Self-similar Properties Research Articles

The Grounding Problem of Equal Respect

In this paper, I explore three theories of value to illuminate how nontheistic and theistic accounts may differ in grounding human dignity: neo- Aristotelian ethical naturalism, Kantian constructivism, and a theistic account of good simpliciter. The theistic account of good simpliciter that I offer adapts Robert Adams’s notion of the transcendent Good as the Excellent. In this account, I explain how Adams’ thesis that goodness is a property consisting in a sort of resemblance to God may be understood in a new way, using ideas drawn from contemporary mathematics and quantum mechanics. On my account, we must value human beings neither because such valuing would be beneficial or necessary for human flourishing nor because it is a logical outcome of anything we care to value. Rather it is because we recognize the property of self-similarity in all of us, which may be understood as a resemblance to God as good simpliciter.

УМОВИ ВИКОРИСТАННЯ КОНФОРМНИХ ВІДОБРАЖЕНЬ ДЛЯ ВИВЧЕННЯ ФРАКТАЛЬНОГО СТИСНЕННЯ ІНФОРМАЦІЇ

To describe the properties of self-similarity and invariance observed in various physical situations, the theory of fractals and multifractals is actively used in modern science. We consider a conformal mapping of the first kind, which is given by the entire linear function w = az + b, where w, z are complex variables, a, b are complex constants, a ≠ 0 . With its help, affine transformations are performed to construct the Sierpinski triangle and the Koch curve by specifying the functions w3(z) and finding the coefficients Re аі, Im аі, Re bi, Im bi, і = 1,2,3 or і = 1,2,3,4. These coefficients encode the image of the object and it can be unambiguously reconstructed by them. It is determined that the method in which self-similar regions in an object are detected and conformal mapping coefficients are found for them works under the condition that each such mapping is compressive. Only then will Banach’s fixed point theorem ensure image collection during decompression. This method of compressing graphic information is called fractal.

Multifractal parameterization of a periodic surface microrelief formed at the face milling. 2. Distribution of elements volume of surface relief

This study represents the first application of multifractal analysis to characterize spatial microforms’ volumes resulting from face milling on flat surfaces, emphasizing its novelty. Photographs of samples of C35 steel and AA22024 aluminum alloy, subjected to various cutting conditions during milling, were utilized to apply the multifractal approach for describing the machined surface microrelief. Multifractal spectra and their key parameters were computed for the samples’ surfaces generated at various feed rates and cutting depths. It is demonstrated that the characteristic functions of the performed multifractal analysis align with their canonical forms. The results offer compelling evidence that machining results in the formation of a system exhibiting self-similarity and fractal symmetry properties on the specimen’s surface. The practical significance of this research lies in the establishment of quantitative dependences between the parameters of the multifractal spectrum of surface microforms generated during the face milling under varying cutting depths and feed rates. This facilitates the identification of technological and physical factors responsible for observed variations in surface fractal properties. The formation of the multifractal spectra from surface element volumes is discussed in connection with the underlying physical phenomena associated with cutting conditions.

A new modern scheme for solving fractal–fractional differential equations based on deep feedforward neural network with multiple hidden layer

The recent development of knowledge in fractional calculus introduced an advanced superior operator known as fractal–fractional derivative (FFD). This operator combines memory effect and self-similar property that give better accurate representation of real world problems through fractal–fractional differential equations (FFDEs). However, the existence of fresh and modern numerical technique on solving FFDEs is still scarce. Originally invented for machine learning technique, artificial neural network (ANN) is cutting-edge scheme that have shown promising result in solving the fractional differential equations (FDEs). Thus, this research aims to extend the application of ANN to solve FFDE with power law kernel in Caputo sense (FFDEPC) by develop a vectorized algorithm based on deep feedforward neural network that consists of multiple hidden layer (DFNN-2H) with Adam optimization. During the initial stage of the method development, the basic framework on solving FFDEs is designed. To minimize the burden of computational time, the vectorized algorithm is constructed at the next stage for method to be performed efficiently. Several example have been tested to demonstrate the applicability and efficiency of the method. Comparison on exact solutions and some previous published method indicate that the proposed scheme have give good accuracy and low computational time.

A study on the evolutionary model of legal disputes based on fractal theory

Abstract In this paper, a legal dispute mediation game model is constructed based on fractal theory, characterized by self-similarity properties in disputes. Secondly, the legal dispute evolution game model is constructed, the dynamic differential equation of frequency or frequency is used, and an adaptive degree dispute evolution stabilization strategy (ESS) is proposed according to the evolution principle. Finally, the simulation and analysis of the dispute evolution game model are carried out with the medical-patient dispute as an example. The results show that when D<0 takes the values of −4 and -2, both sides of the doctor-patient dispute will rapidly move toward the norm and form a benign evolutionary trend. When D>0 takes the values of 2 and 4, the two sides of the doctor-patient dispute will rapidly gather toward the challenge and form a vicious evolutionary trend. The study of this paper helps to understand the evolutionary process of legal disputes better and provides a reference for resolving disputes.

Fractal Dimension Analyses to Detect Alzheimer’s and Parkinson’s Diseases Using Their Thin Brain Tissue Samples via Transmission Optical Microscopy

Biological tissues in nature are fractal due to their self-similarity and porosity properties. These properties change with the progress of some diseases, including brain tissue in leading neurological disorders such as Alzheimer’s disease (AD) and Parkinson’s disease (PD). Thus, there is an unmet clinical need to develop a tool for accurate and early diagnosis of AD and PD conditions. Although the whole brain tissues in AD and PD have been extensively studied, their local structural alterations at the nano-to-submicron levels have not been explored. In this paper, we measure the local structural alterations in different brain regions of AD and PD patients by measuring their change in fractal dimensions via optical microscopy. Our results show an increase in the fractal dimension value of ~5–10% in the affected regions of the brain tissues relative to their respective controls. For AD cases, the structural alteration is attributed to the aberrant deposition of amyloid beta protein and neurofibrillary tangles in the brain, and for PD, the gradual loss of dopaminergic neurons and abnormal accumulation of α-synuclein in the brain. The work will enhance the further understanding of alterations in the brain structures in AD and PD and its detection.

Fractal-Based Hybrid Cryptosystem: Enhancing Image Encryption with RSA, Homomorphic Encryption, and Chaotic Maps.

Protecting digital data, especially digital images, from unauthorized access and malicious activities is crucial in today's digital era. This paper introduces a novel approach to enhance image encryption by combining the strengths of the RSA algorithm, homomorphic encryption, and chaotic maps, specifically the sine and logistic map, alongside the self-similar properties of the fractal Sierpinski triangle. The proposed fractal-based hybrid cryptosystem leverages Paillier encryption for maintaining security and privacy, while the chaotic maps introduce randomness, periodicity, and robustness. Simultaneously, the fractal Sierpinski triangle generates intricate shapes at different scales, resulting in a substantially expanded key space and heightened sensitivity through randomly selected initial points. The secret keys derived from the chaotic maps and Sierpinski triangle are employed for image encryption. The proposed scheme offers simplicity, efficiency, and robust security, effectively safeguarding against statistical, differential, and brute-force attacks. Through comprehensive experimental evaluations, we demonstrate the superior performance of the proposed scheme compared to existing methods in terms of both security and efficiency. This paper makes a significant contribution to the field of digital image encryption, paving the way for further exploration and optimization in the future.

Three-Dimensional Reconstruction and Seepage Simulation of Vermiculite Materials Based on CT Technology

The purpose of this paper is to investigate the 3D reconstruction and seepage simulation of vermiculite-based sealing materials in order to clarify the relationship between material structure and fluid leakage in order to improve their performance and reliability and to provide a foundation for accurate leakage rate calculation under various conditions. CT scanning and 3D reconstruction techniques were used to examine the pore structure and fractal properties of vermiculite-based sealing materials. The seepage mechanism under compression was examined using a porous medium model. The findings demonstrate that the vermiculite-based sealing materials have self-similarity and scale-invariant fractal properties. The porous media model in compression exhibits a low leakage rate, opening up a new avenue for realizing microscopic leakage studies with a leakage rate of 1 × 109 in compression.

Revealing the fractal and self-similarity of realistic collective human mobility

Revealing the fractal scaling laws of collective human mobility is benefit to understand the complexity of collective human mobility behaviours and has broad application prospects, such as infectious disease prevention, because fractal networks are potentially more robust against targeted attacks than non-fractal networks. Although many statistical indicators of individual human mobility had been observed in power-law distributions, there is still no evidence so far to verify the fractal or self-similarity of collective human mobility behaviours. This study uncovered the fractal and self-similarity properties on realistic weighted human mobility networks at world-, nation-, and city-wide scales. After observing the necessary conditions of network fractal, i.e., the evident power-law probability distributions of weight-related metrics, the fractal dimensions of weighted human mobility networks were measured to be about 0.67∼0.88. Furthermore, experimental results demonstrated their distinct statistical self-similarity and approximately structural self-similarity. These exciting findings can enhance the fire-new understandings of collective human mobility behaviours, and provide novel solutions for many real-world applications.

Quantitative ECG based emotion state recognition using Detrended Fluctuation Analysis

Abstract Wearable emotion recogniton system is essential in identifying mental health disorders by early detection and continuous monitoring of human emotions to provide proper treatment care. Electrocardiogram (ECG) signals can be used for emotion recognition for its noninvasiveness and easy usability. In this study, Detrended Fluctuation Analysis (DFA) and Extreme Gradient Boost (XG Boost) classifier is used to classify the scary and boring emotion from the ECG signals. For this, ECG signal corresponding to these emotions are obtained from public database. The preprocessing is performed by adding the video IDs to the signal and annotating it. This preprocessed signal is subjected to DFA to understand the power-law correlations and similarity property. Further, from the power law correlations, features namely Hurst exponent and DFA intercept are extracted. These features are subjected to XG Boost classifier to differentiate the two emotions. Results shows that the log-log plot of power law correlation is linear in nature which indicates that ECG signals of both the emotions have long range correlations and self-similarity property. The extracted scaling exponent indicates variations between scary and boring with a mean and standard deviation of 0.81±0.13 and 0.68±0.07 respectively. Similarly, DFA intercept provides mean and standard deviation for scary and boring 0.15±0.06 and 0.17±0.07 respectively, showing less variability in the ECG signal. XG Boost classifier gives accuracy of 80.5% for classifying scary and boring emotion. Thus, the proposed approach can be used for wearable emotion recognition system to differentiate scary and boring emotion.

Fractional Calculus in Signal Processing: The Use of Fractional Calculus in Signal Processing Applications, Such as Image Denoising, Filtering, and Time Series Analysis

Fractional calculus has transformed signal processing by providing a more flexible representation of complex signals. This article explores its applications in image denoising, filtering, and time series analysis, highlighting its potential to revolutionize signal processing methodologies. Fractional calculus, dealing with derivatives and integrals of non-integer orders, captures long-memory and self-similar properties in real-world signals. Image denoising benefits from fractional derivatives and integrals, preserving essential details while removing noise. Filtering signals with fractional calculus allows for dynamic adaptability to changing signal characteristics. Time series analysis benefits from more accurate modelling, enhancing predictive capabilities. The future scope of fractional calculus in signal processing promises further advancements, making it a valuable tool for researchers and practitioners seeking to analyze complex signals across various domains.

Air–Sea Enthalpy and Momentum Exchange Coefficients from GPS Dropsonde Measurements in Hurricane Conditions

The intensity of tropical cyclones is highly dependent on air–sea enthalpy and momentum exchange. At extreme wind speeds, the values of the enthalpy, CK, and momentum, CD, exchange coefficients are characterized by high uncertainty. The present study aims to expand the previously used algorithm for CD retrieval to obtain the values of CK from wind speed measurements and the enthalpy profiles obtained from NOAA GPS dropsondes in hurricane conditions. This algorithm uses concepts from technical hydrodynamics, describing turbulent boundary layers on flat plates and pipes. According to this approach, the velocity (and enthalpy) defect profiles are self-similar in the entire boundary layer, including the layer of constant fluxes and the “wake” part, where the airflow adapts to the undisturbed flow region. By using the self-similarity property, the parameters of the constant flow layer (the roughness parameter, friction velocity, and the enthalpy and exchange coefficients CD and CK) could be obtained from measurements in the “wake” part for wind speeds from 20 m/s to 72 m/s. The estimates of the CK/CD ratio revealed values of 0.7 and 0.96 (depending on the self-similar approximation limits), and the results suggest that there are slight variations with the wind speed.

Income distribution in Thailand is scale-invariant.

This study examines whether income distribution in Thailand has a property of scale invariance or self-similarity across years. By using the data on income shares by quintile and by decile of Thailand from 1988 to 2021, the results from 306-pairwise Kolmogorov-Smirnov tests indicate that income distribution in Thailand is statistically scale-invariant or self-similar across years with p-values ranging between 0.988 and 1.000. Based on these empirical findings, this study would like to propose that, in order to change income distribution in Thailand whose pattern had been persisted for over three decades, the change itself cannot be gradual but has to be like a phase transition of substance in physics.

Probabilistic Machine Learning Methods for Fractional Brownian Motion Time Series Forecasting

This paper explores the capabilities of machine learning for the probabilistic forecasting of fractional Brownian motion (fBm). The focus is on predicting the probability of the value of an fBm time series exceeding a certain threshold after a specific number of time steps, given only the knowledge of its Hurst exponent. The study aims to determine if the self-similarity property is preserved in a forecasting time series and which machine learning algorithms are the most effective. Two types of forecasting methods are investigated: methods with a predefined distribution shape and those without. The results show that the self-similar properties of the fBm time series can be reliably reproduced in the continuations of the time series predicted by machine learning methods. The study also provides an experimental comparison of various probabilistic forecasting methods and their potential applications in the analysis and modeling of fractal time series.

Self-similarity in homogeneous stationary and evolution problems

We analyse self-similarity properties related to linear elliptic and evolutionary problems involving homogeneous operators in several spaces including measures. We employ these techniques to analyse in particular 2mth-order diffusion equations and the associated fractional problems.

A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties).

Morphology differences between fractional Brownian motion and the Weierstrass-Mandelbrot function and corresponding Hurst evaluation

Fractional Brownian motion (FBM) and the Weierstrass-Mandelbrot (W-M) function are two important methods for constructing self-affine objects. Accurately characterizing their features, such as the morphology and fractal geometry, is fundamental for their follow-up applications. However, due to the differences between self-similar and self-affine properties, the fractal dimension evaluation is difficult and sometimes unconvincing. In addition, the sampling length and diversity of calculation methods both lead to the non-uniqueness of the fractal dimension. This study compared the morphology differences between FBM and the W-M function and then analyzed the effects of each parameter on their morphology. Comparisons indicated that FBM has fewer control parameters than the W-M function. Finally, from basic fractal geometric properties, we derived the relationship between the measurement scale r and profile lengths L(r) for self-affine profiles and found that it is a complicated form rather than L(r)sim c r^{H-1}, where H is the Hurst exponent. Meanwhile, the equivalent vertical height v(r) has an intuitive and clear power law relationship to the measurement scale v(r)sim sigma r^H, which provides a method to estimate the Hurst exponent.

SEPARATION OF MAGNETIC ANOMALIES USING FRACTAL METHOD IN THE ESFORDI REGION FOR IRON EXPLORATION, CENTRAL EAST OF IRAN

Link for citation: Amin Karimi Kalvarzi, Aref Shirazi, Adel Shirazy, Amin Beiranvand Pour, Ardeshir Hezarkhani, Hamed Nazerian, Timkin T.V., . Voroshilov V.G. Separation of magnetic anomalies using fractal method in the Esfordi region for iron exploration, central east of Iran. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 3, рр. 197-209. In Rus.
 The relevance of research is determined by the possibilities of measuring the potential magnetic field, which has self-similar (fractal) properties, as well as a practical tool for prospecting and exploration of iron ores. In the Esfordi area, this method was used by us for the first time to identify, separate and interpret geophysical (magnetic) anomalies. The main aim of this thematic and practical study is the qualitative interpretation of geophysical data, the application of new methods in the prospecting and exploration of mineral deposits, modeling the geological environment and forecasting new promising areas. Object: Esfordi region, Yazd province, Iran. Methods. To obtain additional information about the subsurface, magnetometric data were used with their interpretation by the RTP (reduction to the north magnetic pole) method. For modeling purposes, an artificial sample was made, consisting of a sphere, a cube and a cuboid, and it was found that the fractal method can be used to separate anomalies for unipolar models (cube and cuboid). Results. The results of the study were applied to the Esfordi region, where it was found that at a survey scale of 1:100000, there is a direct relationship between the fractal method and the 3D model, which can be used to locate iron ore mineralization.

Shear-band cavitation determines the shape of the stress-strain curve of metallic glasses

Metallic glasses are known to have a remarkably robust yield strength, admitting Weibull moduli as high as for crystalline engineering alloys. However, their postyielding behavior is strongly varying, with large scatter in both flow stress levels and strains at failure. Using x-ray tomography, we reveal how a strain-dependent internal evolution of shear-band cavities underlies this unpredictable postyielding response. We demonstrate how macroscopic strain softening coincides with the first detection of internal shear-band cavitation. Cavity growth during plastic flow is found to follow a power law, which yields a fractal dimension and a roughness exponent in excellent agreement with self-similar surface properties obtained after fracture. These findings demonstrate how internal microcracking coexists with shear-band plasticity along the plastic part of a stress-strain curve, rationalizing the large variability of plastic flow behavior seen for metallic glasses.

On the Self-Similarity in an Annular Isolator under Rotating Feedback Pressure Perturbations

In this paper, the transient flow simulation in an annular isolator under rotating feedback pressure perturbations simplified from the rotating denotation wave (RDW) is performed. The instantaneous flow characteristics and the self-similarity of the isolator flow-field are investigated in detail. It is found that a helical moving shock wave (MSW) and a quasi-toroidal terminal shock wave (TSW) are induced in the isolator. Hence, the flow-fields on the meridian planes could be classified into three zones, i.e., the undisturbed zone, the terminal shock wave/moving shock wave/boundary layer interaction (TSW/MSW/BLI) zone and the moving shock wave/boundary layer interaction (MSW/BLI) zone. The TSW/MSW/BLI zone is characterized by the coupling of the TSW/BLI and the MSW/BLI due to their small axial distance, which intensifies the adverse pressure gradient on the meridian planes, thus rolling up large separation bubbles developing along the MSW driven by the circular pressure gradient. In the MSW/BLI zone, the shock induces the boundary layer to separate, forming a helical vortex located at the foot of the MSW. During the upstream propagation process, the pattern of the MSWs transforms from a moving normal shock wave to a moving oblique shock wave with decreased strength. Moreover, after the collision with the MSWs, P, Temp and S of the flow elevate with the prompt decrease of va, while vθ increases to a higher level. Despite the deflection effect of the MSWs on the streamlines, the flow direction of the air still maintains an almost axial position at the exit, except in the adjacent region of the MSW. Likewise, three types of zones can be determined in the flow pattern at the exit: the rotating detonation wave/boundary layer interaction (RDW/BLI) zone, the expansion zone, and the vortices discharge zone. Comparing the transient flow patterns at different moments in one cycle and between adjacent cycles, an interesting discovery is that the self-similarity property is observed in the flow-field of the annular isolator under rotating feedback pressure perturbations. The global flow structure of the isolator at different moments shows good agreement despite its rotation with the RDW, and the surface pressure profiles of the corresponding meridian planes all match perfectly. Such a characteristic indicates that the rotation angular velocity of the TSW and the MSW are equal and hold invariant, and the isolator flow could be regarded as a quasi-steady flow. On this basis, the theoretical model of the inclination angles of the MSW by the coordinate transformation and velocity decomposition is developed and validated. The relative errors of the inclination angles between the predicted and measured results are below 3%, which offers a rapid method to predict the shape of the MSW, along with a perspective to better understand the physical meaning of the shape of the MSW.