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A Direct Numerical Simulation (DNS) database of statistically planar flames ranging from the wrinkled flamelets to the thin reaction zones regime and DNS data for a Bunsen premixed flame representing the wrinkled flamelets regime have been utilised to evaluate the fractal dimensions of flame surfaces using the filtering dimension method, the box-counting algorithm and the correlation dimension approach. The fractal dimension evaluated based on the fully resolved three-dimensional data has been found to be reasonably approximated by adding unity to the equivalent fractal dimension evaluated based on two-dimensional projections irrespective of the methodology of extracting fractal dimension. This indicates that the flame surface can be approximated as a self-similar fractal surface for the range of Karlovitz and Damköhler numbers considered here. While all methods, provide results identical to each other for benchmark problems, it has been found that the fractal dimension evaluation based on box-counting method provides almost identical results as that obtained using the filtering dimension method for both three and two dimensions, while the fractal dimensions based on the correlation dimension tend to be slightly smaller. The findings of the current analysis have the potential to be used to reliably estimate the actual fractal dimension in 3D based on experimentally obtained 2D binarised reaction progress variable field. The inner cut-off scales estimated based on all three methodologies yield comparable results in terms of order of magnitude with the box-counting method predicting a smaller value of inner cut-off scale in comparison to other methods. The execution times for fractal dimension extraction based on filtering dimension and box-counting methodologies are found to be comparable but the correlation dimension method is found to be considerably faster than the two alternative approaches and provides results consistent with theoretical bounds in all cases. [ABSTRACT FROM AUTHOR]

This paper explores the fractal properties of token embedding spaces in GPT-2 language models by analyzing the stability of the correlation dimension, a measure of geometric complexity. Token embeddings represent words or subwords as vectors in a high-dimensional space. We hypothesize that the correlation dimension D 2 remains consistent across different vocabulary subsets, revealing fundamental structural characteristics of language representation in GPT-2. Our main objective is to quantify and analyze the stability of D 2 in these embedding subspaces, addressing the challenges posed by their high dimensionality. We introduce a new theorem formalizing this stability, stating that for any two sufficiently large random subsets S 1 , S 2 ⊂ E , the difference in their correlation dimensions is less than a small constant ε. We validate this theorem using the Grassberger–Procaccia algorithm for estimating D 2 , coupled with bootstrap sampling for statistical consistency. Our experiments on GPT-2 models of varying sizes demonstrate remarkable stability in D 2 across different subsets, with consistent mean values and small standard errors. We further investigate how the model size, embedding dimension, and network depth impact D 2 . Our findings reveal distinct patterns of D 2 progression through the network layers, contributing to a deeper understanding of the geometric properties of language model representations and informing new approaches in natural language processing. [ABSTRACT FROM AUTHOR]

Combining biomechanics and motor control, the aim of this study was to investigate the limit cycle dynamics during the high bar longswing across the UK elite gymnastics pathway age groupings. Senior, junior and development gymnasts (N = 30) performed three sets of eight consecutive longswings on the high bar. The centre of mass motion was examined through Poincaré plots and recurrence quantification analysis exploring the limit cycle dynamics of the longswing. Close to one-dimensional limit cycles were displayed for the senior (correlation dimension (CD) = 1.17 ±.08), junior (CD = 1.26 ±.08) and development gymnasts (CD = 1.33 ±.14). Senior elite gymnasts displayed increased recurrence characteristics in addition to longer longswing duration (p <.01) and lower radial angular velocity of the mass centre (p <.01). All groups of gymnasts had highly recurrent and predictable limit cycle characteristics. The findings of this research support the postulation that the further practice, experience and individual development associated with the senior gymnasts contribute to the refinement of the longswing from a nonlinear dynamics perspective. These findings support the idea of functional task decomposition informing the understanding of skill and influencing coaches' decisions around skill development and physical preparation. [ABSTRACT FROM AUTHOR]

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the strain-hardening exponent by experimental research and fractal geometry analysis. From a geometric perspective, the correlation dimension of grains is consistent with that of the equivalent plastic strain field, which represents the correlation dimension of the material manifold. According to the definition of the Hausdorff measure and Ludwik constitutive model, the strain-hardening exponent represents the exponent derived from the Dth root of the measure relationship. This universal law indicates that the strain-hardening behaviors are fractal geometrized and that the strain-hardening exponent represents a geometrical parameter reflecting the multifractal characteristics of grain structures. This conclusion can enhance the comprehension of the relationship between microstructure and mechanical properties of materials and highlights the importance of designing materials with non-uniform grain distributions to achieve desired hardening properties. [ABSTRACT FROM AUTHOR]

Abstract The shale’s multi-scale mechanical behaviors were investigated to understand the time-dependent damage characteristics induced by shale–fluid interaction. The test results indicate that, as fluid–shale interaction proceeds, the mechanical strength of shale has undergone a weakened to an enhanced process as interaction proceeds, and the size distribution of fragments tends to be more uniform, leading to a positive correlation between the mechanical strength and fractal dimension. Within 2 weeks of fluid–shale interaction show a decreasing trend for the fractal dimension of fragments in post-failure and cause less than 2 mm for the dominant size of fragments. The dominant size increases to greater than 30 mm when the fluid–shale interaction is over 2 weeks. Finally, the correlation dimension associated with ring counts of acoustic emission (AE) at each loading stage is determined in terms of the G-P algorithm to predict the damage degree of shale.

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term operation in reduced and difficult weather conditions, wind turbine blades are always seriously damaged. Hence, damage detection in blade structure is essential to evaluate its operational condition and ensure its structural integrity and safety. We aim to use fractal, entropy, and chaos concepts as descriptors for the diagnosis of wind turbine blade condition. They are, respectively, estimated by the correlation dimension, approximate entropy, and the Lyapunov exponent. Formal statistical tests are performed to check how they are different across wind turbine blade conditions. The experimental results follow. First, the correlation dimension is not able to distinguish between all conditions of wind turbine blades. Second, approximate entropy is suitable to distinguish between healthy and erosion conditions and between healthy and mass imbalance conditions. Third, chaos is not a discriminative feature to distinguish between wind turbine blade conditions. Fourth, wind turbine blades with either erosion or mass imbalance exhibit less irregularity in their respective signals than healthy wind turbine blades. [ABSTRACT FROM AUTHOR]

Land use/land cover (LULC) conversion and droughts impact river water quality. Since the river water quality is significant, it was attempted to investigate the effect of hydrological drought (HD) and LULC changes on the water quality parameters of the Helle river in Iran. All the time series (1969–2020) were analyzed using dynamic analysis applications. The results indicated that due to the alteration of LULC and HD patterns, water hardness and salinity were increased by 13.7% and 16.1%, respectively. Moreover, the water quality parameters concentration was intensified. However, the river water potential of hydrogen (pH) decreased by 0.4%. The number of random elements in the time series of the pH, bicarbonate, sulfate, magnesium, and sodium was increased noticeably, and their predictability and long-term memory were reduced. The total dissolved solids (TDS), electrical conductivity (EC), chloride, potassium, and calcium's time series became more chaotic, and their predictability and long-term memory were improved. All the time series faced an increase in the number of noises. The alteration of HD patterns induced a reduction of correlation between HD and sulfate, sodium, magnesium, bicarbonate, pH, and TDS concentration and an augmentation of correlation between HD and chloride, calcium, potassium, and salinity concentration. The Helle river water quality deteriorated due to the alteration of HD patterns and LULC. Future studies can address the strategies to improve the water quality of the Helle river. [ABSTRACT FROM AUTHOR]

Depression, a debilitating mental illness, profoundly impacts an individual's cognition, behaviour, and emotions. Despite efforts to quantify depression rates through electro-encephalo-gram (EEG) signal analysis, classification remains challenging due to inherent noise. This paper introduces a Bat based UNET signal analysis, aimed at accurately classifying depression rates using a normal EEG dataset. It comprises of pre-processing, feature extraction, feature selection, and classification stages. The framework excels at noise reduction during pre-processing, enhancing dataset integrity. Feature extraction leverages band power and correlation dimension to extract crucial features. Furthermore, feature selection optimizes classification accuracy by refining the fitness function of bats in the classification layer. Utilizing a standardized EEG dataset implemented in Matrix Laboratory (MATLAB), the proposed technique demonstrates superior performance compared to existing methods, as evidenced by metrics such as accuracy, area under the curve, precision, and recall (or sensitivity). This innovative framework represents a significant advancement in the classification of depression rates. [ABSTRACT FROM AUTHOR]

Sea surface temperature (SST), with its complex and dynamic behavior, is a major driver of ocean–atmosphere interactions. The purpose of this study is to investigate the behavior of SST and its prediction using a chaotic approach. Average mutual information (AMI) and Cao methods were used to reconstruct the phase space. The Lyapunov exponent and correlation dimension were used to investigate chaos. The Lyapunov exponent index was used to predict SST with a 5-year average prediction horizon using the local prediction method between 2023 and 2027. The results showed a 3-month delay time for the Pacific and Antarctic Oceans, and a 2-month delay time for the Atlantic, Indian, and Arctic Oceans. The optimal embedding dimension for all oceans is between 6 and 7. Our analysis reveals that the dynamics of SST in all oceans exhibit varying degrees of chaos, as indicated by the correlation dimension. The local prediction method achieves relatively accurate short-term SST predictions due to the clustering of SST points around specific attractors in the phase space. However, in the long term, the accuracy of this method decreases as the points in the phase space of SST can spread randomly. The model performance ranking with a Percent Mean Relative Absolute Error shows that the Indian Ocean has the best performance compared to other oceans, while the Atlantic, Pacific, and Antarctic and Arctic Oceans are in the next ranks. This study contributes to understanding the dynamics of SST and has practical value for use in the development of climate models. [ABSTRACT FROM AUTHOR]

The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model's accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality. [ABSTRACT FROM AUTHOR]

Abstract This study examines the market efficiency in the prices and volumes of transactions of 41 cryptocurrencies. Specifically, the correlation dimension (CD), Lyapunov Exponent (LE), and approximate entropy (AE) were estimated before and during the COVID-19 pandemic. Then, we applied Student’s t-test and F-test to check whether the estimated nonlinear features differ across periods. The empirical results show that (i) the COVID-19 pandemic has not affected the means of CD, LE, and AE in prices, (ii) the variances of CD, LE, and AE estimated from prices are different across pre-pandemic and during pandemic periods, and specifically (iii) the variance of CD decreased during the pandemic; however, the variance of LE and the variance of AE increased during the pandemic period. Furthermore, the pandemic has not affected all three features estimated from the volume series. Our findings suggest that investing in cryptocurrencies is advantageous during a pandemic because their prices become more regular and stable, and the latter has not affected the volume of transactions.

Automatic voice pathology detection enables an objective assessment of pathologies that influence the voice production strategy. By utilizing the conventional pipeline model as well as the modern deep learning-centric end-to-end methodology, numerous pathological voice analyzing techniques have been developed. The conventional methodology is still a valid choice owing to the lack of enormous amounts of training data in the study region of pathological voice. In the meantime, obtaining higher precision, higher accuracy, and stability is still a complicated task. Therefore, by amalgamating the nonlinear measure, the pathological voices are analyzed to abate such risks. The viability of six nonlinear discriminating measures derived from the phase space realm, involving healthy and pathological voice signals, is studied in this work. The analyzed parameters are Singularity spectrum coefficients ( α min , α max , γ 1 and γ 2 ). Correlation entropy at optimum embedding dimension ( K 2 m ) and correlation dimension at optimum embedding dimension ( D 2 m ). Analyzing the pathological voices with better accuracy rates is the major objective of the proposed methodology. Here, the Support Vector Machine (SVM) was utilized as the classifier. Experimentations were performed on VOiceICarfEDerico (VOICED) databases subsuming 208 healthy, as well as pathological voices, amongst these 50 samples, were utilized. Here, the model obtained 97% of accuracy with 99% as of the classifier with Gaussian kernel function. Therefore, to differentiate normal as well as pathological subjects, the six proposed characteristics are highly beneficial; in addition, they will be supportive in pathology diagnosis. [ABSTRACT FROM AUTHOR]

In order to explore the mesoscopic mechanism of rock damage and the characteristics of precursor to destruction, the shear failure acoustic emission test was carried out on carbonaceous shale, the phase-space reconstruction and correlation dimension calculation of the three acoustic emission parameters were carried out by Grassberger-Procaccia(G-P) algorithm, and the evolution law of correlation dimension in the whole process of carbonaceous shale destruction was analyzed by using fractal theory. The results show that the acoustic emission activity is active in different stages of rock deformation, and there is a calm period. The acoustic emission ringing count, energy, and amplitude sequences all have fractal characteristics, and the correlation dimensions of the three show the change law of “ ascending dimension-dynamic fluctuation-dimensionality reduction ” in the whole process of destruction. The dimensionality reduction phenomenon occurs before the destruction of the rock and ends after the destruction of the rock sample, so it can be used as the precursor information of the rock instability failure as the calm period, and the dimensionality reduction phenomenon of the associated dimension of the ringing count and energy is more significant and lasts longer, which is more suitable as the precursor information of the rock instability failure than the amplitude. [ABSTRACT FROM AUTHOR]

The Kaiser effect exhibits stress-induced changes in the acoustic properties of rock, so accurately identifying the Kaiser point of rock under complex conditions is essential for understanding the damage and failure mechanisms. In this paper, triaxial loading and unloading tests were conducted on salt rock using MTS815 material testing machine and PCI-II acoustic emission (AE) system. By introducing the Kaiser point calculation approach involving AE signal optimisation, Grassberger-Procaccia (GP) algorithm, K-means clustering, and random sample consensus (RANSAC) algorithm, we achieved enhanced accuracy in analysing the Kaiser effect for salt rock. The cumulative AE energy exhibited a distinct surge point during  the cycle number (N) was less than 21, closely aligned with the optimised key AE energy point. For N > 21, the cumulative energy displayed stable changes, rendering conventional method ineffective. By utilising screened AE signals (8796 samples) within the GP algorithm, the surge point of cumulative energy consistently aligned with the sudden drop point of the correlation dimension when N<21, with a maximum error of 6 s. Remarkably, the GP algorithm identified the prominent Kaiser point even when N > 21, suggesting its applicability throughout the loading cycle. [ABSTRACT FROM AUTHOR]

These days, physical health concerns are gradually elevated to the same degree of significance as essential issues such as education, health, and protection as national living standards improve. Enhancing the physical health of university students and encouraging healthy growth are significant moments for China, due to which each university controls a physical examination every year. Utilizing wireless sensors for data capture in university students' physical fitness evaluation systems has played an essential role in various areas. Therefore, this paper presents a body domain network multimodal sensor and decision algorithm-based physical health assessment system for university students. The proposed system uses multiple body domain network sensors to collect physiological indicators data by using decision algorithms to combine multiple assessment indicators for analysis and prediction, ultimately generating a health assessment report and providing customized health advice and health plans for each user. In addition, to improve the physical health evaluation of university students in the context of sports medicine integration, this paper also proposes a physical health evaluation method based on the ID3 decision tree algorithm. It constructs a physical health information mining and feature extraction model for university students in the context of sports medicine integration. It uses a decision tree optimization method for classification detection. The proposed method also uses an associative multidimensional feature detection algorithm to evaluate university students' physical fitness and health status. It establishes a decision indicator function to investigate information fusion and constrained feature decomposition through the significant difference and balanced training fusion methods. In the ID3 decision tree, the branching system of college students' physical fitness health information in the context of sports medicine fusion is designed. It examines the entropy weight index parameters for managing test data management to assess university students' physical fitness in fusing sports medicine. The experimental results show that the system can accurately assess the physical health status of university students and provide corresponding recommendations and plans, which is essential for improving the health status of university students. In addition, these results proved that the proposed method has good output stability and strong optimization ability, which improves the classification management and information integration of college students' physical fitness and health information. [ABSTRACT FROM AUTHOR]

This study examines the market efficiency in the prices and volumes of transactions of 41 cryptocurrencies. Specifically, the correlation dimension (CD), Lyapunov Exponent (LE), and approximate entropy (AE) were estimated before and during the COVID-19 pandemic. Then, we applied Student's t-test and F-test to check whether the estimated nonlinear features differ across periods. The empirical results show that (i) the COVID-19 pandemic has not affected the means of CD, LE, and AE in prices, (ii) the variances of CD, LE, and AE estimated from prices are different across pre-pandemic and during pandemic periods, and specifically (iii) the variance of CD decreased during the pandemic; however, the variance of LE and the variance of AE increased during the pandemic period. Furthermore, the pandemic has not affected all three features estimated from the volume series. Our findings suggest that investing in cryptocurrencies is advantageous during a pandemic because their prices become more regular and stable, and the latter has not affected the volume of transactions. [ABSTRACT FROM AUTHOR]

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Given a dynamical system, we prove that the shortest distance between two n -orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit. [ABSTRACT FROM AUTHOR]

Local transient solid holdup signals were measured by using an optical fiber probe at different axial and radial positions in the gas-solid cyclone reactor. The chaotic time series analysis of transient solid holdup was adopted to characterize the dynamic characteristics. The maximum and minimum solid holdup indicates that the high variability of gas-solid interaction. With the increase of the gas / solid ratio, the frequency of solid holdup fluctuation increased while the mean value of amplitude decreased; besides, the level of detail in the power spectrum fluctuations increased. In the diffusion chamber, the overall Kolmogorov entropy (K entropy) and correlation dimension (D c) were higher indicating the strong chaos characteristic of gas-solid flow. In the agglomeration chamber, higher solid holdup initially reduced gas-solid interaction, decreasing flow randomness, but a subsequent increase led to irregular particle collisions and an upturn of K entropy and D c. In the annulus chamber, K entropy and D c were lower around the inner wall and outer wall compared to the middle region. In the separation chamber, K entropy and D c tended to increase and then decrease as the radial position increased. Finally, the relationship between K entropy, D c , and time-averaged solid holdup was analyzed. [Display omitted] • Kolmogorov entropy and correlation dimension were studied in the cyclone reactor. • A subsequent increase of solid holdup leads to an upturn of chaotic properties. • The higher chaotic properties in the diffusion chamber may favor gas-solids mixing. [ABSTRACT FROM AUTHOR]

ABSTRACTThe potential earthquake and tsunami hazard along the Sumatra subduction margin, especially around the coast of West Sumatra-Bengkulu, was investigated based on the availability of pre-seismic surface displacement data and shallow crustal earthquake catalogue data from 1907 to 2016. The pre-seismic surface displacement data is based on the displacement data prior to and corrected displacement data after major earthquakes. Using the results of our previous study on the local covariance function and the relationship of Correlation Dimension (DC) with the b-value of Gutenberg-Richter (GR) Law, we estimated the maximum horizontal crustal strain rate (SHmax) and DC around the study area. Least squares prediction based on horizontal displacement data using the local covariance function is used to estimate the displacement model in the entire gridding study area with a 10 km × 10 km size. Furthermore, DC is calculated based on the b-value using the maximum likelihood method based on the input of a constant number of earthquake samples, assuming the regional b-value of GR Law equals 1. Furthermore, the spatial correlation of SHmax and DC can define the area of possible earthquake hazard potential. The identification results are then linked with previous stress reconstruction results for seismic hazard study and analysis. Based on the finding, we then estimate the Seismic Hazard Function (SHF) and Tsunami Height simulation to estimate the possible hazard risk at several observation points. We suggest that the result of this study could be beneficial to understand better the potential seismic and tsunami hazard in the future, mainly to support mitigation purposes.

This paper explores the fractal properties of token embedding spaces in GPT-2 language models by analyzing the stability of the correlation dimension, a measure of geometric complexity. Token embeddings represent words or subwords as vectors in a high-dimensional space. We hypothesize that the correlation dimension D2 remains consistent across different vocabulary subsets, revealing fundamental structural characteristics of language representation in GPT-2. Our main objective is to quantify and analyze the stability of D2 in these embedding subspaces, addressing the challenges posed by their high dimensionality. We introduce a new theorem formalizing this stability, stating that for any two sufficiently large random subsets S1,S2⊂E, the difference in their correlation dimensions is less than a small constant ε. We validate this theorem using the Grassberger–Procaccia algorithm for estimating D2, coupled with bootstrap sampling for statistical consistency. Our experiments on GPT-2 models of varying sizes demonstrate remarkable stability in D2 across different subsets, with consistent mean values and small standard errors. We further investigate how the model size, embedding dimension, and network depth impact D2. Our findings reveal distinct patterns of D2 progression through the network layers, contributing to a deeper understanding of the geometric properties of language model representations and informing new approaches in natural language processing.

The acoustic characteristics of the rock failure process can reflect the evolution of crack development and expansion. To study the evolution of different types of internal cracks during the rock failure process, the acoustic emission (AE) signal of the yellow sandstone under uniaxial compression was collected with AE monitoring technology, and the acoustic characteristics, crack types and evolution of the correlation dimension of the rock failure process were studied. The parameter analysis of RA (rise time/amplitude) and AF (counts/duration) for classifying different cracking modes during loading revealed that tensile cracks are dominated in the crack closure stage and linear elastic deformation stage, and the proportion of shear cracks increased in the plastic deformation stage and the post-peak failure stage. High-energy events were more widely distributed in the areas of shear cracks. The average AE energy of shear cracks is higher than that of tensile cracks, and the AE energy ratio for shear cracks and tensile cracks is within the range of 4.29–4.78. The correlation between stress and AE data was calculated using fractal theory methods and Grassberger–Procaccia (G–P) algorithms to study the evolution of different types of internal cracks. The correlation dimension displayed a stable stage, a descending stage and a fluctuating low value stage. The descending stage and the fluctuating low value stage appeared during the plastic deformation stage and the post-peak failure stage, respectively, indicating that the type of rock internal micro-crack changed greatly in the above two stages. The findings in this study contribute to better understanding of the rock failure mechanism. Highlights: The evolution of proportion of tensile and shear cracks during the rock failure process were analyzed. The average AE energy of shear cracks is higher than that of tensile cracks, and the AE energy ratio for within the range of 4.29–4.78. The evolution of the type of micro-cracks in rock failure process is analyzed by Grassberger–Procaccia (G–P) algorithms. [ABSTRACT FROM AUTHOR]

Introduction: It is proved that there are differences between gait performed by females and males, which appear in movements of selected body parts. Despite numerous state-of-the-art studies related to the discriminative analysis of motion capture data, the question of whether measures of signal complexity and uncertainty can extract valuable features for the problem of sex distinction still remains open. It is the subject of the paper. Methods: Correlation dimension, as well as approximate and sample entropies, are selected to describe motion data. In the numerical experiments, the collected dataset with 884 samples of 25 females and 30 males was used. The measurements took place in the Human Motion Laboratory (HML), equipped with a highly precise motion capture system. Two variants of data representation were investigated-time series that contain joint rotations of taken skeleton model as well as positions of the markers attached to the human body. Finally, a comparative analysis between the populations of females and males using descriptive statistics, non-parametric estimation, and statistical hypotheses verification was carried out. Results: There are statistically significant sex differences extracted by the taken measures. In general, the movements of lower limbs result in greater values of correlation dimension and entropies for females, while selected upper body parts play a similar role for males. The dissimilarities are mainly observed in hip, ankle, shoulder, and head movements. Discussion: Correlation dimension and entropy measures provide robust and explainable features of motion capture data with a valuable description of the human locomotion system. Thus, beyond the importance of discovered differences between females and males, their interpretation and understanding are also known. [ABSTRACT FROM AUTHOR]

The main objective of this work is the characterization of the seismic activity linked to the Pinar Fault in the region of Western Cuba, through the use of mathematical methods that allow characterizing and/or quantifying the space-time relationship in said seismic series. Along with this analysis, an interpretation and correlation of the results obtained with those derived from previous research is carried out. To solve the problem of this work, four fundamental parameters are determined through the application of mathematical-statistical methods: the Poisson entropy, the estimation of the generalized dimensions Dq, the b value of the Gutenberg-Richter relationship and the Hurst exponent. The integration of the criteria offered by these parameters contributes to the interpretation of the neotectonics of the region from a seismic point of view and in turn allows the creation of more complete seismic records than the current ones. Among the main results obtained is the demonstration that the seismic series of the region complies with Hurst's hypothesis, where most natural phenomena behave as statistical trends with stochastic noise and not as random walks. This facilitates the relationship between the research results and the recent phenomena that govern neotectonics in the area. [ABSTRACT FROM AUTHOR]

Fractals are widely used in different fields of science. The main characteristic of a fractal set is its fractal dimensions. The presented work addresses the accuracy of estimation of the correlation fractal dimension D2 and corresponding confidence interval. For example, in geosciences D2 is applied for characterizing surfaces or the spatial distribution of fracture systems. Two qualitatively different problems are studied. The first problem is related to the analysis of the fractal sets assigned on regular grids. The fractals consisting of a family of irregular points are considered in the second problem. The Monte Carlo method is proposed to estimate the confidence intervals of correlation fractal dimension for the first problem. This approach is based on the statistical generation of the fractal sets realizations with a given D2. Numerical testing using synthetic models showed the reliability of this method. A comparison with PBM and BCA bootstrap methods is performed. The paper also investigates the accuracy of the D2 and the corresponding confidence intervals estimate for the second studied problem. An algorithm employing the jackknife method is suggested. Its accuracy has been studied and proved numerically. [ABSTRACT FROM AUTHOR]

There are numerous types of cognitive tasks classified as mental tracking (MT), working memory (WM), reaction time (RT), discrimination and decision-making and verbal fluency (VF). However, limited studies have investigated the effects of cognitive task type on postural control in older adults. s: The aim of this study was to investigate the effect of aging and several types of cognitive tasks on postural control in terms of nonlinear analysis. Postural control was investigated under 6 conditions (single task and dual-task with RT; easy and difficult VF; easy and difficult WM; easy and difficult MT. Outcome measurements were the max Lyapunov, entropy, and correlation dimension at anteroposterior (AP) and mediolateral (ML) directions. The results revealed that within the older group, the AP & ML max Lyapunov at dual-task with difficult WM and MT was significantly higher than all other conditions. In addition, the older group had lower AP entropy at dual-task with easy VF, difficult WM, and easy as well as difficult MT. The results can be useful to understand the postural control mechanisms and to detect the alterations following aging and applying different types of cognitive tasks. In addition, the investigated parameters can be a basis for identifying postural control deficiencies. [ABSTRACT FROM AUTHOR]

Bubble columns (BCs) are widely used in the chemical industry. In many industrial applications, these important gas-liquid contactors operate in a churn-turbulent flow regime. In principle, it is essential to determine the operating conditions in every BC reactor, in which local isotropic turbulence is established. In this work, it was demonstrated that several different parameters (Kolmogorov entropy, correlation dimension and novel hybrid index) follow a monotonic decreasing trend. This finding could be explained by the constantly increasing coalesced bubble size, which brings more order into the gas-liquid system and thus any entropic or chaotic parameter should decrease with the increase in the superficial gas velocity Ug. The profiles of the new parameters in various gas-liquid systems were studied. They were extracted from different pressure signals (gauge or absolute). In this research, BCs of different diameter and equipped with different gas distributors were used. It was demonstrated that the studied parameters could be successfully correlated with the length scale of the micro eddies and thus the Ug range of applicability of the local isotropic turbulence theory under various operating conditions was indirectly determined. The overall gas holdup profiles were analyzed and, based on the exponent of the Ug value, it was found that in the aqueous solutions of alcohols studied, the conditions in the bubble bed (BB) are homogeneous, whereas in the air-tap water system aerated in different BCs, the conditions in the BB are heterogeneous. This result implies that the local isotropic turbulence conditions predominate mainly around the corresponding measurement positions. [ABSTRACT FROM AUTHOR]

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the strain-hardening exponent by experimental research and fractal geometry analysis. From a geometric perspective, the correlation dimension of grains is consistent with that of the equivalent plastic strain field, which represents the correlation dimension of the material manifold. According to the definition of the Hausdorff measure and Ludwik constitutive model, the strain-hardening exponent represents the exponent derived from the Dth root of the measure relationship. This universal law indicates that the strain-hardening behaviors are fractal geometrized and that the strain-hardening exponent represents a geometrical parameter reflecting the multifractal characteristics of grain structures. This conclusion can enhance the comprehension of the relationship between microstructure and mechanical properties of materials and highlights the importance of designing materials with non-uniform grain distributions to achieve desired hardening properties.

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term operation in reduced and difficult weather conditions, wind turbine blades are always seriously damaged. Hence, damage detection in blade structure is essential to evaluate its operational condition and ensure its structural integrity and safety. We aim to use fractal, entropy, and chaos concepts as descriptors for the diagnosis of wind turbine blade condition. They are, respectively, estimated by the correlation dimension, approximate entropy, and the Lyapunov exponent. Formal statistical tests are performed to check how they are different across wind turbine blade conditions. The experimental results follow. First, the correlation dimension is not able to distinguish between all conditions of wind turbine blades. Second, approximate entropy is suitable to distinguish between healthy and erosion conditions and between healthy and mass imbalance conditions. Third, chaos is not a discriminative feature to distinguish between wind turbine blade conditions. Fourth, wind turbine blades with either erosion or mass imbalance exhibit less irregularity in their respective signals than healthy wind turbine blades.

Musical symbols are a good way of expressing human emotions and communication. It is a kind of nonlinear dynamic system with hysteresis. It isn’t easy to analyze its nonlinear characteristics by conventional methods. This paper divides it into several parts from the form of music. Secondly, this paper studies the local attributes of music signals and generalizes and deduces their general features. Then the Lyapunov index of music is numerically simulated, and its energy spectrum is analyzed. It is proved that the method is weak to some extent. Through the music correlation dimension analysis, we find its inherent nonlinear characteristics. The nonlinear music signals corresponding to classical piano works are given in this paper. Finally, the correlation dimension is numerically simulated by differential operation. The results show that the nonlinear characteristics of this method are stable.

IntroductionIt is proved that there are differences between gait performed by females and males, which appear in movements of selected body parts. Despite numerous state-of-the-art studies related to the discriminative analysis of motion capture data, the question of whether measures of signal complexity and uncertainty can extract valuable features for the problem of sex distinction still remains open. It is the subject of the paper.MethodsCorrelation dimension, as well as approximate and sample entropies, are selected to describe motion data. In the numerical experiments, the collected dataset with 884 samples of 25 females and 30 males was used. The measurements took place in the Human Motion Laboratory (HML), equipped with a highly precise motion capture system. Two variants of data representation were investigated-time series that contain joint rotations of taken skeleton model as well as positions of the markers attached to the human body. Finally, a comparative analysis between the populations of females and males using descriptive statistics, non-parametric estimation, and statistical hypotheses verification was carried out.ResultsThere are statistically significant sex differences extracted by the taken measures. In general, the movements of lower limbs result in greater values of correlation dimension and entropies for females, while selected upper body parts play a similar role for males. The dissimilarities are mainly observed in hip, ankle, shoulder, and head movements.DiscussionCorrelation dimension and entropy measures provide robust and explainable features of motion capture data with a valuable description of the human locomotion system. Thus, beyond the importance of discovered differences between females and males, their interpretation and understanding are also known.

Principal Component Analysis (PCA) is a widely used technique for fault detection and diagnosis. PCA works well when the data set has linear characteristics. However, most industrial processes have nonlinear characteristics in their data. Kernel PCA (KPCA) is an alternative solution for such types of data sets. This solution doesn't come without a cost since one of KPCA's disadvantages is a large number of observations which results in more occupied storage space and more execution time than the PCA technique. Furthermore, if the data is too large it may minimize the monitoring performance of the KPCA model. Reduced KPCA (RKPCA) is a solution for the conventional KPCA limitations. Firstly, RKPCA can deal with nonlinear characteristics without crucial problems because it is based on the KPCA algorithm with a data reduction part where it keeps most of the data's information. Thus, by reducing the number of observations RKPCA reduces the occupied storage space and execution time while preserving tolerable monitoring performance. The proposed RKPCA algorithm consists of two parts. First, the large-sized training data set is reduced using the fractal dimension technique (correlation dimension). Afterward, the KPCA model is developed through the obtained reduced training data set. The proposed scheme is applied to the Tennessee Eastman Process and the Cement Plant Rotary Kiln data sets to evaluate its performance in comparison with other algorithms. [ABSTRACT FROM AUTHOR]

Dynamic failure of surrounding rock often causes many casualties and financial losses. Predicting the precursory characteristics of rock failure is of great significance in preventing and controlling the dynamic failure of surrounding rock. In this paper, a triaxial test of granite is carried out, and the acoustic emission events are monitored during the test. The fractal characteristics of acoustic emission events' energy distribution and time sequence are analyzed. The correlation dimension and the b value are used to study the size distribution and sequential characteristics. Furthermore, a rock failure prediction method is proposed. The correlation dimension is chosen as the main index and the b value is chosen as a secondary index for the precursor of granite failure. The study shows that: (1) The failure process can be divided into an initial stage, active stage, quiet stage, and failure stage. (2) The b value and correlation dimension both can describe the process of rock failure. There is a continuous decline before failure. Because of the complexity of the field, it is difficult to accurately estimate the stability of surrounding rock using a single index. (3) The combination of the b value and correlation dimension to establish a new method, which can accurately represent the stability of the surrounding rock. When the correlation dimension is increasing, the surrounding rock is stable with stress adjusting. When the correlation dimension is decreasing and the b value remains unchanged after briefly rising, the surrounding rock is stable, and stress is finished adjusting. When the correlation dimension and b value are both decreasing, the surrounding rock will be destroyed. [ABSTRACT FROM AUTHOR]

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The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model’s accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality.

The purpose of this study was to assess the effects of transverse friction massage (TFM) on the electromechanical delay components and complexity of the surface electromechanical activity in the rectus femoris (RF) and vastus medialis (VM) muscles and to identify possible mechanisms behind TFM-induced alterations in the dynamics of RF and VM activity. Seven female and five male healthy subjects participated in this study. The subjects generated five maximal voluntary isometric contractions (MVICs) consecutively before and after TFM. Meanwhile, electromyography (EMG), mechanomyography (MMG), and force were recorded. The onset times of the recorded signals were detected offline by setting the threshold to three times the SD of the baseline. The delays between EMG and MMG (Δt(EMG–MMG)), MMG and force (Δt(MMG–Force)), and EMG and force (Δt(EMG–Force)) were computed from the detected onsets. The fractal dimension (FD) of the EMG time series was computed using the correlation dimension method. TFM increased Δt(MMG–Force) and Δt(EMG–Force) significantly in the RF but decreased Δt(EMG–MMG) and increased Δt(MMG–Force) in the VM. TFM decreased the FD in the RF and increased it in the VM. The results imply that TFM decreased the stiffness of both the RF and VM and decreased the duration of the electrochemical processes in the VM. It is proposed that the decrease in EMG complexity in the RF may be associated with the decreased stiffness of the RF, and the increase in EMG complexity in the VM may be associated with the decreased electrochemical processes in this muscle. It is also suggested that the opposite changes in EMG complexity in the RF and VM can be used as a discriminating parameter to search for the effects of an intervention in the quadriceps muscles. The present study also demonstrates how to discriminate the nonlinear dynamics of a complex muscle system from a noisy time series. [ABSTRACT FROM AUTHOR]

To verify whether fractal characteristics are measured by infrared radiation (IR) experiments, IR and AE monitoring of sandstone samples under uniaxial compression was carried out. The relationship between the infrared radiation count correlation dimension (DIR) based on the G-P algorithm and fractal theory was established by these tests. The fractal characteristics and precursors of IR during sandstone fracture were obtained by comparing the correlation between the IR and AE characteristics. The following relationships were established: (1) infrared radiation counts (IRC) and acoustic emission ringing-down count (AEC) have simultaneous and sudden changes in all sandstones before final failure. (2) during the compression process, the DIR shows a trend of fluctuating increase → rising fluctuation amplitude → peak → sudden drop, and this trend is similar to the development of the AEC count correlation dimension (DAE). This indicates that the IR on the sandstone surface has fractal characteristics and is consistent with the internal fracture process, which is an orderly process of dimensionality reduction. (3) The average stress at the DIR sudden drops is 90.3% of the peak value, while the time leading to the peak stress and final failure is 30.5 s and 80.8 s on average, respectively, suggesting that the sudden drop of DIR after reaching the peak can be regarded as the precursor information of sandstone fracture and stability loss. These research results provide a theoretical basis and data support for the use of IR to monitor and predict sudden and brittle rock mass instability and damage. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of the time-series of water temperature of the Skokomish River (2019–2020) at hourly time scale by employing well-known nonlinear methods of chaotic data analysis including average mutual information, false nearest neighbors, correlation exponent, and local divergence rates. The delay time and the embedding dimension were calculated as 1400 and 9, respectively. The results indicated that the thermal regime in this river is chaotic due to the correlation dimension (1.38) and the positive largest Lyapunov exponent (0.045). Furthermore, complex networks have been applied to study the periodicity of thermal time-series throughout a year. A special algorithm is then used to find the so-called communities of the nodes. The algorithm found three communities which have been called Cold, Intermediate, and Warm. The temperatures in these three communities are, respectively, in the intervals (0.8, 5.8), (5.8, 11.63), and (11.63, 15.8). This analysis indicates that highest variations in water temperature occur between warm and cold seasons, and complex networks are highly capable to analyze hydrothermal fluctuations and classify their time-series. [ABSTRACT FROM AUTHOR]

The multifractal theory is applied in an analysis of bridge disturbance signals with the aim of investigating their nonlinear characteristics, and then the recognisable fault features are extracted from them. By calculating the box dimension and correlation dimension of the bridge disturbance signal, the dimensional characteristics of the disturbance data are analysed to distinguish the health-state of the bridge. Finally, taking the bridge disturbance data as an example, and by using the multifractal spectrum analysis of the disturbance data, it is concluded that the multifractal method can accurately identify the fault state and realise the bridge health monitoring.

Abstract Burnishing experiments with different burnishing parameters were performed on a computer numerical control milling machine to characterize the surface roughness of an aluminum alloy during burnishing. The chaos theory was employed to investigate the nonlinear features of the burnishing system. The experimental results show that the power spectrum is broadband and continuous, and the Lyapunov exponent λ is positive, proving that burnishing has chaotic characteristics. The chaotic characteristic parameter, the correlation dimension D, is sensitive to the time behavior of the system and is used to establish the corresponding relationship with the surface roughness. The correlation dimension was the largest, when the surface roughness was the smallest. Furthermore, when the correlation dimension curve decreases, the roughness curve increases. The correlation dimension and surface roughness exhibit opposite variation trends. The higher the correlation dimension, the lower the surface roughness. The surface roughness of the aluminum alloy can be characterized online by calculating the correlation dimension during burnishing.

The development of personalized medicine is determined by the synergy of scientists from several fields of medicine, mathematics and computer science. Approaches based on modern methods of measurement, signal processing and machine learning complement the main methods of studying biological processes, allow us to identify the mechanisms of the disease and personalize the treatment strategy. The article is devoted to the development of a new method for monitoring the system of autoregulation of cerebral circulation in patients in real time. The process of cerebral autoregulation is determined by the signals of blood flow velocity in the arteries of the base of the brain and systemic arterial pressure in the range of Mayer waves, recorded by non-invasive methods of photoplethysmography and transcranial dopplerography. The existing methods are based on the cross-correlation function, the phase shift function between the signals of blood flow velocity and pressure, or on the assessment of the transfer function of the autoregulation system. The article proposes to use fractal methods that use the calculation of the Hölder multifractal spectra of signals and the determination of the correlation dimension of the system. The advantage of fractal methods is that they can be applied to scale-invariant signals. The combination of multifractal and traditional spectral-correlation methods in the information-measuring system will improve the quality of monitoring of cerebral autoregulation directly at the patient's bedside, take into account the characteristics of the patient and, thereby, develop the principle of personalized medicine.

ObjectiveA ship's system is a complex mechanism composed of multiple pieces of equipment. Due to the dynamic and non-linear characteristics of the parameters of each component, fault diagnosis is complicated. This paper proposes a dynamic feature fusion method for performing efficient fault diagnosis on the system.MethodsFractal theory, dynamic theory and the kernel principal component analysis (KPCA) method are used to reconstruct, map and filter the system state data, and obtain the principal component characteristic data matrix, square prediction error (SPE) and corresponding control limits. An offline monitoring model based on the health data of a marine diesel engine intake and exhaust system is then constructed and used to diagnose and analyze ship system faults. In order to verify the validity of the model, the fault data of the intake and exhaust system of a marine diesel engine is selected for verification and analysis. ResultsThe results show that this method can effectively realize the accurate analysis of a system's dynamic nonlinear state data and efficient analysis and diagnosis of faults, with better fault diagnosis performance than the KPCA and support vector machine (SVM) method.ConclusionsThe method proposed in this paper can realize the detection and diagnosis of marine diesel engine intake and exhaust system failures, and improve the reliability and safety of system operation.