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A surface profilometry technique was used to characterize the surfaces of paper products. A stylus-contact type profilometer capable of simultaneously generating both surface roughness- and friction-profiles was used. As a stylus for the profilometer, a conical shape whose tip was rounded to have a 0.5 mm curvature radius was designed and successfully employed in both printing & writing (P&W) papers and hygiene papers such as bathroom tissues and kitchen towel. From the profiles, the mean absolute deviation (MAD) from the averages, i.e., R-MAD from the roughness average and F-MAD from the average coefficient of friction, were suggested as the new surface characterization parameters. To elucidate the surface roughness profiles by fractal dimension analysis, the variogram method was applied to get the fractal dimensions of the paper products. Generally, the value of the fractal dimension increased as the surface roughness increased. The surface profilometry technique with the fractal dimension analysis with the variogram method looks promising to gain additional insight on the surface characteristics of paper products. [ABSTRACT FROM AUTHOR]

The insulating paper of the transformer is affected by many factors during the operation, meanwhile, the surface texture of the paper is easy to change. To explore the relationship between the aging state and surface texture change of insulating paper, firstly, the thermal aging experiment of insulating paper is carried out, and the insulating paper samples with different aging times are obtained. After then, the images of the aged insulating paper samples are collected and pre-processed. The pre-processing effect is verified by constructing and calculating the gray surface of the sample. Secondly, the texture features of the insulating paper image are extracted by box dimension and multifractal spectrum. Based on that, the extreme learning machine (ELM) is taken as the classification tool with texture features and aging time as the input and output, to train the algorithm and construct the corresponding relationship between the texture feature and the aging time. After then, the insulating paper with unknown aging time is predicted with a trained ELM algorithm. The numerical test results show that the texture features extracted from the fractal dimension of the micro image can effectively characterize the aging state of insulating paper, the average accuracy can reach 91.6%. It proves that the fractal dimension theory can be utilized for assessing the aging state of insulating paper for onsite applications. [ABSTRACT FROM AUTHOR]

Although paper shredding is widely used to prevent confidential papers from being misused, still it cannot be considered a convenient process. The size of paper shreds became smaller and smaller, as new methods of shredded paper reassembly and reconstruction are evolving. This paper focuses on clustering, which is a possible phase in the assembly process. This work considers real strip-cut shreds, in addition to images shredded by a simulator in one direction to make strip-cut shreds of different sizes, from wide to narrow shreds, and images shredded in two directions, possibly reflecting cross-cut and micro-cut shreds. K-means is used to cluster shreds, the features tested are gray-level ranges, and the well-known gray-level co-occurrence matrix, invariant moments, segmentation-based fractal texture analysis algorithm, and color moments. The number of shreds grouped in the same cluster with originally adjacent neighbors is used to indicate clustering effectiveness, in addition to the overall accuracy of strip-cut shreds clustering. When the number of clusters is 5, and the k-means experiments run 100 times for 38 images, the overall accuracy of gray-level ranges in simulated strip-cut shreds is 84.87, 89.27, and 93.5 percent in the three different sizes tested, also in cross-cut and micro-cut shreds, gray-level ranges achieve a relatively high number of shreds with 3 and 4 originally adjacent neighbors found in the same cluster. [ABSTRACT FROM AUTHOR]

Surface textures of a large collection of photographic papers dating from 1896 to the present were measured using a laser scanning confocal microscope (LSCM) with four different objective lenses. Roughness characterization parameters were calculated from the texture measurements and were compared with gloss measurements. Characterization by the area-scale fractal dimension (Das) and the area-scale fractal complexity (Asfc) provided the strongest correlations between gloss reflectance and surface texture. The measurements with the 5× and 10× objectives, which contained many large-scale, spiky measurement artifacts that distorted the measurement, resulted in the strongest correlations (R2 > 0.8) compared to the 20× and 50× (R2 < 0.5). The presence of spiky artifacts in the measurements, which increases when the magnification of the objective lens is decreased, appears to amplify surface features in such a way to improve the correlations. SCANNING 37:204-217, 2015. © 2015 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Passive micromixers are important microdevices in the field of microfluidics. It is necessary to increase the mixing efficiency of the micromixer. In this paper, six micromixer models with co-lateral and different side Koch fractal obstacles are designed based on COMSOL5.2a. This paper studies the effects of Koch fractal obstacles on mixing performance and mixing mechanism. The results show that the Koch fractal obstacle enhances the chaotic convection of the fluid in the microchannel and improves the mixing efficiency. In addition, when Re = 0.1, when the Koch obstacle is located on the same side of the microchannel, the larger the number of Koch fractals, the higher the mixing efficiency, and the mixing efficiency reaches 98.2%. The Koch Obstacle Micromixer has better mixing performance on different sides than other micromixers. The research work in this paper provides a novel design and theoretical basis for the design of micromixers. [ABSTRACT FROM AUTHOR]

Abstract: Paper properties such as gloss, friction or printability strongly depend on the surface roughness. However, this dependence on the roughness in relation to the measurement scale is not systematically taken into account. Paper surface topography is here studied in a multi-scale approach with the help of a focus variation device. For each measurement scale, statistical parameters were calculated to describe the surface. We isolated among the long list of parameters calculated those that were relevant for paper surface topography analysis. A new method of characterization of paper surface roughness is proposed based on a scaling analysis taken from either profile or surface data. Depending on the scale of analysis different fractal stages were exhibited. The influence of the step of discretization on the roughness parameters was also investigated. [Copyright &y& Elsevier]

The challenge for a non-linear vibration system in a fractal space is more fractal dimensions than frequency-amplitude relationship, the system energy consumption depends upon its fractal property, so its best-case scenario is to establish a relationship among the fractal dimensions, frequency and amplitude. For this purpose, this paper studies a fractal-fractional vibration system of a mass attached to a stretched elastic wire in a fractal space, and its asymptotic periodic property is elucidated, the effect of the fractal dimensions on the vibration system is discussed. This paper offers a new road to fast and reliable analysis of fractal oscillators with high accuracy. [ABSTRACT FROM AUTHOR]

For the first time, paper-based SWB fractal monopole antenna is presented. A miniaturized SWB monopole, which deploys a fractal radiator, is the smallest reported paper-based SWB monopole antenna to date. It is shown that fractal made of iterations of a flower shaped slot inside a circular metallic patch provides a successful method of reducing size while retaining or enhancing antenna performance. This paper presents the design of SWB monopole antenna for flexible applications and is validated for UWB applications. The proposed antenna has a wide impedance bandwidth (-10 dB bandwidth) from 3.3 GHz to frequency over 30 GHz. The aforementioned antenna with a compact size 18 mm × 24 mm is designed and fabricated on a paper substrate of 0.5 mm thickness. The main features like environment friendly, flexibility, green electronics applications and low cost of fabrication are the key factors for this proposed antenna. The proposed antenna has a 14-dB bandwidth from 3.9 GHz to 13.1 GHz, which is suitable for UWB outdoor propagation. Bending measurements have been performed to validate the flexibility of the prototype under extremely bent condition of bent radius 10 mm. The possible applications cover WLAN, C-band, Ku-band, K-band operations, WiMAX and ISM band. [ABSTRACT FROM AUTHOR]

This paper discusses the introduction of quantum ideas in general physics and education. Considering the fragile environment of physics education, the paper suggests three possible ways to help teaching and learning of quantum physics, (1) quantum metrics (2) simple experiments demonstrating quantum quantities (3) fractal and quantum analogies between worlds. From this perspective, this paper is also meant to be a test indicator of quantum popularity in physics and education. [ABSTRACT FROM AUTHOR]

In this paper, we consider a combined technique for a fractal modification of the attachment oscillator arising from nanotechnology. This technique is called as TSFT-GRHBM by coupling the two-scale fractal transformation and the global residue harmonic balance method. The approximations and frequencies of this fractal attachment oscillator are given without linearization. Numerical results are provided to confirm its efficiency. [ABSTRACT FROM AUTHOR]

Currently, countries worldwide are embracing digital strategies, enabling enterprises to utilize digital technology, digital supply chains, blockchain, and additional digital measures to increase their competitiveness. This paper analyzed the correlation between the digital transformation of manufacturing export enterprises and their business and export performance, focusing on China's manufacturing export enterprises through empirical analysis. The study investigated the influence of digital transformation on enterprise performance. Using the Resource Based View theory and Trade theory, hypotheses were proposed and regression models were developed to analyze a sample of 1007 enterprises listed on the Shanghai and Shenzhen Stock markets from 2012 to 2019. The study conducted regression analysis, intermediate effect test, robustness test, stage lag, and heterogeneity analysis. The study found that (1) Manufacturing export enterprises listed in the stock market implemented digital transformation, leading to a significant positive impact on their overall performance. (2) Digital transformation led to cost reduction, improved R&D intensity, and enhanced human resources, among other benefits for enterprise performance. (3) According to the fractal analysis, non-state-owned enterprises exhibited more favorable effects on enterprise performance, and the digital transformation of manufacturing export companies in developed regions had a more significant impact on their performance. Finally, the study's empirical results yielded pertinent proposals for digital transformation. [ABSTRACT FROM AUTHOR]

In recent years, the development of two offshore low-permeability oil fields has revealed unexpected challenges. The actual productivity of these fields significantly deviates from the designed capacity. Some wells even outperform the expectations for low-permeability limestone fields. This discrepancy primarily stems from a lack of accurate understanding of natural fractures before and after drilling, resulting in substantial errors in capacity assessment. This paper addresses these challenges by proposing a new production capacity model and evaluation method for both vertical and horizontal wells in low-permeability limestone reservoirs. The method leverages logging curve data, incorporating vertical gradation and fractal analysis to effectively represent the fracture's complexity and connectivity. It uniquely considers factors such as fracture fractal dimensions, threshold pressure, and stress sensitivity, significantly enhancing prediction accuracy. Furthermore, by analyzing the longitudinal gradient in logging curves, the method effectively identifies strong heterogeneity, leading to more accurate capacity evaluations in actual fields. The results demonstrate that our model reduces the average prediction error to less than 15%, markedly outperforming traditional methods. Calculation results of the newly developed capacity formula align closely with actual production data and tracer test results, showcasing its practical applicability and potential for widespread use. This study notably advances the evaluation of reasonable production capacity in similar offshore reservoirs. [ABSTRACT FROM AUTHOR]

Historical texts incorporate important characteristics that need to be assessed including genre, text structure and content. Often overlooked are characteristics of handwritten manuscripts commonly divided into legibility, readability and aesthetics. To determine the scientific feasibility of classification of handwritten texts an objective approach is developed to describe twenty handwritten pages of an 1819 Greek manuscript, that refers to the initiation to the Greek secret "friendly society" (Philike Hetaereia) organization, established as part of the Greek independence against the Ottoman Turks. It is investigated through a fractal and RGB image analysis approach. Fractal Minkowski Dimension was applied on the handwritten text and the RGB color analysis on the ink and paper and both were used as a non-invasive manner and revealed interesting results. The novel RGB image analysis and the fractal analysis of the manuscript identified respectively, five iron gall inks and four scribes from the ink content and handwritten styles, of the compact five lines text and whole text pages. The novel approach was verified with another old manuscript of known ink pigments, as well as with thirteen known handwritten texts of that period and four prints representing modern and similar period texts substantiating the findings of the novel methods. [ABSTRACT FROM AUTHOR]

Two miniaturized wire antennas using a 3D space-filling Hilbert-curve (HC) fractal geometry were constructed on a Kodak photo paper using inkjet printing. The proposed antennas each had the 3D dimensions of an 8 cm3 box and operate at very high-frequency band frequencies of 238 and 147 MHz. We wrapped 4- and 2-mm-thick planar wires around a 3D box made of photo paper using second- and third-order HC space-filling fractal designs. The electrical volumes of the proposed second-order HC and third-order HC fractal antennas were 0.004 λ0 × 0.004 λ0 × 0.004 λ0 and 0.002 λ0 × 0.002 λ0 × 0.002 λ0, respectively. The operation of the proposed antennas was verified using simulation and measurement results. [ABSTRACT FROM AUTHOR]

Wagyu beef is gaining worldwide popularity, primarily due to the fineness of its marbling. Currently, the evaluation of this marbling is performed visually by graders. This method has several issues: varying evaluation standards among graders, reduced accuracy due to long working hours and external factors causing fatigue, and fluctuations in grading standards due to the grader's mood at the time. This paper proposes the use of fractal analysis for the grading evaluation of beef marbling to achieve automatic grading without the inconsistencies caused by human factors. In the experiments, cross-sectional images of the parts used for visual judgment were taken, and fractal analysis was performed on these images to evaluate them using fractal dimensions. The results confirmed a correlation between the marbling evaluation and the fractal dimensions, demonstrating that quantitative evaluation can be achieved, moving away from qualitative visual assessments. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to analyzing the components of centrifuge rotor vibrograms using a 2D trajectory fractal analysis based on the Detrended Moving Average method. The method identifies the different noise oscillatory behavior of the rotor depending on the rotation frequencies, ranging from non-stationary unbounded and 1/f pink noise to correlated and uncorrelated noise. Fractal characteristics of the vibrograms were computed for the first time and demonstrated differences for rotation frequencies close to the eigenfrequencies and far from them. This paper also discusses the influence of gyroscopic effects on the natural frequencies of centrifuge oscillations and the excitation of second harmonics when the centrifuge rotates at higher frequencies. The main cause of rotor vibration is identified as the mass imbalance of the rotors, and this paper proposes a vibration classification according to source nodes to diagnose serviceable and faulty technical systems. Finally, the possibility of anisotropy of the vibrogram is discussed, and the oriented fractal scaling components analysis method is applied to pave the way for further investigation. [ABSTRACT FROM AUTHOR]

This paper reports kHz EM observations recorded by the Kardamas station in Ilia, Greece a few days before four near-field earthquakes occurred within a 24 km radius, with epicentral depths below 29 km. This work investigated the fractal features hidden in the EM observations via power-law analysis. All EM signals exhibited characteristic fractal epochs with organisation in space and time. A significant number of accurate fractal segments were delineated in the majority of the EM observations. A significant number of fractal areas corresponded to predictable Class I fBm category with 1 ≤ b ≤ 3 (0 ≤ H ≤ 1). Numerous persistent key-periods are reported with 2 < b ≤ 3 (0.5 ≤ H ≤ 1) which are deemed as signs of impeding earthquakes according to the literature. Numerous segments were found with strong persistent b-values in the range (2.3 ≤ b ≤ 3) (0.65 ≤ H ≤ 1) and b-values corresponding to switching between antipersistency and persistency with (1.7 ≤ b < 2.3) (0.35 ≤ H < 0.65). These are deemed as the most significant precursory signs. Interpretations are given via the asperity model. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of efficient algorithms for the computation of the Riemann zeta function on the complex plane. We introduce two precalculated arrays-based modifications of MB-method. We perform numerical experiments with these algorithms using Zetafast as a benchmark and apply the algorithms for the visualizations of fractal structures associated with the Riemann zeta function. [ABSTRACT FROM AUTHOR]

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function's differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math.76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts α after α -order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of f (x) does not increase after α -order Hadamard fractional integral D H − α f (x) , i.e. dim ¯ B Υ (D H − α f , I) ≤ dim ¯ B Υ (f , I). Particularly, while dim B Υ (f , I) = 1 , for example, a function is of unbounded variation and/or infinite length, it holds dim B Υ (D H − α f , I) = dim B Υ (f , I) = 1 , which answers completely this conjecture for one-dimensional fractal function. The published paper could verify Conjecture 1.1 under Riemann–Liouville or Wyel fractional integral, for only constructed functions with one-dimensional fractal [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. Li and W. Xiao, Fractal dimensions of Riemann–Liouville fractional integral of certain unbounded variational continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500475; Y. S. Liang, Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal.21 (2018) 1651–1658; Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin.3 (2014) 517–524; J. Wang and K. Yao, Construction and analysis of a special one-dimensional continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500207; X. Liu, J. Wang and H. L. Li, The classification of one-dimensional continuous functions and their fractional integral, Fractals26 (2018), doi:10.1142/S0218348X18500639]. [ABSTRACT FROM AUTHOR]

Fractal calculus is an excellent tool to explaining natural phenomena in porous media. In this paper, we first give a simple introduction on He's fractal derivative, and then it is used to establish a model for thermal conduction of silkworm cocoon reveal its biomechanism. The theoretical results obtained in this paper are helpful for the biomimetic design. [ABSTRACT FROM AUTHOR]

During rock drilling, a drill bit will wear as it breaks the rock. However, there is no uniform grading standard for rock abrasiveness. To solve this problem, the wear mechanisms of a polycrystalline diamond compact (PDC) bit and the formation it is drilling into are analyzed in depth, and an abrasiveness evaluation method based on the fractal dimension of the rock surface topography is established. Initially, a three-dimensional digital model is generated from a scanning electron microscope image of the rock after drilling; next, an evaluation of the irregularities on the rock surface is performed using an adapted Weierstrass–Mandelbrot (W-M) function to ascertain the fractal dimensionality. Then, the microcontact characteristics of the contact surface between the formation and the PDC bit are analyzed, and the distribution of the microconvex contact points of the two-body friction pair in a region is obtained. Because the sliding friction between the drill bit and the rock produces a large amount of heat, according to the contact area formula of the friction surface and heat conduction theory, the temperature rise and overall temperature distribution of the formation and PDC bit under the condition of sliding friction are revealed, and the real contact area between the formation and the drill bit within a certain temperature range is obtained. Finally, the evaluation index of rock abrasiveness under sliding conditions is established by adopting the wear weight loss of the rock cutting tool per unit volume as the index of rock abrasiveness, and the model is verified by a microdrilling experiment. The research in this paper is highly important for improving the rock-breaking efficiency and bit service life during drilling. [ABSTRACT FROM AUTHOR]

In this paper, the capillary is divided into several regions, and the effective pore radius is proposed. The relationship between the effective hole radius and the maximum radius is analyzed. The fractal models of fractal flow rate, average velocity and effective permeability of power-law fluid flow in fractal soil composed of circular cross section capillaries are derived. The effects of different power exponent, viscosity coefficient and fractal dimension on the average flow velocity and effective permeability model are analyzed, and the comparative analysis of different sections is carried out. The reasonable calculation results of the fractal model given in this paper verify the correctness of the model. The introduction of the effective pore radius concept and the derivation of fractal models have advanced our ability to characterize and predict flow behavior. We anticipate that these findings will stimulate further research and practical applications in various domains, benefiting both scientific knowledge and technological advancements. [ABSTRACT FROM AUTHOR]

This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let g(x) and h(x) have the same box dimension t (1 < t ≤ 2), the box dimension of g(x) + h(x) may or may not exist. If it exists, it can take an arbitrary real number γ satisfying 1 < γ ≤ t. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers t1 and t2 that satisfy 1 < t1 < t2 < t ≤ 2. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function Wt(x), we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals. [ABSTRACT FROM AUTHOR]

Objectives: Endocardial trabeculae undergo varicose changes and hyperplasia in response to hemodynamic influences and are a variable phenotype reflecting changes in disease. Fractal analysis has been used to analyze the complexity of endocardial trabeculae in a variety of cardiomyopathies. The aim of this paper was to quantify the myocardial trabecular complexity through fractal analysis and to investigate its predictive value for the diagnosis of heart failure with preserved ejection fraction (HFpEF) in patients with multivessel coronary artery disease (CAD). Methods: The retrospective study population consisted of 97 patients with multivessel CAD, 39 of them were diagnosed with HFpEF, while 46 healthy volunteers were recruited as controls. Fractal dimension (FD) was obtained through fractal analysis of endocardial trabeculae on LV short-axis cine images. Logistic regression analyses were used to confirm the predictors and compare different prediction models. Results: Mean basal FD was significantly higher in patients with HFpEF than in patients without HFpEF or in the healthy group (median: 1.289; IQR: 0.078; p < 0.05). Mean basal FD was also a significant independent predictor in univariate and multivariate logistic regression (OR: 1.107 and 1.043, p < 0.05). Furthermore, adding FD to the prediction model improved the calibration and accuracy of the model (c-index: 0.806). Conclusion: The left ventricular FD obtained with fractal analysis can reflect the complexity of myocardial trabeculae and has an independent predictive value for the diagnosis of HFpEF in patients with multivessel CAD. Including FD into the diagnostic model can help improve the diagnosis. Critical relevance statement: Differences show in the complexity of endocardial trabeculae in multivessel coronary artery disease patients, and obtaining fractal dimensions (FD) by fractal analysis can help identify heart failure with preserved ejection fraction (HFpEF) patients. Key Points: The complexity of myocardial trabeculae differs among patients with multivessel coronary artery disease. Left ventricular fractal dimensions can reflect the complexity of the myocardial trabecular. Fractal dimensions have predictive value for the diagnosis of heart failure with preserved ejection fraction. [ABSTRACT FROM AUTHOR]

Cement-based composites', as the most widely used building material, macroscopic performance significantly influences the safety of engineering structures. Meanwhile, the macroscopic properties of cement-based composites are tightly related to their microscopic structure. The complexity of cement-based composites' microscopic structure is challenging to describe geometrically, so fractal theory is extensively applied to quantify the microscopic structure of cement-based composites. However, existing studies have not clearly defined the quantification methods for various microscopic structures in CCs, nor have they provided a comprehensive evaluation of the correlation between the fractal dimensions of different microscopic structures and macroscopic performance. So, this study categorizes the commonly used testing methods in fractal theory into three categories: particle distribution (laser granulometry, etc.), pore structure (mercury intrusion porosity, etc.), and fracture (computed tomography, etc.). It systematically establishes a detailed process for the application of testing methods, the processing of test results, model building, and fractal dimension calculation. The applicability of different fractal dimension calculation models and the range of the same fractal dimension established by different models are compared and discussed, and the advantages and disadvantages of different models are analyzed. Finally, the research delves into an in-depth analysis of the relationship between the fractal dimension of cement-based composites' microscopic structure and its macroscopic properties, such as compressive strength, corrosion resistance, impermeability, and high-temperature resistance. The principle that affects the positive and negative correlation between fractal dimension and macroscopic performance is discussed and revealed in this study. The comprehensive review in this paper provides scholars with methods and models for quantitative research on the microscopic structural parameters of cement-based composites and offers a pathway for the non-destructive assessment of the macroscopic performance of cement-based composites. [ABSTRACT FROM AUTHOR]

This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes. [ABSTRACT FROM AUTHOR]

Structural health monitoring (SHM) for early diagnosis of damage in structures has become vital in improving the structural integrity and safety of the structure and thereby providing a smooth operation for most of the important in-service civil structures such as buildings and bridges. SHM also contributes to the substantial reduction in the maintenance cost of the structure. During serious structural failures, SHM can safeguard human life through an early warning system. This paper presents an output-only model-free technique by extracting damage features from the energy profiles derived from measured time-history responses. The proposed algorithm is organized into two phases i.e. the learning phase and the monitoring phase. While the learning phase can optionally be executed offline, the monitoring phase can preferably be online. The damage feature is extracted by carrying out a fractal dimension analysis of the continuous waveform of energy associated with the jolt (i.e. rate of change of acceleration) evaluated from the acceleration responses of the structure. A novel damage index is proposed, to localize the damage, using the damage features extracted from the fractal dimension, while handling operational and environmental variability (EoV). The proposed algorithm is evaluated through numerical simulations on a typical span of 220 m RCC bridge across river Amaravathi in Tamil Nadu, India and also a 20-story frame structure. Experimental investigations are also carried out by testing a laboratory-scale 10-story frame structure and also an RCC bridge model of a 7.7 m span to evaluate the practical amenability of the proposed algorithm. The experimental and numerical studies carried out in this paper, suggest that the proposed energy-based diagnostic technique can effectively localize the damage even in the presence of confounding factors like EoV. Conclusively, the proposed diagnostic technique can be an effective tool for online SHM, as it is model-free with less tedious computations and can perform well even in the presence of noise. [ABSTRACT FROM AUTHOR]

This paper derives a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation based on the conformable fractional derivative for the first time. Some new types of the fractal traveling wave solutions are successfully constructed by applying a novel approach which is called the fractal semi-inverse variational method. To our knowledge, the obtained results are all new and have not reported in the other literature. In addition, the dynamic characteristics of the different solutions on the fractal space are discussed and presented via the 3D plots, 2D contour and 2D curves. It can be found that: (1) The fractal order can not only affect the peak value of the fractal traveling waves, but also affect the wave structures, that is, the smaller the fractional order value is, the more curved the waveform is, and the slower waveform changes. (2) In the fractal space, the fractal wave keeps its shape unchanged in the process of the propagation and still meets the energy conservation. The methods in this paper can be used to study the other fractal PDEs in the physics, and the findings are expected to bring some new thinking and inspiration toward the fractal theory in physics. [ABSTRACT FROM AUTHOR]

This paper adopts the method of visual characteristics of the double random matrix to conduct in-depth research and analysis on the design of fractal art patterns. For the practical application needs in fractal graphic design, a method is proposed to automatically extract the core base pattern based on fractal graphic content and generate a four-sided continuous pattern. The method first uses the Canny operator for edge detection to analyze the area of the main pattern. Then, it uses a grayscale cogeneration matrix to extract and analyze the graphic texture features, based on which the best splicing method is selected to splice the extracted pattern, and then, it achieves two splicing methods of flat row and staggered four-sided continuous pattern. The method has good practicality with low complexity and high versatility under the premise of ensuring the beauty of the generated four-sided continuous pattern. It can assist designers to design patterns, improve efficiency, and save design costs. In this paper, we improve the existing image segmentation methods, adopt two segmentation methods, namely quadratic tree segmentation and HV segmentation, propose a new local codebook selection strategy, and study the degree of self-adaptation of different methods to images in terms of segmentation methods and local codebook selection strategies. It makes the network unable to train efficiently on long text prediction problems. Finally, the improved algorithm of this paper is tested on the standard and living image libraries, and the experimental results show that the use of local codebooks makes the image coding speed significantly improved compared with the fixed fractal. In the face of the image to be retrieved, it is only necessary to perform the coding operation to obtain the fractal code to perform similarity matching, which can meet the requirement of real-time retrieval. Applying the improved distance formula, the search accuracy obtained on the test gallery is significantly better than that of the grayscale histogram algorithm. [ABSTRACT FROM AUTHOR]

This paper explores the working hypothesis that fractal patterns that closely match those found in nature are more likely to convey a strong sense of genius loci to humans by comparison with 'Euclidean' patterns that do not occur in nature frequently. A part of this survey is concerned with showing the pattern-conscious thinking, regarding the façade composition and material textures, of historical buildings compared to different ecological or geological scenes. We also examine the background of pattern-design from architectural theory, and extrapolate the matter to certain questions about spatial quality, tectonics, and the phenomenon of place. Our most important concern is an attempt to enhance architectural arguments regarding place and character with mathematical calculations. We introduce 'relief method' as a possible way to capture the haptic nature of architecture beyond the patterns of its two-dimensional projections. Through this approach, façades are considered as reliefs and pictures at the same time, thus reflecting the tension between their materiality and visual representation. Fractal geometry also helps to understand how architectonic layers define scale, and by which means architecture could be translated into the human level of physical existence. [ABSTRACT FROM AUTHOR]

Purpose: The nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle. Design/methodology/approach: Appropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique. Findings: This paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems. Practical implications: Nonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems. Originality/value: The variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding. [ABSTRACT FROM AUTHOR]

The research object of this paper is the mixed (κ , s) -Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed (κ , s) -Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under σ = (σ 1 , σ 2) order of the mixed integral is 3 − min { σ 1 κ , σ 2 κ } where  κ > 0. [ABSTRACT FROM AUTHOR]

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In this paper, the possibility of using monofractal and multifractal analysis of acoustic signals of pipelines to detect leaks is considered. An experimental stand has been created to study the fractal characteristics of acoustic signals of pipelines with "slit" type defects. During the experiments, defects of the "slit" type pipeline with dimensions of 2 mm, 8 mm, and 20 mm were modeled. Detrended fluctuation analysis (DFA) and the multifractal detrended fluctuation analysis (MF-DFA) were used. As a result of the experimental studies, it was found that the occurrence of leakage leads to the occurrence of anticorrelated vibrations in a pipeline with multifractal properties. The analyses of acoustic signals by DFA and MF-DFA methods make it possible to reliably determine the leakage. The Hurst exponent and the width of the multifractal spectrum can serve as indicators of the occurrence of leaks in pipelines. [ABSTRACT FROM AUTHOR]

Broken rock masses with the complexity and concealment widely exist in nature such as underground mine, collapse column, and zone. It is extremely difficult to model fracture networks and to simulate water diffusion for broken rock masses. To explore a reasonable fracture network model for broken rock masses, a new method for modeling a two-dimensional planar fracture network model is proposed in this paper. It includes packer test, empirical relationship, fractal width description, and symmetric expansion modeling. Then, the fluid-solid coupling is used to simulate the diffusion properties of water in the two-dimensional planar fracture network model. It is found that the diffusion velocities v max and v min do not appear in the fracture widths λ max and λ min . It indicates that the fracture widths λ max and λ min in the fracture network model for broken rock mass have little impact on the diffusion velocity. Furthermore, the fracture distribution pattern in the fracture network model is an important factor affecting the diffusion velocities v max and v min . The simulation results of water diffusion in the currently proposed model are almost consistent with the actual process of the packer test. Also, the validity of the two-dimensional planar fracture network model is verified by comparing the simulation results with the existing research. [ABSTRACT FROM AUTHOR]

Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes. [ABSTRACT FROM AUTHOR]

Based on the combination of landscape ecology theory and GIS technology, this paper selects the landscape pattern index at the patch type level and landscape level from the two aspects of unit characteristics and landscape heterogeneity, and quantitative analysis of the landscape pattern of parks and green spaces in the main urban area of Wuhan City and suggestions for optimization are put forward. The research results show that: from the patch level, the park green space in the main urban area mainly presents a landscape pattern dominated by comprehensive parks, supplemented by specialized parks, and supplemented by community parks and strip parks. Among them, Hongshan District Park has the largest green space patch area, and Jianghan District has the smallest area; in terms of the average patch area of park green space, the strip park is the largest and the community park is the smallest. From the perspective of landscape level, the diversity index and evenness index of park green space in each district in the main urban area are generally low, among which Jianghan District is the lowest and Hongshan District is the highest; the degree of fragmentation is relatively high in Jianghan District and the lowest in Hongshan District.The patch boundary shape of comprehensive park is complex and has the highest fractal dimension, while that of community park is regular and has the lowest fractal dimension. On this basis, this paper puts forward targeted optimization suggestions based on the research results. [ABSTRACT FROM AUTHOR]

Fractal theory, as a new research method, has been widely studied in various fields. Based on fractal theory and edge computing, the numerical simulation of ballistic limit of protective structure is deeply studied in this paper. In this paper, the ballistic limit of spherical aluminum projectiles colliding with Whipple protective structure under the condition of encounter velocity of 1~12 km/S is numerically simulated. By combining the existing experimental data with the data obtained by numerical simulation, the fitted impact limit equation is obtained by multiple linear regression, and the linear relationship of the fitted equation is tested for significance. According to the critical failure kinetic energy discrimination method proposed in this paper, the high-speed impact protection performance evaluation of the above two structures is given, and the experimental verification is carried out in the ballistic range. The experimental results are consistent with the theoretical predictions. It is proved that the ballistic limit of Whipple protective structure in high-speed region is significantly higher than that predicted by traditional ballistic limit equation, which has reference value for spacecraft protective structure design and spacecraft in-orbit safety analysis. [ABSTRACT FROM AUTHOR]

In this paper, two different types of plasma models with variable coefficients are described by using the fractal derivative. This paper aims at establishing fractal variational principles for two different types of discontinuous plasma physics by the fractal semi-inverse method, so that it can be applied to numerical analysis of the effect of the discontinuous time and discontinuous space on the solution properties. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to propose a fractal model of thermal contact conductance (TCC) of rough surfaces based on cone asperity. Design/methodology/approach: A detailed numerical study is conducted to examine the effects of contact load, fractal dimensional, fractal roughness and material properties on the TCC of rough surfaces. Findings: The results indicate that when the fractal dimension D is less than 2.5, the TCC of rough surfaces increases nonlinearly with the increase of the contact load. However, when the fractal dimension D is greater than or equal to 2.5, the TCC of rough surfaces increases linearly with the increase of the contact load; the TCC of the rough surfaces increases with the increase of the fractal dimension D and the decrease of the fractal roughness G; the material parameters also have an influence on the TCC of the rough surfaces, and the extent of the effect on the TCC is related to the fractal dimension D. Originality/value: A fractal model of TCC of rough surfaces based on cone asperity is established in this paper. Some new results and conclusions are obtained from this work, which provides important theoretical guidance for further study of TCC of rough surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we model a heterogeneous dielectric medium exhibiting fractal geometry or disordered random structures by applying non-integer dimensions to determine its capacitance between two parallel plates. The capacitance depends on the fractional dimensions of the fractal or disordered dielectric slab, which may be obtained from the theoretical fractal dimension or box-counting method. The findings are verified by CST Studio Suite (Electromagnetic field simulation software), experimental measurements, and the equivalent capacitance method. Five common types of fractals (Cantor bars/plates, Sierpinski carpet, Sierpinski triangle, Haferman carpet, and Menger sponge) and random structures are tested with good agreement. There is also an effective gain of capacitance in using less amount of dielectric materials, which may be useful in material-savings of dielectrics. This research shows a useful tool in modeling the capacitance of heterogeneous materials, where fractals and disordered structures may be commonly encountered in organic materials and any dielectrics where precision and fabrication are not perfect. [ABSTRACT FROM AUTHOR]

This paper proposes a fractal viscoelastic element via He's fractal derivative, its properties are analyzed in details by the two-scale transform for the first time. The element is used to establish a fractal Maxwell-rheological model, which unifies the fractal creep equation and relaxation equation, and includes the classic elastic model and the classical Maxwell-rheological model as two special cases. This paper sheds a bright light on viscoelasticity, and the model can find wide applications in rock mechanics, plastic mechanics, and non-continuum mechanics. [ABSTRACT FROM AUTHOR]

In this paper, a new spatiotemporal chaotic Parameter Uncertainty Mixed Coupled Map Lattice (PUMCML) is proposed. The Cantor diagonal matrix is generated from the Cantor set, and the ordered rotation scrambling strategy for this matrix is used to generate the scrambled image. Cantor set is a fractal system, so the Cantor set has a good effect on chaotic image encryption. The dynamic behavior of the PUMCML system is analyzed. The system has good chaotic property, so it is very suitable for chaotic image encryption. Using the PUMCML system, a diffusion strategy based on the mixture of Arnold and Logistic is proposed. Compared with other algorithms, the encryption effect of the proposed method is better and more secure. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the influence of microscopic electric field enhancement caused by microprotrusions on microparticle impact phenomena based on fractal modeling. In this paper, microprotrusions with a radius of 5~\mu \textm and heights 0, 1, 2, 5, and 10~\mu \textm are considered, according to the height–width ratios 0, 0.2, 0.4, 1, and 2, respectively. The microparticles and contacts are assumed to be made of copper. The radius of the microparticle is assumed to be 0.1~\mu \textm and the applied voltage is 60 kV. First, microprotrusions with different height–width ratios are modeled based on fractal theory. Then, the microscopic electric field distortions caused by microprotrusions on the contact surface are simulated based on fractal models of microprotrusions. The impact velocities of microparticles under different cases are calculated based on basic microparticle theories. Finally, the microparticle impact phenomena are simulated using the smoothed particle hydrodynamics method under the microscopic electric field caused by microprotrusions on the contact surface. The results show that the microscopic electric field enhancement caused by microprotrusions will have a significant influence on impact velocities, the characteristics of secondary particles, and the craters produced on the target contact surface. The results of this paper may provide some useful information to further understand the vacuum breakdown initiated by microparticles. [ABSTRACT FROM AUTHOR]

This paper's main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics. [ABSTRACT FROM AUTHOR]

Acoustic emission (AE) technology has the advantage of online localization to study the change laws of AE in the process of coal spontaneous combustion and to reveal the generation mechanisms of AE signal during the process of heating and rupture of coal body from a microscopic perspective. This paper first constructs a large-scale coal spontaneous combustion AE test system and conducts experimental tests on the AE signal in the process of coal spontaneous combustion. The results show that with the increase of temperature in the process of coal spontaneous combustion, the AE signal shows a trend of increasing fluctuations. Low-temperature nitrogen adsorption experiments studied the pore structure of coal spontaneous combustion, and the results showed a correspondence between the development of pores and the temperature of coal spontaneous combustion. Further, through the analysis of the evolution of the pore structure of coal through Fourier transform and fractal theory, it is found that the high-frequency main frequency AE signal and average frequency are continuously enhanced with the increase of temperature. The fractal dimension of the pore structure and the fractal dimension of the AE count of the coal body first rise and then decline. The mechanism of coal spontaneous combustion AE of coal is revealed, and the pore development caused by thermal stress when coal heats up is the main source of AE signal generation. The research in this paper is of great significance for applying AE technology to detect the position of coal spontaneous combustion. [ABSTRACT FROM AUTHOR]

Antenna arrays consist of more than one patch element and are used in various wireless applications. If a single patch element is defective then degraded radiation pattern is obtained which affect the performance of an antenna array. So it has become a critical issue to detect faulty patch in antenna array. In this paper, ANN and FFT based approach has been used to detect number and position of defective patch in designed arrays. Several optimization methods have been applied to detect partial and complete fault in linear and planar antenna arrays. These techniques include Firefly Algorithm (FA), Bat Algorithm, Bacteria Foraging Optimization (BFO) and Cuckoo Search Algorithm (CSA). A cost function has been developed for error between faulty power patterns and estimated one. Minimum value of this cost function will detect the location of defective patch in array. A virtual instrument based model had also been developed for fault analysis in a fractal array. [ABSTRACT FROM AUTHOR]