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1

Investigation of alterations in droughts and floods patterns induced by climate change

Rahmani Farhang, Fattahi Mohammad Hadi

Acta Geophysica

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2024

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Vol. 72, no. 1

405–-418

EN

Every year, droughts and floods cause significant damage to the economy and water resources of the UK. Numerous studies have been explored droughts and floods from various points of view, however few have pointed the variations in the patterns induced by climate change. The precipitation data of Central England in the UK was gathered from 1931 to 2020. The analysis was performed by application of fractal dimension, noise variance, Lyapunov exponent, approximate entropy, extreme climate indices, and Standard Precipitation Index. The cross-correlation results indicated the study area warming owing to CO2 emissions on a global and local scale, implicating the climate change in the study area. Moreover, the mean maximum and minimum temperatures were affected by CO2 emissions on global and local scales, respectively. The nonlinear dynamic analysis indicated that the duration and intensity of the dry and wet spells were increased due to climate change. In other words, the droughts’ intensity and duration were augmented. However, the number of annual droughts and wetness’s have remained unaffected by climate change. The results signified a weakening in the flash floods possibility and an increment in the flash floods severity owing to climate change. Moreover, climate change brought about an intensification in the rivers’ inundation (fluvial floods) probability. The findings of the present study contribute to the understanding of the mechanism of climate change impacts on droughts and floods (flash, pluvial, and fluvial) patterns and furnished references for nonlinear dynamic studies of droughts and floods patterns.

2

The effect of energies on the impact breakage characteristic of magnetite ores

Si Liang, Cao Yijun, Li Guosheng

Physicochemical Problems of Mineral Processing

|

2023

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Vol. 59, iss. 1

art. no. 159098

EN

The energy applied during breakage is the key to enhancing the magnetite liberation degree and improving quality. The relationship between energy and liberation properties remains unclear due to various complicated factors affecting mineral liberation. Therefore, this work aims to study the effect of energy on the breakage characteristics of magnetite ores; the impact breakage test was conducted on magnetite particle groups at different energies using a drop weight impact tester; the statistical analysis was performed based on the fractal theory to research the particle size distribution; the fracture morphology and liberation properties of these ores were analyzed using scanning electron microscope and mineral liberation analyzer. Results show that the particle size distribution of magnetite after breakage conforms to the fractal law. The larger the energy, the greater the fractal dimension for this distribution, showing a linear relation between them, which implies that the fractal dimension can evaluate the breakage degree. The fracture morphology of magnetite ores indicates that as the energy increases, the intergranular fracture evolves into transgranular fracture, proving the influence of energy on fracture modes. It is found that the magnetite liberation degree first increases and then decreases with the rising of energy, indicating that the magnetite liberation can be improved at an appropriate amount of energy. The above conclusions provide a theoretical reference for optimizing energy and improving broken product quality.

3

Fractal dimensions in the Gromov–Hausdorff space

Ishiki Yoshito

Bulletin of the Polish Academy of Sciences. Mathematics

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2023

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Vol. 71, no 2

147--168

EN

We first show that for any four non-negative real numbers, there exists a Cantor ultrametric space whose Hausdorff dimension, packing dimension, upper box dimension, and Assouad dimension are equal to the given four numbers, respectively. Next, using a direct sum of metric spaces, we construct topological embeddings of an arbitrary compact metrizable space into the two subsets of the Gromov–Hausdorff space: the set of all compact metric spaces possessing prescribed topological dimension and the aforementioned four dimensions, and the set of all compact ultrametric spaces.

4

Deformation and failure laws and acoustic emission characteristics of low‑strength molybdenum ore

Zhao Kang, Yang Jian, Song Yufeng, Wang Qing, Zhao Kangqi, Ji Yongbo

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 2

art. no. e74, 2023

EN

Uniaxial compression acoustic emission (AE) tests were conducted on low-strength molybdenum ore (LSMO) to investigate its deformation and failure laws and AE characteristics. The stress-strain curve and AE parameter data of LSMOs were obtained by uniaxial compression AE test, and the relationships of stress, AE parameter, amplitude fractal dimension, and AE b value with loading time were analyzed accordingly to obtain the general law of their deformation and failure and a series of AE characteristics. The research shows that under the action of uniaxial stress, the failure mode of LSMOs mainly shows brittle failure, and the failure form mainly shows monoclinic shear failure. The stress-strain curve shows obvious plastic-elastic deformation, the plastic deformation time is long, and the division of each stage of deformation failure is not obvious. The simultaneous occurrence of large surges in ringing count and energy to higher orders of magnitude can be used as precursor information for failure destabilization of LSMOs. The evolution process of AE parameters of LSMO corresponds well with its deformation and failure process, and the variation pattern of ringing counts and energy shows a high consistency. With increasing stress, the amplitude correlation dimension and b value are mainly in the form of "falling-rising-falling-fluctuating". The results of the study can provide some theoretical basis for the assessment of the stability of the mine surrounding rock and the determination of a reasonable and effective reinforcement plan.

5

Effects exerted by average particle size and non uniformity on bed surface fractal properties

Pan Yunwen, Xia Junqiang, Cai Tun, Yang Kejun

Acta Geophysica

|

2023

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Vol. 71, no. 1

517--529

EN

Sediment bed surfaces exist widely in natural rivers, and many aspects in river dynamics are closely relevant to bed surface roughness, such as flow structure, river resistance and sediment transport. As two important parameters for quantifying bed surface roughness, how average particle size and non-uniformity affect bed surface structure is unknown. Therefore, nine groups of sediment samples with different average particle sizes or different non-uniformities were firstly prepared by screening dry natural sediments. Then, the prepared sediment samples were used to manually pave nine groups of bed surfaces, and the high-precise bed surface digital elevations were obtained by a handheld 3D laser scanner. Finally, the effects exerted by the average particle size and non-uniformity on the bed surface fractal properties were discussed. The results showed that there is only a scale-free range in a profile or a two-dimensional specific direction of a bed surface with normal-distributed particle gradation. The averaged scale-free upper limit in the two-dimensional specific directions and that related to many profiles are less affected by the non-uniformity, but more affected by the average particle size. For the bed surfaces with the same non-uniformity, when the average particle size is smaller than 15 mm, the larger the average particle size is, the smaller the fractal dimension is, but the larger the scale coefficient is; when the average particle size is larger than 15 mm, the larger the average particle size is, the larger the fractal dimension and the scale coefficient are, while for the bed surfaces with the same average particle size, the non-uniformity has no significant effects on the fractal dimension and the scale coefficient. The averaged scale coefficient in the two-dimensional specific directions of an isotropic bed surface and that related to many profiles are approximately equal, but the averaged fractal dimension in the two-dimensional specific directions is obviously larger than that plus 1 related to many profiles.

6

Investing in dividend vs. non-dividend stocks – efficiency assessment using fractal measures

Buła Rafał, Jabłoński Bartłomiej

Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska

|

2022

|

z. 160

109--128

EN

Purpose: The purpose of the paper is to reveal potential differences in risk and profitability of investment in dividend and non-dividend stocks. Design/methodology/approach: The scientific aim of the paper is achieved by conducting a scrupulous literature analysis. Moreover, the authors use methods of comparative analysis to investigate the characteristics of dividend and non-dividend stocks and reveal similarities and differences. Study of fractal features of chosen stocks and comparisons between abovementioned groups of shares are conducted using the ANOVA methods. Findings: The results of the empirical analyses conducted in this paper prove that dividends paid by US dividend companies grow at significantly lower rate than dividends distributed by Polish dividend stocks. Additionally, rates of return on Polish dividend stocks are more heavily influenced by dividend pay-outs than rates of return on US ones. Taking into account riskiness of investments there are no differences in risk level between dividend and non-dividend stocks in USA and Poland, independently whether the risk measure exploited is stock volatility or its fractal dimension. Research limitations/implications: The research was based on limited number of companies analyzed. As a result, there could be present a bias introduced by the deterministic method of choosing a sample of stocks. It is recommended to enlarge the analyzed set in future research. Practical implications: Knowledge about similarities and differences among dividend and non-dividend companies is highly relevant to investors as well as corporate managements. As a consequence, better financial decisions could be taken leading to increased final wealth. Social implications: Among the social implications of the paper the possible change in investors’ attitude towards dividend and non-dividend companies seems most important. This could influence companies’ boards to adjust their payout policies to satisfy the investors. Finally, the improvement in investor’s needs fulfillment can be achieved. Originality/value: The novelty of the paper is the comparison of dividend and non-dividend stocks taking into account classical and modern risk measures. Moreover, it compares the efficiency of investing in dividend and non-dividend stocks during period 2015-2021, i.e. partially catching the effect of SARS-CoV-2 pandemic filling a gap in our knowledge.

7

Influence of measuring system noise on the fractal dimension of the chaotic signal attractor

Bolek Karol, Urbański Michał K.

Przegląd Elektrotechniczny

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2022

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R. 98, nr 12

68--71

EN

The present paper investigates the influence of stochastic noise on the estimation of the fractal dimension of the chaotic signal attractor for additive and multiplicative noises in the frequency and time domain. A simplified analogue measuring system noise model was proposed as the amplitude and phase noise in the Fourier domain, which was the equivalent of multiplicative and additive noise in the time domain. A numerical experiment was performed, which introduced noise of various intensities into the chaotic signal from the Chua system. It has been shown that in the logarithmic diagram of the correlation integral, additional scaling regions appear, the range of which increases with increasing noise intensity, causing dimension estimation errors. It has also been shown that without a thorough analysis of the correlation integral, deterministic noise can be easily confused with stochastic noise in the frequency domain.

PL

W pracy przedstawiono wpływ szumu stochastycznego na estymacje wymiaru fraktalnego atraktora sygnału chaotycznego dla szumów addytywnych i multiplikatywnych w dziedzinie częstości i czasu. Zaproponowano uproszczony model szumów analogowego toru pomiarowego jako szum amplitudowy i fazowy w dziedzinie Fouriera, stanowiący odpowiednik szumu multiplikatywnego oraz addytywnego w dziedzinie czasu. Wykonano eksperyment numeryczny, za którego pomocą wprowadzano do sygnału chaotycznego pochodzącego z numerycznego układu Chua szum o różnych natężeniach. Pokazano, że na wykresie logarytmicznym całki korelacyjnej pojawia się dodatkowe obszary skalowania, których zakres rośnie wraz ze wzrostem natężenia szumów, powodując błędy estymacji wymiaru. Pokazano również, że bez wnikliwej analizy całki korelacyjnej szum deterministyczny łatwo pomylić z szumem stochastycznym w dziedzinie częstości.

8

Study on the effect of coal microscopic pore structure to its spontaneous combustion tendency

Dong Xianwei, Li Gaojing, Dong Xuanmeng, Wang Fusheng

Journal of Sustainable Mining

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2022

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Vol. 21, iss. 2

120--127

EN

Coal is a porous medium. Due to the large number of pores in coal and the pore size on its surface, usually ranging from millimeter to nanometer, it is difficult to measure and analyze the microscopic pore structure of coal. In order to investigate the effect of the microscopic pore structure of coal on its spontaneous combustion tendency, coal samples from different coal mines of the Kailuan Group were selected as the research objects, and the data of the microscopic pore distribution of three different coal samples were measured by using mercury injection apparatus. The regression analysis of microscopic pore data of coal samples obtained in the mercury injection experiment shows that the correlation coefficients of the regression curves are all greater than 0.94 and the fitting degree is good, indicating that there is a good correlation between the pressure, mercury intake and pore size of the coal samples, indicating that the fractal dimension of pore distribution is very effective. The fractal dimension is generally between 2 and 3, indicating that the microscopic pores of coal samples have good fractal characteristics and meet the fractal theory to describe the distribution characteristics of microscopic pores in porous media. Through the simulation system of natural combustion of coal, the simulation experiment of temperature rise oxidation of different coal samples (gas coal, fat coal, and coke coal) was carried out, and the curve of the concentration of gas products CO and CO2 in the process of temperature rise and oxidation of coal samples was drawn in the experiment. The experimental results show the relationship between the distribution structure of coal pores and its spontaneous combustion tendency, and the coal with a good distribution dimension has a stronger combustion tendency.

9

Explainable COVID-19 detection using fractal dimension and vision transformer with Grad-CAM on cough sounds

Sobahi Nebras, Atila Orhan, Deniz Erkan, Sengur Abdulkadir, Acharya U. Rajendra

Biocybernetics and Biomedical Engineering

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2022

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Vol. 42, no. 3

1066--1080

EN

The polymerase chain reaction (PCR) test is not only time-intensive but also a contact method that puts healthcare personnel at risk. Thus, contactless and fast detection tests are more valuable. Cough sound is an important indicator of COVID-19, and in this paper, a novel explainable scheme is developed for cough sound-based COVID-19 detection. In the presented work, the cough sound is initially segmented into overlapping parts, and each segment is labeled as the input audio, which may contain other sounds. The deep Yet Another Mobile Network (YAMNet) model is considered in this work. After labeling, the segments labeled as cough are cropped and concatenated to reconstruct the pure cough sounds. Then, four fractal dimensions (FD) calculation methods are employed to acquire the FD coefficients on the cough sound with an overlapped sliding window that forms a matrix. The constructed matrixes are then used to form the fractal dimension images. Finally, a pretrained vision transformer (ViT) model is used to classify the constructed images into COVID-19, healthy and symptomatic classes. In this work, we demonstrate the performance of the ViT on cough sound-based COVID-19, and a visual explainability of the inner workings of the ViT model is shown. Three publically available cough sound datasets, namely COUGHVID, VIRUFY, and COSWARA, are used in this study. We have obtained 98.45%, 98.15%, and 97.59% accuracy for COUGHVID, VIRUFY, and COSWARA datasets, respectively. Our developed model obtained the highest performance compared to the state-of-the-art methods and is ready to be tested in real-world applications.

10

Fractal Dimension as Robust Estimate of Low Carbon Steels Hardness

Zając Krzysztof, Płatek Karolina, Wachel Paweł, Łatka Leszek

Advances in Science and Technology. Research Journal

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2022

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Vol. 16, no 5

335--344

EN

Application of computational methods in engineering and science constantly increases, which is also visible in sector of material science, often with promising results. In following paper, authors would like to propose fractal dimension, a mathematical method of quantifying self-similarity and complexity of spatial patterns, as robust method of hardness estimation of low carbon steels. A dataset of microstructure images and corresponding Vickers hardness measurements of S235JR steel under different delivery conditions was created. Then, three different computational methods for evaluation of materials hardness based on microstructure image were tested. In this paper those methods are called: (i) Otsu-based index, (ii) fractal dimension index and (iii) vision transformer index. The results were compared with method used in literature for similar problems. Comparison showed that fractal dimension performs better than other evaluated methods, in terms of median absolute error, which value was equal to 4.12 HV1, which is significantly lower than results achieved by Otsu-based index and vision transformer index, which were 4.49 HV1 and 5.07 HV1 respectively. Those results can be attributed to the relative robustness of fractal dimension index, when compared to other methods. Robust estimation is preferable, due to the high amount of noise in the dataset, which is a consequence of the nature of used material.

11

Identification of volcanic lithofacies using box-counting method calculating fractal dimension of logging data

Mou Dan, Wang Zhuwen, Tan Xili, Shi Shaoyun

Acta Geophysica

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2022

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Vol. 70, no 6

2647--2658

EN

Identification of volcanic lithofacies is critical for reservoir exploration and a major difficulty in China's Liaohe oil field. In this paper, we present a fractal analysis method for identifying volcanic facies by estimating the fractal dimension of logging data. The fractal properties of lithofacies logging curves are explored, as well as the categorization scheme of volcanic lithofacies in the eastern sag of the Liaohe basin. Five logging curves impacted by volcanic lithofacies from four wells in the Liaohe basin's eastern depression were chosen. The Box-counting dimension is used to develop a logging lithofacies identification criterion. Furthermore, we calculated the fractal dimension of logging curves using box-counting dimension methods.

12

Możliwość zastosowania wymiaru fraktalnego do określania kształtu kryształów na przykładzie wybranych minerałów skałotwórczych intruzji Monczegorska (NE Fennoskandia)

Huber Miłosz

Przegląd Geologiczny

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2021

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Vol. 69, nr 7

434--442

EN

A petrographic description of olivine, ortho-, clinopyroxene, and plagioclase was obtained for rocks derived from a Paleoproterozoic layered intrusion of the Monchepluton. Based on these results, calculations were made of the fractal box dimension, which determines the degree of development of the boundaries of these minerals. After the calculations, they were illustrated on maps showing the correlation of this coefficient with the location of the sample in the field. These results were then correlated with petrographic observations. Based on this method, the analogies between the result of the fractal dimension and the petrographic description of rock-forming minerals were assessed. The data show that such an analogy to a limited extent may be a significant indicator for the interpretation of the results. The use of this method is not universal and strictly depends on the mineral association and the knowledge obtained by classic petrographic studies.

13

Applying chaos theory to risk management in a game of chance

Jóźwiak Ireneusz J., Mariański Aleksander, Switana Jan, Szczepanik Michał

Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska

|

2020

|

z. 144

195--211

EN

Purpose: The purpose of the paper is to illustrate the usage of techniques known from chaos theory to analyze the risk Design/methodology/approach: In this case the objects of application are winnings graphs of different poker players. Two types of players are presented; winning players (those with positive expected value) and breaking even players (expected value close to zero). Findings: Charts were analyzed with a fractal dimension calculated with the box method. Originality/value: Relation between fractal dimension and Hurst exponent is shown. Relation between risk in sense of chaos theory and players’ long-term winning is also described. Further applications of chaos theory to analyze the risk in games of chance are also proposed.

14

Fractal Geometry in Designing and Operating Water Networks

Iwanek Małgorzata, Kowalski Dariusz, Kowalska Beata, Suchorab Paweł

Journal of Ecological Engineering

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2020

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Vol. 21, nr 6

229--236

EN

Fractals are self-similar sets that cannot be easily described by classical geometry. Fractal sets have been implemented in almost all areas of human activity since they were introduced to science by Mandelbrot in 1982. For the last 10 years, the interest in fractal geometry has increased by the issues connected with water distribution networks (WDNs). The aim of this paper was to review the application of fractal geometry in designing and operating WDNs. Treating a WDN as a fractal pattern enables its description and classification, simplifies the assessment of a network reliability, helps to solve the problems of routing and dimensioning WDN, as well as enables to select the places to locate measurement points in a network to control water quality, pressure in pipes and water flow rate. Moreover, the application of tree-shaped fractal patterns to reflect WDNs helps to solve the problems of their optimization. Fractal geometry can be also applied to investigate the results of WDNs failures connected with leakage of water to the ground. Using fractal dimension of a pattern created by points reflecting places of water outflow on the soil surface after a prospective pipe breakage enables to determine the zone near a pipe, where the outflow of water on the soil surface is possible. It is an important approach for the security of humans and existing infrastructure. Usage of fractal geometry in description, optimisation and operation analysis of WDNs still continues, which confirms the efficiency of fractal geometry as a research tool. On the other hand, it can be supposed that fractal geometry possibilities have still not been fully used.

15

Diagnosis of Attention Deficit Hyperactivity Disorder with combined time and frequency features

Altınkaynak Miray, Dolu Nazan, Güven Aysegül, Pektas Ferhat, Özmen Sevgi, Demirci Esra, Izzetoglu Meltem

Biocybernetics and Biomedical Engineering

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2020

|

Vol. 40, no. 3

927--937

EN

The aim of this study was to build a machine learning model to discriminate Attention Deficit Hyperactivity Disorder (ADHD) patients and healthy controls using information from both time and frequency analysis of Event Related Potentials (ERP) obtained from Electroencephalography (EEG) signals while participants performed an auditory oddball task. The study included 23 unmedicated ADHD patients and 23 healthy controls. The EEG signal was analyzed in time domain by nonlinear brain dynamics and morphological features, and in time-frequency domain with wavelet coefficients. Selected features were applied to various machine learning techniques including; Multilayer Perceptron, Naïve Bayes, Support Vector Machines, k-nearest neighbor, Adaptive Boosting, Logistic Regression and Random Forest to classify ADHD patients and healthy controls. Longer P300 latencies and smaller P300 amplitudes were observed in ADHD patients relative to controls. In fractal dimension calculation relative to the control group, the ADHD group demonstrated reduced complexity. In addition, certain wavelet coefficients provided significantly different values in both groups. Combining these extracted features, our results indicated that Multilayer Perceptron method provided the best classification with an accuracy rate of 91.3% and a high level of reliability of concurrence (Kappa = 0.82). The results showed that combining time and frequency domain features can be a useful and discriminative for diagnostic purposes in ADHD. The study presents a supporting diagnostic tool that uses EEG signal processing and machine learning algorithms. The findings would be helpful in the objective diagnosis of ADHD.

16

On a formula finding fractal dimension

Kamilov Kh., Zaitov A., Tulaganov A.

Archives of Materials Science and Engineering

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2020

|

Vol. 104, nr 1

19--22

EN

Purpose: The article is devoted to the determination of the fractal dimension of cellular concrete, in particular foam concrete, and the further clarification of the relationship between fractal dimension and porosity and average density of cellular concrete. Design/methodology/approach: In the theoretical description of disordered systems, the fundamentals of fractal theory are actively used, which allow obtaining statistical indicators of chaotic natural and artificially disordered systems, which include cellular concrete. The parameters of the pore structure are difficult to quantify by conventional methods because of the complexity and irregularity of the pore structure due to their random distribution. Findings: Formulas for calculating the fractal dimension and average density of highly porous material are calculated and proposed. The formula for calculating the average density takes into account the density of the material between the pore walls. Research limitations/implications: The calculation of the fractal dimension is one of the main factors affecting the practical application of the theory of fractals, a natural problem arises on a theoretical basis to justify these calculations. Practical implications: The formulas proposed in this work for calculating the fractal dimension and density of a highly porous structure improve research on methods for producing substances with a controlled fractal structure, which will help create materials with unusual mechanical properties, density, and porosity. Originality/value: The formula for calculating the fractal dimension obtained in the work improves the well-known Hausdorff-Bezikovich formula. On the other hand, it makes it possible to obtain a highly porous structure with a given density of the material under study.

17

FRACTALS – return to origin

Such Piotr

Nafta-Gaz

|

2019

|

R. 75, nr 2

89--93

EN

Fractals have become fashionable. Therefore, in recent years there have been many articles in which the authors support something that they call a fractal account, including, for example, the sum of fractal dimensions. This paper is a recapitulation of what a fractal account is in the Earth sciences, what are its uses and boundaries. The definition of fractals is: it has a non trite structure in all scales, it is very hard to describe fractal structure in the Euclidean geometry, it is self-similar (directly or statistical), its Hausdorf dimension is greater than its topological dimension, it is described by recurrent formula, its dimension is not an integral number. In the face of such a wide and imprecise formula, various fields of science have introduced their definitions of fractal. It only has to meet most of the conditions included in the definition. In the analysis of geological objects in Earth sciences and in oil and gas industry, fractals are defined by the recurrence formula with its range of applicability, fractal dimension share a part of the space occupied by the fractal object, so the highest value of fractal dimension is equal to 3. Fundamental work in which the name of fractals for self- similar objects were introduced was The Fractal Geometry of Nature by Mandelbrot (1977). In the Earth sciences, statistical fractals (pseudofractals) are used. The straight line in the log-log plot is the indicator of fractal structure. In other words, the fractal structures are associated with the power patterns obtained during the analysis of geological objects. Generally, in the analysis of geological objects the Menger sponge and box methods of fractal dimension calculations are used. Fractals provide a unique opportunity to characterize complicated objects with the use of a single number, nevertheless, in order for the obtained results to be applicable and comparable with the results of other analyzes, both the model of the analyzed object and the method of calculation of the fractal dimension should be given, as well as the scope of applicability of this dimension.

PL

Fraktale stały się modne. W związku z tym w ostatnich latach obserwuje się wiele artykułów, w których autorzy wspierają się czymś, co nazywają rachunkiem fraktalowym, łącznie np. z sumowaniem wymiarów fraktalowych. Niniejszy artykuł stanowi rekapitulację tego, czym jest rachunek fraktalowy w naukach o Ziemi, jakie są jego zastosowania i granice. Co jest, a co nie jest fraktalem. Zasadniczo na definicję wymiaru fraktalnego składają się następujące warunki: nie jest prostą i taką samą strukturą we wszystkich skalach, bardzo trudno opisać go w geometrii euklidesowej, jest strukturą samopodobną (wprost lub statystycznie), jego wymiar Hausdorffa jest większy od jego wymiaru topologicznego, jest opisany formułą rekurencyjną oraz jego wymiar nie jest liczbą całkowitą. Wobec tak szerokiej i nieprecyzyjnej formuły różne dziedziny nauki wprowadziły swoje definicje fraktala. Ma on jedynie spełniać większość warunków zapisanych w definicji. W naukach o Ziemi fraktale definiowane są przez wzory rekurencyjne z analizą obszaru stosowalności. W analizie obiektów geologicznych wymiar fraktalny wskazuje na część przestrzeni zajmowaną przez dany obiekt, w związku z czym jego wartość nie może przekraczać 3. Mandelbrot w swojej fundamentalnej pracy The Fractal Geometry of Nature (1977) wprowadził nazwę fraktala jako obiektu samopodobnego. W naukach o Ziemi stosowane są fraktale statystyczne, zwane również pseudofraktalami. Wskaźnikiem struktury fraktalnej jest linia prosta na wykresie typu log-log. Inaczej mówiąc, fraktalne struktury są związane ze wzorami potęgowymi uzyskanymi podczas analizy obiektów geologicznych. Zasadniczo w analizie obiektów geologicznych stosujemy model gąbki Mengera oraz wymiar pudełkowy dla obiektów dwuwymiarowych. Fraktale dają unikalną możliwość scharakteryzowania skomplikowanych struktur za pomocą jednej liczby. Tym niemniej, aby otrzymane wyniki były stosowalne i porównywalne z wynikami innych analiz, należy zarówno podać model analizowanego obiektu i sposób wyliczenia wymiaru fraktalnego, jak też określić zakres stosowalności tego wymiaru. Tylko wtedy wymiar fraktalny będzie miał sens fizyczny.

18

Methods of comparison of surface texture based on fractal dimension and Hotelling’s T2 test

Gogolewski Damian

Journal of Machine Engineering

|

2019

|

Vol. 19, No. 4

82--90

EN

Comparative analysis of the surface texture of machine parts can be successfully carried out using statistical tests. The paper presents a methodology of method used to compare the surface texture by applying Hotelling’s T2 test as well as a method used to evaluate surface topography by applying fractal dimension. The tests were carried out on samples produced with the use of face milling process for four types of materials. The following types of steel were used: 40HM, C45, NC6 and WCL. For each type of material, four areas were machined with the same machining parameters. Based on these results a decision was made whether the surfaces, despite the same machining conditions, were significantly different from each other. Furthermore, the analysis indicated that the fractal dimension enabled to characterise signal irregularities in quantitative and qualitative way.

19

Fractal analysis of tropical karst relief : South China Karst case study

Andreychouk Viacheslav, Blachowicz Tomasz, Dłużewski Maciej

Geological Quarterly

|

2019

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Vol. 63, No. 4

729--740

EN

Areas of tropical karst create the most spectacular earth landscapes from a geomorphological perspective. These areas are characterized by a variety of specific forms resulting from the long-term karst-erosion dismemberment of terrains in favourable humid tropical conditions. Tropical karst areas are extremely diverse from a geomorphological point of view both in terms of local conditions of development and developmental stages. Among the many types of karst relief, the following two basic types can be recognized: fenglin (tower karst) and fengcong (cone karst). The other types can be treated as a mixture of these two basic types. To find potential quantitative rates characterizing the two main types, as well as the mixed types, we calculated fractal dimensions and cover factors of 17 areas located within the two well-known regions of South China Karst - Guilin and Huanjiang. The calculations show that the numerical characteristics obtained, especially the cover factor parameter, can be useful as complementary tools in the recognition and typology of tropical karst relief and landscapes.

20

Fractal Analysis of Noise Signals of Sampo and John Deere Combine Harvesters in Operational Conditions

Boroujeni Farzad Mahdiyeh, Maleki Ali

Archives of Acoustics

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2019

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Vol. 44, No. 1

89--98

EN

Combine harvesters are the source a large amount of noise in agriculture. Depending on different working conditions, the noise of such machines can have a significant effect on the hearing conditio of drivers. Therefore, it is highly important to study the noise signals caused by these machines and find solutions for reducing the produced noise. The present study was carried out is order to obtain the fractal dimension (FD) of the noise signals in Sampo and John Deere combine harvesters in different operational conditions. The noise signals of the combines were recorded with different engine speeds, operational conditions, gear states, and locations. Four methods of direct estimations of the FD of the waveform in the time domain with three sliding windows with lengths of 50, 100, and 200 ms were employed. The results showed that the Fractal Dimension/Sound Pressure Level [dB] in John Deere and Sampo combines varied in the ranges of 1.44/96.8 to 1.57/103.2 and 1.23/92.3 to 1.51/104.1, respectively. The cabins of Sampo and John Deere combines reduced and enhanced these amounts, respectively. With an increase in the length of the sliding windows and the engine speed of the combines, the amount of FD increased. In other words, the size of the suitable window depends on the extraction method of calculating the FD. The results also showed that the type of the gearbox used in the combines could have a tangible effect on the trend of changes in the FD.