics through the Box-Counting method, being nine normal, eight low-intraepithelial lesions, 16 high-intraepithelial lesions, eight carcinomas in situ, 20 squamous cell carcinomas and six endocervical carcinomas. the values of fractal dimension of the nuclear and cytoplasmic borders with respect to the totality varied between 0.719 to 1128 and 0.81 to 1024 while the occupation spaces in the 2 pixels grid were between 293 to 1606 and 64 to 693 respectively and in the 4 pixels grid oscillated between 153 to 894 and 36 to 361, respectively. Exocervical cells values had sensitivities between 78.3% to 100% in order to differentiate them from different types of cervical lesions. according to the results obtained, the mathematical values found are suggestive of being able to differentiate between normality and some colposcopy-guided cervical biopsy lesions.

БИОСИСТЕМНАЯ САМООРГАНИЗАЦИЯ И ФРАКТАЛЬНАЯ СТРУКТУРА ЧАСТОТНО-ТАКСОНОМИЧЕСКИХ ПРОФИЛЕЙ МИКРОБИОТЫ КИШЕЧНИКА БРОЙЛЕРОВ ПОД ВЛИЯНИЕМ КОРМОВЫХ ПРОБИОТИКОВ

The article provides theoretical and empirical data about the probiotic effect on the microbiota bioconsolidation in the broiler intestines of the Smena cross. The results of such studies can be used to improve the quality and volume of meat products in large-scale production. The probiotics have been used to improve feed digestibility and accelerate bird development. To stimulate the transformation of plant substrates in the bird intestines, probiotics were used instead of antibiotics. The probiotics contained the bacteria Lactobacillus plantarum and Lactobacillus fermentum. The study goal is to develop a methodology for fractal analysis of the frequency-taxonomic profiles of operational taxonomic units (OTUs) of the microbiota into the bird intestines. Using the fractal methodology, the index of microbiom bioconsolidation of the bird intestines was calculated, which characterizes the biosystem self-organization of microflora and the efficiency of biochemical transformations of plant substrates in the bird intestines. In the experiment, the microflora was studied in one control and two experimental groups of birds. The OTUs profiles were obtained by the molecular genetic NGS method (Next Generation Sequencing). The key concept of fractal analysis of OTUs profiles was the concept of the elementary OTU fractal. The elementary OTU fractal is three OTUs, the frequencies of which form a geometric numerical sequence (for example, {0.5; 0.25; 0.125}). The OTU profiles may contain several elementary OTU fractals combined into one larger OTU megafractal. We assume that if the number of OTUs combined into the OTU megafractal increases, then biochemical transformations of plant substrate are carried out more efficiently and on a large scale, and the bird macroorganism receives more nutrients. Therefore, we define the bioconsolidation index of the broiler microbiome as the ratio of the number of OTUs in the OTU megafractal to the total number of OTUs in the OTU profiles. The fractal portraits of OTU profiles were used to identify elementary OTU fractals. The elementary OTU fractals were identified by the linear arrangement of three OTU images on fractal portrait. The fractal analysis confirmed that the bacterial probiotics increase the microbiom bioconsolidation in the bird intestines. The microbiome bioconsolidation index in the experimental groups of birds (0.82...0.86) was higher than this index in the control group of birds (0.55). According to the results of fractal analysis, probiotic No. 1 (with Lactobacillus plantarum) is not recommended for use, and probiotic No. 2 (with Lactobacillus fermentum) is recommended for use.

FRACTAL ANALYSIS OF FREQUENCY-TAXONOMIC PROFILE OF BROILER’S GUT MICROBIOTA FOR STUDYING THE INFLU-ENCE OF PROBIOTICS ON BIRD DEVELOPMENT

The article provides theoretical and empirical data about the probiotic effect on the microbiota bioconsolidation in the broiler intestines of the Smena cross. The results of such studies can be used to improve the quality and volume of meat products in large-scale production. The probiotics have been used to improve feed digestibility and accelerate bird development. To stimulate the transformation of plant substrates in the bird intestines, probiotics were used instead of antibiotics. The probiotics contained the bacteria Lactobacillus plantarum and Lactobacillus fermentum. The study goal is to develop a methodology for fractal analysis of the frequency-taxonomic profiles of operational taxonomic units (OTUs) of the microbiota into the bird intestines. Using the fractal methodology, the index of microbiom bioconsolidation of the bird intestines was calculated, which characterizes the biosystem self-organization of microflora and the efficiency of biochemical transformations of plant substrates in the bird intestines. In the experiment, the microflora was studied in one control and two experimental groups of birds. The OTUs profiles were obtained by the molecular genetic NGS method (Next Generation Sequencing). The key concept of fractal analysis of OTUs profiles was the concept of the elementary OTU fractal. The elementary OTU fractal is three OTUs, the frequencies of which form a geometric numerical sequence (for example, {0.5; 0.25; 0.125}). The OTU profiles may contain several elementary OTU fractals combined into one larger OTU megafractal. We assume that if the number of OTUs combined into the OTU megafractal increases, then biochemical transformations of plant substrate are carried out more efficiently and on a large scale, and the bird macroorganism receives more nutrients. Therefore, we define the bioconsolidation index of the broiler microbiome as the ratio of the number of OTUs in the OTU megafractal to the total number of OTUs in the OTU profiles. The fractal portraits of OTU profiles were used to identify elementary OTU fractals. The elementary OTU fractals were identified by the linear arrangement of three OTU images on fractal portrait. The fractal analysis confirmed that the bacterial probiotics increase the microbiom bioconsolidation in the bird intestines. The microbiome bioconsolidation index in the experimental groups of birds (0.82...0.86) was higher than this index in the control group of birds (0.55). According to the results of fractal analysis, probiotic No. 1 (with Lactobacillus plantarum) is not recommended for use, and probiotic No. 2 (with Lactobacillus fermentum) is recommended for use.

Approximation by shape preserving fractal functions with variable scalings

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions $$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$ , where $$\varvec{\alpha }$$ is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the $$\varvec{\alpha }$$ -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the $$\varvec{\alpha }$$ -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the $$\varvec{\alpha }$$ -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Muntz theorem in $${\mathcal {C}}[0,1]$$ .

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y′′(x)+Q(x)y′(x)+P(x)y(x)=R(x) with the Dirichlet’s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y′′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.

Overtourism and online short-term rental platforms in Italian cities

Although Italian cities have undergone several waves of touristification, concerns about overtourism have only recently become widespread. In the article, we suggest that the diffusion of short-term rental platforms is not merely a concomitant factor, but is crucial to understanding the how and where of contemporary overtourism. To this end we apply a fractal methodology to identify, map and compare those parts of the city that are most affected, and measure the pressure short-term rentals have on city centres as places of residence. By allowing the conversion of residential apartments into tourist accommodation, we argue, short term rentals contribute to the displacement of residents more directly than a generic process of gentrification or touristification. Second, platforms such as Airbnb not only contribute to increasing the accommodation capacity of urban areas, but radically change the morphology of the tourist city. The growing concerns about overtourism are not due to the rising number of tourists per se, but to their increasing penetration into the residential city. We suggest, therefore, that to conceive of overtourism merely as overcrowding is not only inadequate but counterproductive. Even though the depopulation of city centres is difficult to reverse, the coronavirus emergency is an opportunity to plan a different city where tourism coexists with other urban uses and functions.

A Novel Method for Characterizing the Aggregation of Wax Crystals and An Improvement in Wax Deposition Modeling

Abstract The aggregation behavior and the subsequent deposition behavior of wax crystals own undesirable effects on the production and transportation of waxy crude oil. The understanding and prediction of these behaviors are essential to ensure economic and uninterrupted flow of waxy crude oil when the oil temperature decreases below the wax appearance temperature (WAT). In this paper, a novel method of fractal dimensional analysis was introduced to elucidate the aggregation behavior of wax crystals in different shear flow fields. The fractal methodology for characterizing wax crystal aggregation was then developed, and a blanket algorithm was introduced to compute the fractal dimension of the aggregated wax crystals. Considering the flow characteristics of waxy crude oil in a pipeline can be correlated with the shearing stress work, a modified wax deposition model focusing on shearing energy analysis was established. The results indicate that a quantitative interpretation of the wax crystal aggregation behavior can be realized using the fractal methodology. The aggregation behavior of the wax crystals is closely related to the temperature and shearing experienced by the waxy crude oil. The aggregation behavior will be intensified with decreasing temperature and shearing effect, and a wider fractal dimension distribution appears at lower temperatures when the same shear rate range is used. Furthermore, the improved model provides a method for discussing the effects of the operating conditions on wax deposition. The average relative deviation between the improved model prediction results and experimental results from the literature is 3.01–5.32%.

More General Fractal Functions on the Sphere

In this article, a family of continuous functions on the unit sphere $$S\subseteq \mathbb {R}^{3}$$ generalizing the spherical harmonics, is considered. Using fractal methodology, the fractal version of this family of continuous functions on the sphere S is constructed. To do this, an iterated function system (IFS) and a linear bounded operator that maps classical functions to its fractal analogues is defined. Some approximation properties of fractal functions on the sphere are investigated. Restricting the scale vector involved in the IFS, a fractal Hilbert basis is established for the functions on the sphere.

Fractal description of fouling deposits in boiling heat transfer modelling

A novel methodology is developed for predicting the thermal impact of fouling in Steam Generators (SG). The originality of this methodology is to resort to fractal and statistical theories to depict the porous structure of the deposits. The proposed Statistical Fractal methodology (SF) accounts for the heat transfer driven by the liquid-vapor phase change inside the deposits. It simulates the complex intricate networks of sinuous open pores of different scales, with liquid inflows (capillaries) and vapor outflows (steam-chimneys). The multi-layered representation of fouling deposits allows to mimic aging mechanisms such as densification which occur during SG operation.The SF predictions are consistent with experimental data. The deposit thickness and the profile of porosity are found to be the most influential fouling properties on the heat exchange. The methodology is capable to simulate the experimentally observed heat transfer enhancement for thin and porous deposit as well as the heat exchange decline for thick and dense deposit.

Geometrical Evaluation of Cervical Cells. Fractal and Euclidean Diagnostic Methodology of Clinical Application

Background: The concomitant use of fractal and Euclidian measurements has led to the development of new methodologies of cell evaluation, including a diagnosis of cervical cells that set up differences between normality and various degrees of lesion, to carcinoma. Aim: To confirm the diagnostic capacity of the methodology based on fractal and Euclidian geometry for the mathematical diagnosis through a blind study of normal cells and with different types of lesion, as atypia of undetermined significance (ASCUS), low grade squamous intra-epithelial lesion (LGSIL) and high grade squamous intra-epithelial lesion (HGSIL). Methods: 100 cells of Papanicolaou tests were analyzed and divided into 4 groups according to conventional parameters: 25 normal, 25 ASCUS, 25 LGSIL and 25 HGSIL. By means of the Box-counting Fractal Space, we calculated the fractal dimension and occupying spaces of the border and surface in pixels of the cell nucleus and cytoplasm. The diagnostic parameters of the previously developed methodology were applied and compared with the conventional diagnosis, setting up sensibility, specificity, negative likelihood ratio and Kappa coefficient. Results: The values of the occupation of the border and surface of the cell nucleus and cytoplasm were consistent with the values found by the diagnostic methodology previously found. The subtraction of the nucleus and cytoplasm frontiers presented values between: 189 and 482 for normality; 159 and 432 for ASCUS; 126 to 401 for LGSIL and 39 to 122 for HGSIL. A sensitivity and specificity of 100%, and a Kappa coefficient of 1 were obtained. Conclusions: The capacity of the methodology to diagnose quantitatively the different stages in the evolution of the cervical cells observed in Papanicolaou tests was confirmed, from normality to HGSIL.

Numerical Determinationof Critical Condensate Saturation in Gas Condensate Reservoirs

Critical condensate saturation, Scc, is a key parameter for the evaluation of well deliverability in gas condensate reservoirs. We propose a new method to determine Scc by performing three-phase flow simulations with three-dimensional (3D) pore network model. First, we establish a network model with random fractal methodology. Second, based on the condensation model in the literature of Li and Firoozabadi, we develop a modified condensation model to describe the condensation phenomenon of gas with connate water in the porous medium. The numerical model is verified by experimental measurements in the literature. Then, we investigate the influence of different factors on the critical condensate saturation, including micro pore structure (pore radius and fractal dimension), condensate gas/oil interfacial tension (IFT), and flow rate at different irreducible water saturation, Swi. The simulation results show that Scc decreases with increasing of average pore radius, but increases with increasing of fractal dimension. In the case of the same gas/oil interfacial tension, the higher the connate water saturation, the higher the critical condensate saturation. There is a critical gas/oil interfacial tension, below the critical value, the critical condensate saturation increases drastically with increasing of interfacial tension while it keeps almost unchanged when the interfacial tension is above the critical value. The critical condensate saturation decreases with increasing in the gas flow rate. High capillary number results in low critical condensate saturation. Reasonable increase in producing pressure drop can effectively improve the flow capacity of condensate oil.

An Analysis of the Multiscale Structure of Surfaces with Various Finishes

ABSTRACTEven with the various surface finishing techniques, all surfaces are rough with different structures and geometric characteristics over multiple scales. In this work, a profilometer is utilized to measure the profiles of different rough surfaces, and the profiles are characterized using a variety of statistical, spectral, and fractal methodologies. Three different methods are implemented in calculating the fractal dimension, D, and these three methods are then compared. The relationship between the fractal dimension and the fractal scaling constant, G, is investigated as well. The measured rough profiles are also compared with Weierstrass-Mandelbrot (W-M) function–generated rough profiles using the characterization results of the measured profiles. After comparing a series of statistical and fractal parameters, which are calculated based on the surface profile data of the original and regenerated profiles, it can be found that the W-M function does not always appear to be very suitable for representing measured rough profiles. Another important conclusion is that many measured profiles are not always consistent with the quality of self-affinity that many of the popular fractal models assume. Therefore, a discrepancy exists between idealized fractal equations and real profiles. However, these findings are limited to the measurement resolution and specific surfaces considered in this work.

Fractal methodology analysis on microstructure of bionics bone through three-dimensional biofabrication

Fractal methodology analysis on microstructure of bionics bone through three-dimensional biofabrication

Multifractal behavior of Electricity Bid Price in Indian Energy Market

Abstract The bid price of power generation in a country is of extreme importance due to its obvious impact on the industry although it is difficult to explore correlation among cost of a product in any industry with fluctuations of biding price of power. This is due to the fact that the fluctuations are nonlinear and extremely complex. Recent years have witnessed the study of this type of complex systems with powerful tool involving fractal methodology. This paper reports for the first time in Indian scenario a study on the fluctuation pattern of biding price of power in five different bid areas of India with a robust and rigorous technique based on chaos and fractal behavior. The exhaustive study reveals very interesting patterns of the fluctuation process, namely in all domains, the fluctuations are multifractal in time and degree of multifractality is quite different in different zones. The data is completely new and has serious implications on product cost of industry and ultimately on stock price indices. The paper includes possible future applications including predictability of next day price.

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a \(\mathcal{C}^1\)-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global \(\mathcal{C}^2\)-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1) (2007), pp. 41–53].