Fractal Dimension Values Research Articles

Fractal analysis to determine JRC on sandstones and its correlation to SRF, Ende-Lianunu Regency, East Nusa Tenggara Province, Indonesia

The study area is located in Ende-Lianunu Regency, East Nusa Tenggara Province, Indonesia, with mountainous and hilly morphology, and is part of the Kiro Formation which is composed of tuffaceous sandstones. Discontinuities in the sandstone are in the form of bedding planes and joints, which affect the mechanical properties of the rock mass, reduce its strength, and affect slope stability. The discontinuity condition that affects mechanical behavior is surface roughness. This research aims to define the fractal dimension value of surface roughness using a box-counting method, joint roughness coefficient (JRC), and simulate the strength reduction factor (SRF) value on slopes that have a certain JRC. The fractal dimension of fine-grained samples = 1.0010 and coarse-grained = 1.0056. The average JRC value is 6.25 (Range 4-6). Simulation at JRC 0-20, gives different SRF values. On slopes with JRC = 0, the critical SRF = 1.35. If JRC = 20, the critical SRF = 1.47. It can be inferred, that the fractal dimension of the roughness of the sliding plane correlates with JRC and SRF. As the fractal dimension value increases, so do the JRC and SRF values, resulting in more stable slope conditions.

Macromolecular Hydrodynamics and Fractal Structures of the Lignins of Fir Wood and Oat Husks.

The topological structure of the macromolecules of lignins isolated from oat husk and fir wood was studied by means of macromolecular hydrodynamic methods. The macromolecular properties were analyzed by evaluating the intrinsic viscosity and coefficients of the translational diffusion and the sedimentation velocity of the lignins in dilute dimethylformamide solutions. The average molecular weights (MDη) and polydispersity parameters were calculated based on the results of the fractionation, as follows: Mw = 14.6 × 103, Mn = 9.0, and Mw/Mn = 1.62 for lignins from fir wood and Mw = 14.9 Mn = 13.5 and Mw/Mn = 1.1 for lignins from oat husks. The fractal analysis of the lignin macromolecules allowed us to identify the distinctive characteristics of the fractal and topological structures of these lignins. The measurements indicated that the fractal dimension (df) values of the guaiacyl-syringyl lignins from oat husks were between 1.71 and 1.85, while the df of a typical guaiacyl lignin from fir wood was ~2.3. Thus, we determined that the lignin macromolecules of oat husks belong to the diffusion-limited aggregation-type cluster-cluster class of fractals of the Meakin-Kolb type, with a predominance of characteristics common to a linear configuration. The lignins of softwood fir trees exhibited a branched topological structure, and they belong to the diffusion-limited aggregation-type particle-cluster class of fractals of the Witten-Sander type. Lignins from oat husks have the linear topology of macromolecules while the macromolecules of the lignins from fir wood can be characterized as highly branched polymers.

PENERAPAN DIMENSI FRAKTAL BOX COUNTING DAN K-MEDOIDS UNTUK DETEKSI JENIS HEWAN KINGDOM MAMALIA BERDASARKAN MOTIF TUBUH

Mammals are a group of vertebrate animals that have characteristics such as hair and mammary glands. And the kingdom of mammals is one of the kingdoms with a large number of species. The shape, color, and size of animals belonging to the mammalian kingdom can vary greatly. These differences can be distinguished by the characteristics of animals such as the distinctive shape of their body motifs. For image processing, the method that has been used to identify features in an image is edge detection. In the field of mathematics that is widely used to study the irregular shape of an object is fractal geometry. In this research, grouping of animals based on their body motifs based on fractal dimensions was carried out. 120 animal images were obtained in the mammal kingdom which will be processed through segmentation which obtains the region of animal body motifs. The region obtained will be used to determine the pattern of body motifs with Canny edge detection. Then it will be calculated using the box counting method and produce fractal dimension values ​​for the cluster stage. The results of the experimental K-Medoids Clustering method from six clusters, namely cheetahs, tigers, leopards, hyenas, giraffes, and zebras, have an accuracy of 84.16%.
 Keywords: Mammal Animal Body Motives, Box Counting, K-Medoids.

Application of Fractal Theory to the Analysis of Failure Characteristics of Low-Velocity-Impact Concrete Slabs

The fractal characteristics of low-velocity-impact concrete slabs were studied using fractal theory, and the fractal dimension value of cracks of each concrete specimen plate was calculated using box dimension as the basic principle and digital image analysis technology in conjunction with MATLAB software (R2021b) calculation functions. The energy dissipation of concrete slabs during low-velocity impact is calculated using the elastic sheet theory. The calculation results are realistic, and the energy conversion of concrete slabs during low-velocity impact is analyzed based on these results. The research findings indicate that concrete slab cracks exhibit good fractal characteristics during low-velocity impact, and their values can be utilized as a parameter to determine the extent of concrete slab failure. Moreover, the study found that the fractal dimension value of concrete slab cracks and the associated plastic deformation energy display good exponential function characteristics in the energy dissipation mechanism of low-velocity concrete slabs.

Information Fusion and Metallogenic Prognosis of Gold Deposits in the Qixia Area, Northern Shandong Province, China

Analyzing and fusing information layers of exploratory parameters is a critical step for enhancing the accuracy of identifying mineral potential zones during the reconnaissance stage of mineral exploration. The Qixia area in Shandong Province is characterized by intricate geological structures and abundant mineral resources. Numerous gold polymetallic deposits have been discovered in this region, highlighting the potential for discovering more such deposits in the ore concentration zone and its adjacent areas. In this study, we focus on the Qixia area and employ the box dimension method to analyze the fractal dimension of fault structures. We investigate the relationship between orebody occurrence and fault incidence within the mining region. Furthermore, we combine fractal analysis with Fry analysis to comprehensively predict the metallogenic potential in the area. This study reveals the fractal dimension values of fault structures, demonstrating that fault structures govern the distribution of ore bodies, with NE and NW fault structures being the primary ore-hosting features. Based on thorough analysis, we hypothesize that gold deposits in this area are generally distributed along the northeastern direction. By considering mineral distribution characteristics, this study identifies five potential metallogenic prospect areas within the study region. Capitalizing on advancements in information technology and big data, digital geology has gained prominence in prospecting and prediction. To this end, we construct a multi-information comprehensive prospecting model based on the structure-geochemical anomaly-mineralization alteration, employing the convolutional neural network (CNN) model for quantitative estimation of regional gold mineral resources. The findings validate the CNN model’s robust prediction performance in this area, leading to the determination of five prediction prospects. We observe a notable congruence between the two methods, offering significant insights for subsequent exploration endeavors in the region.

Image simulation and realism evaluation for mammography and tomosynthesis based on the detailed breast phantom

The virtual image system is a key component of virtual clinical trials (VCTs) which can be utilized to evaluate and improve medical imaging devices. The main limitation of VCTs in the Chinese female breast is the lack of detailed structure in the breast model employed in the Chinese specification for testing of quality control in mammography. In this paper, based on Chinese female breast parameters, detailed breast phantoms with different glandularity (fatty, glandular, dense) and compressed breast thicknesses (CBT) (2–7 cm) were generated. Digital mammography (DM) and digital breast tomosynthesis (DBT) projections for these phantoms were simulated with clinical system configuration. Compressed breast volumes were reconstructed through DBT projections with different angles. Power spectrum analysis and fractal dimension measurement were applied for simulated and clinical images. The results show that the average power law exponents and standard deviation (DM, DBT) for clinical and simulated images were (3.26 ± 0.29, 3.33 ± 0.42) and (3.63 ± 0.25, 3.18 ±0.36) respectively. The average fractal dimensions and standard deviation (DM, DBT) for clinical and simulated images were (2.36 ± 0.03, 2.41 ± 0.05) and (2.10 ± 0.01, 2.26 ± 0.06) respectively. We constructed two virtual imaging systems for DM and DBT and obtained the imaging data based on the three-dimensional (3D) detailed breast phantoms. These results indicate that the texture indexes of simulated images are similar to that of the clinical images, and validate that these breast phantoms are suitable for imaging in VCTs requiring realistic anatomy with the Chinese female population. Results also show that the values of power law exponents for clinical and simulated images in DM and DBT are positively correlated with the value of breast glandularity, but the values of fractal dimensions keep steady for different glandularity.

Five-year change of panoramic radiomorphometric indices and fractal dimension values in type 2 diabetes patients.

Diabetes mellitus is a chronic disease characterized by dysregulation of glucose metabolism, with characteristic long-term complications accompanied by changes in bone quality. The purpose of this study is to compare the results with a control group by performing radiomorphometric analyses on panoramic radiographs obtained 5years apart to examine changes in the mandibular bone cortex and microstructures of type 2 diabetes mellitus (T2DM) patients. Two panoramic radiographs that were taken 5years (mean 5.26 ± 0.134) apart from 52 patients with T2DM (n:26) and a control group (n:26) were used. A total of 104 images were evaluated. Analyses were done from the condyle (FD1), angulus (FD2), distal second premolar apex (FD3), and anterior to the mental foramen (FD4) for fractal dimension (FD) in the mandible. Symphysis index (SI), anterior index (AI), molar index (MI), posterior index (PI), and panoramic mandibular index (PMI) measurements were taken for cortical analysis. Three-way ANOVA, three-way robust ANOVA, two-way ANOVA, and two-way robust ANOVA tests were used for statistical analysis (p < 0.05). After a 5-year period, there was a significant decrease in all FD measures of the mandible in both T2DM and control groups (p < 0.05). This resulted in a statistical difference in the main effect of time. After a 5-year period, no significant difference in mandibular cortical measures was identified between the T2DM and control groups (p > 0.05). According to panoramic radiography, the mandibular trabecular structure deteriorated after 5years, whereas cortical values remained the same. It concluded that T2DM had no effect on these results.

In-phase matrix profile: A novel method for the detection of major depressive disorder

Background and ObjectiveMajor depressive disorder (MDD) is the leading cause of disability worldwide. Reliable detection of MDD is the basis for early and successful intervention in treating the disorder and preventing disability. We introduce a novel feature extraction method, the in-phase matrix profile (pMP), which is specifically adapted for electroencephalographic (EEG) signals. Methods: The pMP characterizes general self-similarity of an EEG signal. The method extracts overlapping one-second-long subsegments from an EEG signal segment, calculates Euclidean distances between all possible subsegment pairs, and subsequently uses the distance values, where subsegments are most in phase, to calculate pMP. The method was applied to the resting-state eyes-closed EEG data of an MDD group and age- and gender-matched healthy controls (66 subjects). Higuchi's fractal dimension (HFD) values were calculated for the same groups for comparison. Results: Both pMP and HFD values were higher in MDD. The pMP successfully distinguished MDD and control group in all 30 EEG channels. In contrast, HFD resulted in statistically significant group distinguishability in 13 (43%) channels located mainly in the central region of the head. The highest classification accuracy for pMP was 73% and for HFD 67%. Conclusion: The present article shows that pMP outperforms HFD in detecting MDD and is a promising method for future MDD studies. Significance: The pMP is a sensitive parameter-free method for detecting MDD that can be used in future studies and is a potential method to reach clinical use for diagnosing MDD.

Complexity and Heterogeneity Evaluation of Pore Structures in the Deep Marine Shale Reservoirs of the Longmaxi Formation, China

The structural evolution and sedimentary differentiation of the Sichuan Basin in China are complex, with intricate reservoir pore structures that significantly impact shale gas production. This study examines the complexity and heterogeneity of the microscopic pore structures in the deep marine shale reservoir in the Longmaxi Formation. Pore structure characterization techniques are used to compare deep and shallow–medium marine shales, and siliceous and silty shales. The results reveal the factors influencing pore structure and their impact on exploration and development. The key points are as follows: (1) The pore structure of deep siliceous shale is the most complex due to its diverse range of pore development patterns, pore types, and sizes. (2) The box dimension of full pore size is about 1.52 for deep marine shale and 1.46 for shallow–medium shale. Organic matter (OM) content, the degree of pore development, and inorganic mineral content all correlate positively with the complexity of the pore structure in deep marine shale, which affects the formation of high-quality reservoirs. (3) Lateral heterogeneity of pore structures shows strong regional variations in the study area. Heterogeneity is more pronounced in the deep marine shale than in the medium and shallow shale formations. OM mesopores significantly influence the overall heterogeneity of the shale pore system. The deep marine shale reservoir is situated in an area with strong regional variations. The pore structure of high-quality reservoirs is more complex than those of shallow–medium marine shales, displaying notable heterogeneity. Pore structures with fractal dimension values close to that of the shallow–medium formations (box dimensions within 1.5) offer promising targets for the exploration and development of deep marine shale gas.

Fractal analysis of the effects on mandibular bone of botulinum toxin therapy of the masseter muscle in patients with bruxism

We examined changes in the mandibular angle, ramus, and condylar neck of patients with bruxism after botulinum toxin-A (BTX-A) injection into the masseter muscle as calculated with fractal analysis (FA) on panoramic radiographs (PRs). We examined the PRs of 3 groups of 22 patients each (n=66) obtained upon presentation and 6 months later. One group included healthy controls without bruxism, one group included patients with untreated bruxism, and one group included patients with bruxism who had undergone BTX-A injection into the masseter muscle. We performed FA of the bilateral angle, ramus, and condylar neck of the PRs to calculate fractal dimension (FD). The FD values of the angle on the second PRs of the untreated bruxism group were significantly higher than those of the other groups (P = .026), specifically when compared to the BTX-A injection group (P = .017). The FD values in the angle and ramus of the bruxism group were significantly higher on the second PRs (P ≤..005)) Conversely, the FD values in the angle of the BTX-A injection group were significantly lower on the second PR (P = .039). Masseter muscle hyperactivity due to bruxism increases bone density in masseter muscle attachment regions. BTX-A injection restricts muscle activity, thereby chnging bone structure and decreasing FD.

Fractal dimension: analyzing its potential as a neuroimaging biomarker for brain tumor diagnosis using machine learning.

Purpose: The main purpose of this study was to comprehensively investigate the potential of fractal dimension (FD) measures in discriminating brain gliomas into low-grade glioma (LGG) and high-grade glioma (HGG) by examining tumor constituents and non-tumorous gray matter (GM) and white matter (WM) regions. Methods: Retrospective magnetic resonance imaging (MRI) data of 42 glioma patients (LGG, n = 27 and HGG, n = 15) were used in this study. Using MRI, we calculated different FD measures based on the general structure, boundary, and skeleton aspects of the tumorous and non-tumorous brain GM and WM regions. Texture features, namely, angular second moment, contrast, inverse difference moment, correlation, and entropy, were also measured in the tumorous and non-tumorous regions. The efficacy of FD features was assessed by comparing them with texture features. Statistical inference and machine learning approaches were used on the aforementioned measures to distinguish LGG and HGG patients. Results: FD measures from tumorous and non-tumorous regions were able to distinguish LGG and HGG patients. Among the 15 different FD measures, the general structure FD values of enhanced tumor regions yielded high accuracy (93%), sensitivity (97%), specificity (98%), and area under the receiver operating characteristic curve (AUC) score (98%). Non-tumorous GM skeleton FD values also yielded good accuracy (83.3%), sensitivity (100%), specificity (60%), and AUC score (80%) in classifying the tumor grades. These measures were also found to be significantly (p < 0.05) different between LGG and HGG patients. On the other hand, among the 25 texture features, enhanced tumor region features, namely, contrast, correlation, and entropy, revealed significant differences between LGG and HGG. In machine learning, the enhanced tumor region texture features yielded high accuracy, sensitivity, specificity, and AUC score. Conclusion: A comparison between texture and FD features revealed that FD analysis on different aspects of the tumorous and non-tumorous components not only distinguished LGG and HGG patients with high statistical significance and classification accuracy but also provided better insights into glioma grade classification. Therefore, FD features can serve as potential neuroimaging biomarkers for glioma.

Coupled Fixed Point and Hybrid Generalized Integral Transform Approach to Analyze Fractal Fractional Nonlinear Coupled Burgers Equation

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis.

Multiscale analysis of pore structure in clay due to freeze-thaw

The variation of pore structure due to freeze-thaw (F-T) plays a vital role in revealing the thaw settlement mechanism of soils. In this study, remolded saturated clay specimens experienced unidirectional freezing and natural thawing in an open or closed system (i.e., with or without water supply). X-ray computed tomography (CT) and mercury intrusion porosimetry (MIP) were conducted to analyze pore structural characteristics, including macropores, mesopores, and micropores, along the temperature gradient direction before and after F-T. The longitudinal pore binary images, obtained by processing CT images, were used to assess the changes in porosity, pore shape, and orientation of macropores and mesopores due to F-T. The results indicate that the total longitudinal-sectional porosity and macropore percentage increased, whereas the mesopore percentage decreased. F-T considerably influenced the shape and orientation of macropores but had minimal effects on mesopores. The most significant changes occurred in the unfrozen zone due to F-T in an open system but near the final freezing front in a closed system. Additionally, Micropore porosity, diameter, and thermal fractal dimension were analyzed based on MIP data. The results showed that micropore porosity and diameter decreased, with significant changes occurring in the unfrozen zone of the open and closed systems. The micropore porosity and diameter decreased more in the unfrozen zone and near the final freezing front after F-T in a closed system, whereas a smaller decrease occurred in the frozen zone. The thermal fractal dimension value increases after F-T, indicating that the micropore structure became complex. In general, the multiscale pore analysis demonstrated that pore structure, including macropores, mesopores, and micropores, is significantly affected by F-T in the open system compared to the closed system.

Evaluation of Renal Ischemia–Reperfusion Injury Using Optical Coherence Tomography Based on Fractal Dimension

Renal ischemia–reperfusion (IR) is inevitable in partial nephrectomy and kidney transplantation. Acute tubular necrosis (ATN) induced by renal IR causes the loss of functional units in the kidney, pathologically presenting as the nonhomogeneous distribution of normal uriniferous tubules and usually used to characterize kidney activity. Optical coherence tomography (OCT) has been proven to have the ability to image the kidney microstructure in vivo and in real time. However, a fast image analysis method is needed for clinical application purpose. In this work, a new method for assessing renal IR injury was developed using a fractal-dimension-based texture of the OCT kidney image. Thirty-five Wistar rats were divided into seven groups for renal ischemia–reperfusion: control and various ischemia-time groups. A time series of 3D OCT kidney images was obtained. We calculated the fractal dimension (FD) of OCT en face images and found that the value increased gradually and finally became stable after 90 min of reperfusion. The stable value in the long-ischemia-time group was smaller than that in the short-ischemia-time group. The FD value of the OCT kidney image was highly responsive to renal IR injury. The proposed method is promising for a fast diagnostic application.

The mandibular bone structure in children by fractal dimension and its correlation with pixel intensity values: a pilot study.

To identify a normal pattern of mandibular trabecular bone in children based on the fractal dimension (FD), and its possible correlation with pixel intensity (PI) values, to facilitate the early diagnosis of possible diseases and/or future bone alterations. The 50 panoramic images were selected and divided into two groups, according to the children's age: 8-9 (Group 1; n = 25) and 6-7 (Group 2; n = 25). For FD and PI analyses, three regions of interest (ROIs) were selected, and their mean values were evaluated for each ROI, according to each group, using the t test for independent samples and the model of generalized estimation equations (GEE). Subsequently, these mean values were correlated by the Pearson test. Comparing the groups, FD and PI did not differ from each other for any of the measured regions (p > 0.00). It was observed that in the mandible branch (ROI1), FD and PI means were 1.26 ± 0.01 and 81.0 ± 2.50, respectively. In the mandible angle (ROI2), the means were 1.21 ± 0.02 (FD) and 72.8 ± 2.13 (PI); and in the mandible, cortical (ROI3) values of FD = 1.03 ± 0.01 and PI = 91.3 ± 1.75 were obtained. There was no correlation between FD and PI in any of the analyzed ROI (r < 0.285). The FD means of ROI1 and ROI2 did not differ from each other (p = 0.053), but both were different from ROI3 (p < 0.00). All PI values differed from each other (p < 0.00). The bone trabeculate pattern in 6-9-year-old children presented FD between 1.01 and 1.29. Besides that, there was no significant correlation between FD and PI.

Evaluation of mandibular osseous structure in patients with mandibular asymmetry: a fractal analysis study

We evaluated the effect of mandibular asymmetry on the trabecular structure of the mandible as measured through fractal analysis (FA). We divided a sample of 105 patients with skeletal class I occlusion, cervical vertebral maturation CS6, and skeletal vertical dimension ratios within normal limits into right asymmetric, left asymmetric, and symmetric groups. We performed FA on 6 bilateral regions of interest on panoramic radiographs of the patients and calculated the mean fractal dimension (FD) for each region. We performed the Wilcoxon signed-rank and paired t tests to compare the significance of differences in FD between sides within each group and between groups for all regions. Statistical significance was established at P < .05. We found statistically significant differences in mean FD values between the right and left condyles of the 2 asymmetric patient groups. The FD was greater in the left condyle of patients with right asymmetry and in the right condyle of patients with left asymmetry. The right condyle had a significantly greater FD in left asymmetric patients than in right asymmetric patients. The results of FA indicate that asymmetry is characterized by an increase in the mean FD of the condyle that is responsible for the asymmetry. Fractal analysis can be a valuable diagnostic technique in distinguishing trabecular differences in the condylar areas of asymmetric individuals.

Applicability of Fractal Analysis for Quantitative Evaluation of Midpalatal Suture Maturation.

The treatment of transversal maxillary deficiency usually aims at skeletal expansion. The treatment option highly depends on the maturation stage of the midpalatal suture (MPS), which may vary between individuals at the same chronological age. Therefore, the individual determination of the MPS maturation is crucial. Our aim was to investigate the applicability of fractal analysis for the quantitative evaluation of MPS maturation. Nine experienced orthodontists were asked to evaluate the MPS maturation on 51 cone beam computed tomography (CBCT) scans according to the Angelieri classification method. Intra- and inter-examiner reliability was measured using Cohen's Kappa coefficient. The stages were agreed upon according to the results of the examiners with the highest strength of agreement. Fractal analysis was then performed on the CBCTs and the correlation between the fractal dimension values and maturation stages was then evaluated. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. The cut-off point was found at 1.073, at which the use of fractal dimension for predicting MPS maturation showed 100% sensitivity, 93.7% specificity, 9.5% false positive, 0% false negative rate. Our results provided further evidence that fractal analysis is a reliable tool to determine MPS maturation stage.

Systematic Pore Characterization of Sub-Bituminous Coal from Sohagpur Coalfield, Central India Using Gas Adsorption Coupled with X-ray Scattering and High-Resolution Imaging

Pore characterization helps to estimate the coalbed methane recovery and carbon storage potential of the reservoir. Earlier research on the characteristics of coal pores has shown that coal has high hydrocarbon storage potential in the adsorbed state, but few studies have shown the influence of chemical heterogeneities and depth on the adsorption potential of the coal. With the objective of studying the effect of chemical variation, depth, and surface roughness on gas adsorption potential, this study combines coal composition analysis and adsorption-based pore characterization of coal and shale samples coupled with high-resolution imaging and X-ray scattering measurements. Variation in pore features is correlated with varying depth and composition. A decrease in the mesopore volume and surface area is observed with an increase in the depth and total organic content and inverse behavior is observed for micropores. Scanning electron microscopy images depict the change in the pore shape from semi-spherical OM pores to elongated pores with depth, and samples with high mineral content show a dominance of inter- and intraparticle pores. Fractal dimension values estimated from SAXS are notably higher than N2-LPGA-derived values (i.e.,─DS > DN) due to the incorporation of inaccessible pores, which reflects an increase of up to 62% in SAXS estimated mesopore volume and surface area. This study will provide a better approach to understand the impact of composition, depth, and surface roughness over the gas storage potential in coal reservoirs.

Non-Linear Analyses of Fish Behaviours in Response to Aquatic Environmental Pollutants—A Review

Analysis of fish behaviour is an effective way to indirectly identify the presence of environmental pollutants that negatively affect fish life, its production and quality. Monitoring individual and collective behaviours produces large amounts of non-linear data that require tailor-suited computational methods to interpret and manage the information. Fractal dimension (FD) and entropy are two groups of such non-linear analysing methods that serve as indicators of the complexity (FD) and predictability (entropy) of the behaviours. Since behavioural complexity and predictability may be modulated by contaminants, the changes in its FD and entropy values have a clear potential to be embedded in a biological early warning system (BEWS), which may be particularly useful in Precision Fish Farming settings and to monitor wild populations. This work presents a review of the effects of a wide range of environmental contaminants, including toxic compounds, cleaning and disinfecting agents, stimulant (caffeine), anaesthetics and antibiotics, heavy metals (lead, cupper, and mercury), selenium, pesticides and persistent environmental pollutants, on the FD and entropy values of collective and individual behavioural responses of different fish species. All the revised studies demonstrate the usefulness of both FD and entropy to indicate the presence of pollutants and underline the need to consider early changes in the trend of the evolution of their values prior to them becoming significantly different from the control values, i.e., while it is still possible to identify the contaminant and preserve the health and integrity of the fish.

Fractal dimension used as a proxy to understand the spatial distribution for carlin-type gold deposits

The spatial distribution of ore-forming element grades typically exhibits strong nonlinear characteristics as a result of continuous geological processes. The trend of change in ore grade sequences is a crucial indicator of the degree of enrichment of an ore deposit. Therefore, quantitative evaluation of the spatial distribution of ore-forming elements holds great significance for understanding ore-forming processes as well as for ore exploration. In this work, based on the principles of fractal theory, a MATLAB-based program to calculate fractal dimensions was developed, and fractal analysis was conducted on the gold concentration values (g/t) of 25 drill holes in the Lannigou Gold Mining Area, Guizhou, China. Rescaled range analysis and the fractal correlation integral were utilized to obtain the spatial distribution of gold concentrations in the drill holes. Rescaled range analysis revealed that the Hurst exponent (HR/S) values vary in the range of 0.58–0.97, which reflected the persistent behavior of vertical variation in gold tonnage at the studied wells. Furthermore, the relationship between gold grade distribution and associated fluctuations in the fractal dimension derived from the correlation dimension (DC) was discussed. The DC value was found to vary between 1.01 and 1.59, indicating that high-grade gold deposits were distributed along steep fractal gradients and that fractal dimension values could be used as a reliable proxy to efficiently understand the migration pathways of auriferous fluid. Studies have shown that the Hurst exponent of the Au grade sequence exhibited a strong dependence on the grade of mineralization, with gold mineralized veins persisting throughout the studied drill holes. Thus, fractal dimension gradients can serve as an important proxy for identifying the locations of potential economic gold deposits in unexplored areas. Moreover, this approach can be employed to map the distribution patterns of other types of ore deposits.