Application Of Fractal Geometry Research Articles

PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

Fractal nature of human gastrointestinal system: Exploring a new era.

The morphological complexity of cells and tissues, whether normal or pathological, is characterized by two primary attributes: Irregularity and self-similarity across different scales. When an object exhibits self-similarity, its shape remains unchanged as the scales of measurement vary because any part of it resembles the whole. On the other hand, the size and geometric characteristics of an irregular object vary as the resolution increases, revealing more intricate details. Despite numerous attempts, a reliable and accurate method for quantifying the morphological features of gastrointestinal organs, tissues, cells, their dynamic changes, and pathological disorders has not yet been established. However, fractal geometry, which studies shapes and patterns that exhibit self-similarity, holds promise in providing a quantitative measure of the irregularly shaped morphologies and their underlying self-similar temporal behaviors. In this context, we explore the fractal nature of the gastrointestinal system and the potential of fractal geometry as a robust descriptor of its complex forms and functions. Additionally, we examine the practical applications of fractal geometry in clinical gastroenterology and hepatology practice.

Application of Fractal Geometry in Studies of the Bulgarian Financial Market

The present study presents and summarizes in a condensed manner the basic postulates and concepts arising from the principles and ideas of fractal geometry, applied in the field of behavior and dynamics of financial markets. A comparative analysis has been made between the classic financial and statistical indicators for measuring and summarizing a certain state of the financial system, on one hand, and the main quantitative parameters resulting from the fractal analysis, on the other. In the last part, an analysis and test for fractality of the Bulgarian financial market has been made represented by the main stock index – SOFIX.

Application of various fractal models in characterizing the morphology of pore structures of hydrating cement pastes

The pore structures of cementitious materials are very crucial to the development of the various engineering properties of these materials. Based on the importance of the pore structure, several studies have been conducted on the application of fractal geometry, which can be used as an interpretation tool for the characteristics of the pore structure as well as in various engineering phenomena. Accordingly, various fractal models have been developed from the mercury intrusion porosimetry (MIP) data. However, studies on applying the fractal model considering the characteristics of the material to be analyzed are relatively scarce. In this study, we conducted an in-depth investigation of the characteristics of the pore structures of cementitious materials considering both the volumetric and geometric characterization parameters. Among the three concepts of fractal dimensions of pore-size distribution, the solid mass fractal concept exhibited the highest correlation with the degree of hydration. The fractal dimensions of pore-size distribution and pore tortuousness tended to increase as the complexity of the entire microstructures increased, mainly due to the increase in the solid phase fraction. Furthermore, the results of the correlation analysis (between the volumetric and geometric parameters) and regression analysis (between newly derived parameter and initial sorptivity) showed that fractal dimensions can be utilized as pore structure characterization parameters like the existing volumetric parameters, and these parameters serve as complementary tools for interpreting the transport properties of cementitious materials.

Fractal Feature Based Image Resolution Enhancement Using Wavelet–Fractal Transformation in Gradient Domain

The fractal geometries are applied extensively in many applications like pattern recognition, texture analysis and segmentation. The application of fractal geometry requires estimation of the fractal features. The fractal dimension and fractal length are found effective to analyze and measure image features, such as texture, resolution, etc. This paper proposes a new wavelet–fractal technique for image resolution enhancement. The resolution of the wavelet sub-bands are improved using scaling operator and then it is transformed into texture vector. The proposed method then computes fractal dimension and fractal length in gradient domain which is used for resolution enhancement. It is observed that by using scaling operator in the gradient domain, the fractal dimension and fractal length becomes scale invariant. The major advantage of the proposed wavelet–fractal technique is that the feature vector retains fractal dimension and fractal length both. Thus, the resolution enhanced image restores the texture information well. The texture information has also been observed in terms of fractal dimension with varied sample size. We present qualitative and quantitative analysis of the proposed method with existing state of art methods.

Optimizing academic supervision through CoMent strategy in STEAM-ED approach and application of fractal geometry

This study examines the implementation of academic supervision using the CoMent Strategy of school supervisors in an effort to maximize the STEAM-ED approach and the application of fractal geometry to Batik and Textile Creative Craft instructors at State Vocational School 5 Bengkulu City. The method used was qualitative descriptive method. This research was conducted over two academic years in the academic year 2019-2020 to 2020-2021. In this study, the subjects studied were teachers in the competence of Batik & Textile Creative Crafts, totaling 8 people. The data collection in this study used testimonials in the form of learning logs and documentation. The data analysis was an interactive model of data analysis technique. The results showed that there is a high level of satisfaction in carrying out the learning process using the STEAM-ED approach and the application of fractal geometry, as well as the performance of the trained teachers. The adoption of academic supervision through the CoMent technique proved to be highly effective in maximizing the STEAM-ED approach and the use of fractal geometry for Batik and Textile Creative Craft instructors in their learning. The advantage of the Coment strategy is that it can streamline the implementation of academic supervision of fostered teachers in an effort to provide active, effective, meaningful and fun learning for each student so that they are able to think critically, be creative, innovate, have higher order thinking skills and be creative in solving problems through coaching and mentoring assistance activities.

Обзор геометрических способов повышения удельной прочности конструкций: топологическая оптимизация и фрактальные структуры

The paper is an overview of geometric methods for increasing the specific strength of parts and constructions. In the making of engineering knowledge it had been deduced by theoretical and empirical ways a number of rules for specifying the shape of bodies withstanding the loads applied to them. So, in construction, they prefer to use an I-beam instead of a beam with rectangular section, since the first one is able to withstand a large load with a similar mass and the same material, that is, with a certain loading scheme, the I-beam has a greater specific strength due to the features of its geometry. The basic principles of creating such a geometry have been considered in this paper.
 With the development of the theory of strength of materials, as well as methods for automatization of design and strength calculations, it became possible to create the shape of parts optimized for specific loads. Computer generation of such a form is called topological optimization. A lot of modern research has been devoted to the development and improvement of algorithms for topological optimization (TO). In this paper have been described some of TO algorithms, and has been presented a general analysis of optimized forms, demonstrating their similarity to fractals.
 Despite the rapid development of topological optimization, it has constraints, some of which can be circumvented by using fractal structures. In this study a new classification of fractals is presented, and the possibility of their use to create parts and constructions of increased specific strength is considered. Examples for successful application of fractal geometry in practice are also presented.
 The combination of principles for designing strong parts and fractal shaping algorithms will make it possible in the future to develop the structure of strong elements applicable to increase the constructions’ specific strength. Further research will be devoted to this.

A fractal-based approach for cracking characterization and whole process prediction exploration of PP fiber reinforced ECC containing sustainable ingredients

This study focuses on an eco-friendly engineered cementitious composite (SSPP-ECC) using high-content fly ash (FA) as a cementitious cleaner production alternative, waste superfine sand as silica sand replacement and economical polypropylene (PP) fiber as expensive polyvinyl alcohol (PVA) fiber replacement. The lack of quantitative sustainability assessment and prediction of high ductility SSPP-ECC has impeded the engineering application of SSPP-ECC. This study presents a fractal-based cracking whole process preliminary analysis for SSPP-ECC structural life cycle assessment in order to promote a wider application of SSPP-ECC as large-deformation structural materials, as well as develop the application of fractal geometry in ECC by novel evaluation modes. A new and SSPP-ECC with up to 60% cement replacement by fly ash was designed, followed by a four-point flexural test accompanying with surface crack monitoring. The typical deformation hardening SSPP-ECC has been proved to have engineering potential in improving the structural safety due to good deformation following and crack controlling capacities. It is recommended to combine the fractal dimension and average crack width to reflect the overall fracture energy in the four-point flexural failure process of SSPP-ECC, which would have reliable and effective evaluation effects. Based on the correlation analysis results, the quantitative assessment and prediction models of cracking whole process could be established for eco-friendly SSPP-ECC structural life cycle assessment, which can be benefit to the structural life cycle assessment of SSPP-ECC engineering application based on non-destructive parameters of load and deflection. The eco-friendly SSPP-ECC, as a potential cleaner production, not only is a technically reasonable strategy, but also would lead to significant sustainability benefits with a growing level of environmental awareness.

FRACTALS AND THEIR APPLICATIONS: A REVIEW

In this paper, I have discussed about fractals. The two key properties of fractals have been stated. A brief history about fractals is also mentioned. I have discussed about Mandelbrot fractal and have plotted it using python. A computer-generated fern is compared to a real fern to show how much fractals resemble the real-world objects. Various applications of fractal geometry have also been included.

Application of Fractal Geometry in Gas Sensor: A Review

Increasing the specific superficial area of gas sensing layer is a direction to improve the performance of sensors. Considering the special features, high specific superficial area of fractal geometry and the fractal properties of thin film in the deposition and growth progress, many studies have explored the relationship between fractal geometry and gas sensing characteristics of the sensors. In this paper, the application and development of fractal geometry in gas sensors are reviewed, the superiority of fractal geometry in the application of gas sensors, as well as the formation mechanism and influencing factors of fractal geometry are indicated. The advantages of fractal analysis in the characterization of film morphology and properties are also discussed. This review demonstrates that fractal geometry can be successfully used in the design of macro and micro structure of gas sensors to improve the performance of sensors. Films with appropriate fractal structure and best gas sensing performance can be obtained by controlling the deposition and parameters. 3D printing and MEMS technologies provide an implementation scheme to prepare sensors with fractal structure, such as electrodes and heater based on Hilbert curve and substrates based on Menger sponge.

Городская среда и теория детерминированного хаоса

One of the main directions in the application of fractal geometry is to create complex objects using simple algorithms using computer technology. The use of fractal geometry methods and the theory of deterministic chaos to describe real systems, in particular, in the development of the architectural appearance of the city, will allow solving problems of research and forecasting the development of the urban environment.

FRACTAL GEOMETRY IN SCHOOLS AND UNIVERSITIES

The article raises the question of the actual study of fractal geometry in the framework of schooling. The objective is to show and tell the range of applications of fractal geometry. The result is the use of argumentation of the subject in terms of training for students in the school.

Fractal geometry of internal thread surfaces manufactured by cutting tap and rolling tap

This study presents briefly the application of fractal geometry in internal threads surface characterization to reveal defect fine structures of the internal thread geometry caused by the manufacturing process. The surface texture characteristics of ten internal thread samples M12 × 1.75 of low carbon steel Ç1020, manufactured by cutting tap and rolling tap were analyzed and discussed through comparative analyses of the experimental data. The surface characteristics of samples were analyzed using scanning electron microscopy (SEM) and were found that surface texture characteristics (Str – surface anisotropy ratio and D – fractal dimension) are different for the thread specimens.

TWO-DIMENSIONAL FRACTAL MODEL FOR ULTIMATE CRUSHING STATE OF COARSE AGGREGATES

The research on the ultimate crushing state of coarse aggregates is beneficial to analyze and predict the evolutionary process of crushing. The Growing Path method uses the two-dimensional fractal geometry structure to simulate the size variation of particle size fraction during the particle breakage of coarse aggregates and it serves to investigate the ultimate fractal dimension corresponding to the ultimate crushing state of coarse aggregates. This method manifests the self-growing characteristics of particle size distribution in the process of particle crushing. This study found that the two-dimensional image of ultimate fractal model was precisely similar to that of the Sierpinski gasket of fractal theory when the ultimate crushing state was reached. The results from the model analysis show that the theoretically ultimate fractal dimension is about 2.585, which is consistent with the existing results calculated from the three-dimensional ultimate fragmentation model of cataclastic rock located in the fault zones. The relationship between two fractal models was analyzed. Furthermore, the application of fractal geometry presented in this study will also serve as a reference for the analysis of the other chaos phenomena observed in geotechnical engineering.

Application of fractal geometry to describe reservoirs with complex structures

Abstract This study provides basic instantaneous source functions for different source-reservoir geometries in infinite fractal reservoirs (FRs) and infinite slab FRs. The concept of fractal geometry (FG) has already been adopted to analyse and model well-reservoir systems with complex structures. All these complexities have been modelled through a partial differential equation called fractal diffusivity equation (FDE). In this paper, by use of the source/Green’s function technique, we analytically solve the FDE for different boundary conditions, i.e., different source-reservoir geometries. The wellbore pressure response for horizontal wells in an infinite slab FR (synthetic example) is derived and analysed to illustrate the applications of the provided analytical source solutions and how to use these solutions.

Entropy in multifractal non equilibrium structures of dielectric breakdown

The phenomenon of dielectric breakdown is a clear example of the growth process of a non-equilibrium structure. The introduction of non equilibrium models and the application of fractal geometry has led a good understanding of the morphology of the generated structures known as electrical trees. In this paper we compare the different statistical entropies that can be applied to the fractal simulation of the dielectric breakdown phenomenon such as the entropies of Renyi, Tsallis, Boltzmann–Gibbs and Shannon. We also analyze the behavior of the fractal dimension in terms of the four entropies studied for the simulation of the growth of electrical trees with the dielectric breakdown model.

Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications.

The quantification of brain dynamics is essential to its understanding. However, the brain is a system operating on multiple time scales, and characterization of dynamics across time scales remains a challenge. One framework to study such dynamics is that of fractal geometry; and currently there exist several methods for the study of brain dynamics using fractal geometry. We aim to highlight some of the practical challenges of applying fractal geometry to brain dynamics—and as a putative feature for machine learning applications, and propose solutions to enable its wider use in neuroscience. Using intracranially recorded electroencephalogram (EEG) and simulated data, we compared monofractal and multifractal methods with regards to their sensitivity to signal variance. We found that both monofractal and multifractal properties correlate closely with signal variance, thus not being a useful feature of the signal. However, after applying an epoch-wise standardization procedure to the signal, we found that multifractal measures could offer non-redundant information compared to signal variance, power (in different frequency bands) and other established EEG signal measures. We also compared different multifractal estimation methods to each other in terms of reliability, and we found that the Chhabra-Jensen algorithm performed best. Finally, we investigated the impact of sampling frequency and epoch length on the estimation of multifractal properties. Using epileptic seizures as an example event in the EEG, we show that there may be an optimal time scale (i.e., combination of sampling frequency and epoch length) for detecting temporal changes in multifractal properties around seizures. The practical issues we highlighted and our suggested solutions should help in developing robust methods for the application of fractal geometry in EEG signals. Our analyses and observations also aid the theoretical understanding of the multifractal properties of the brain and might provide grounds for new discoveries in the study of brain signals. These could be crucial for the understanding of neurological function and for the developments of new treatments.

A real attractor non admitting a connected feasible open set

A self-similar set is described as the unique (nonempty) compact subset remaining invariant under the action of a finite collection of similitudes on a complete metric space. Among this kind of fractals, those satisfying the so-called Moran's open set condition are especially appropriate to deal with applications of Fractal Geometry since their Hausdorff dimensions can be easily computed. However, such a separation property depends on an external open set whose properties are not fully known. In this paper, we construct a self-similar set in the real line lying under the open set condition which does not admit a connected feasible open set. This answers an open question posed by Zhou and Li in 2009.

Estudos urbanos e geometria fractal

Muitas metodologias convencionais utilizadas no estudo da forma urbana não têm considerado o grau de irregularidade dos perímetros urbanos nem toda a complexidade morfológica, pois tratam a forma das cidades dentro do âmbito da geometria euclidiana. Este artigo explora a aplicação da geometria fractal ao estudo de áreas urbanas, centrando-se num conjunto de aspetos da forma urbana relacionados com a fragmentação dos tecidos urbanos, a presença de vazios urbanos e, por fim, a ‘auto-semelhança’ em diferentes escalas de observação. Esta análise desenvolve-se a partir do trabalho de investigação desenvolvido por Frankhauser desde os anos 90. A análise fractal é aplicada a um conjunto de cinco cidades no Vale do Rio Pardo, no Rio Grande do Sul, Brasil. Ao longo dos últimos anos, este conjunto de cidades tem crescido de forma desordenada, sendo que o desenvolvimento acelerado e a ampliação do perímetro urbano originaram uma ocupação rarefeita com grandes impactos sobre os tecidos urbanos.

Separation of rock types in reservoir rock using concentration-area multifractal model

Separation and identification of rock type is one of the most important and key stages in reservoir simulation. There are various methods for separation and diagnosis of rock type varying from simple statistical tests (based on statistical parameters of distribution function) to complex fractal (based on special structure of data). Fractal geometry has proved its practical aspects with increasing expansion in various sciences and techniques. One of the main applications of fractal geometry in estimation of threshold and separation of anomaly from background is based on the difference between their fractal dimensions. In this paper, after modeling porosity parameter in a reservoir rock in south of Iran by Kriging method, the area of rock communities was separated using concentration-area fractal model. The results of fractal analysis indicate five rock types based on porosity parameter in this reservoir rock and threshold limit of 10.80% for separation of porosity related to background from anomaly.