Changes In Fractal Dimension Research Articles

Fractal characteristics of overburden fissures in shallow thick coal seam mining in loess gully areas.

The problems of water scarcity and ecological fragility are common in the loess gully area. To research the distribution and evolution of the overburden fissures and quantitatively analyze them have certain theoretical and engineering significance for realizing the evaluation of overburden damage degree and safe and green mining. This paper takes the 6102 working face of Chuancao Gedan Coal Mine as the engineering background. The development law and distribution characteristics of overburden fissures caused by the mining of shallow coal seams in the loess gully area were studied by the combination of physical similarity simulation, numerical similarity simulation and fractal theory. The results show that the fractal dimension change of the overburden fissures caused by the shallow mining of coal seam groups in the loess gully area can be divided into three stages during the mining process of the working face. Repeated mining causes the activation and development of overburden fissures, the fractal dimension increases significantly, and the regularity of changes weakens. The magnitude of the stress near the working face and the fluctuation times of the stress in the goaf have an influence on the change of the fractal dimension of the overburden fissures. According to the development angle and the fractal dimension of the overburden fissures, the overburden rock above the goaf is divided into the collapse fissure area, the compaction fissure area, and the vertical fissure area. Overburden fissures develop violently in the vertical fissure area, the overburden fissures in the compaction fissure area are mostly transverse fissures, and the overburden fissures in the caving fissure area are irregular.

Fractal Seismicity and Coulomb Stress Pattern Analysis for 7 January 2020 Mw 6.4 Puerto Rico Earthquake

Abstract This study aims to understand the variation of the correlation fractal dimension and static stress changes associated with the 2019 Puerto Rico sequence with special reference to the January 2020 Mw 6.4 mainshock. On examining the seismicity data from 1988 to 2020, we find a shift in the seismicity pattern after the 2019 Puerto Rico sequence. A detailed examination of changes in the correlation fractal dimension (Dc) indicates diffused seismicity throughout the study region until the year 2003 that is demonstrated by relatively high Dc values. Toward the end of 2019, a drastic drop in Dc is observed; it is associated with a tight clustering of events to the south-southwest of Puerto Rico Island (PRI), suggesting the onset of the 2019 sequence. We observed a persistent clustering and significant drop in Dc before the Mw 6.4 earthquake, hinting at the presence of a numerical precursor for this event. Over 30 yr of historical earthquakes have contributed positive Coulomb stress distributions mainly in the northern region of PRI. A deflection in this pattern was reflected after the 2019 Puerto Rico sequence. Based on the static stress analysis, we demonstrated that the Mw 6.4 mainshock was triggered by the 2019 Puerto Rico sequence. The new cluster of events, which contains the Mw 6.4 mainshock, exhibits higher stress levels in the south-southwest region of PRI. The eastern end of Septentrional fault (SF), the zone between SF and North Puerto Rico Slope fault, the middle segment of South Puerto Rico Slope fault, South Lajas fault, Punta Montalva fault, and Central fault are the highly stressed regions at present. Stress perturbation induced by the destructive Mw 6.4 event on the adjacent fault system is a potential threat for PRI, where population density is very high. The combined analysis of Dc and static stress changes indicates the presence of a numerical precursor and highly stressed regions southwest of PRI.

Urban growth: Modelling street network growth in Manhattan (1642–2008) and Barcelona (1260–2008)

In this paper, we argue for the case that cities are self-organised complex systems by presenting evidence on positive and reinforcing feedback mechanisms and robust global trends that characterise historical growth patterns. In two case studies; Manhattan and Barcelona, historical stages of urban growth were mapped and analysed. The analyses revealed regularities that may help define the local and global processes that characterise urban growth marked by alternating periods of expansion and pruning in street networks. The global trend marked by a lognormal distribution of segmental integration (closeness) in street networks was consistently restored following planning interventions. The overall street network growth trend appeared to fit an exponential or power law distribution, along with a linear change in fractal dimension. Underlying these global trends, we found evidence for local positive and reinforcing feedback mechanisms; explained by preferential attachment to well-connected street structures, and pruning of weakly integrated local street structures. The findings are likely to improve our understanding of urban growth.

Porous Blasting Morphology and Resistivity Response Characteristics of Soft Coal Seam

The fracturing characteristics of multi-holes blasting in soft coal seam were analyzed quantitatively by using the multi-holes model blasting experiment of soft coal seam and digital image processing (DIP) and electrical resistivity tomography (ERT) techniques, and the electrical resistivity response rules before and after blasting were obtained. The results show the following: (1) At the same time as the formation of blasting-induced cracks, the coal inside the control hole presents two kinds of displacement phenomena: crack and noncrack, partially offsetting the extrusion effect caused by the expansion of the blasting hole. (2) The comprehensive value (S) of blasting-induced cracks, control hole, and blasting hole area change is proposed to analyze the blasting effect. When single-hole blasting is performed, S < 0, the fractal dimension change rate of coal seam crack is only 5%, the control area presents a compaction phenomenon, and the resistivity decreases by about 20%. When multi-hole blasting is performed, the blasting cracks are abundant and the coal displaces sufficiently. The fractal dimension of cracks reaches 21.28%, the negative effect of the blast holes cavity is offset (S > 0), and the resistivity in the controlled area generally increases by 1 to 30 times. (3) Compared with the single-hole blasting model, the multi-hole blasting is more in line with the coal mine site, and can effectively increase the permeability in the control area, and the resistivity generally increases.

Fractality of Borsa Istanbul during the COVID-19 Pandemic

Forecasting price changes is very important for the process of estimating and managing market risk in financial markets. Price changes in financial markets may also depend on non-market factors. Considering this situation, the study investigates the effect of the COVID-19 pandemic on Borsa Istanbul. It tackles changes in the fractal dimensions of the time series obtained with the daily closing prices of stocks traded on Borsa Istanbul (BIST). According to the results of the sector-based analysis, we found that fractal dimension changes were quite effective in price estimation.

Effects of chemical solvents on coal pore structural and fractal characteristics: An experimental investigation

Chemical solvent treatment is one of the most effective means to improve pore structure of low-permeability coal and increase production of coalbed methane. Low-temperature liquid nitrogen adsorption method, X-ray diffraction (XRD) mineral analysis and fractal theory were adopted to study variations in coal pore structural parameters and fractal characteristics affected by hydrochloric acid (HCl), tetrahydrofuran (THF) and carbon disulfide (CS2). Liquid nitrogen adsorption results of coal samples were analyzed using density functional theory (DFT), Brunauer-Emmett-Teller (BET) equation and Barrett-Johner-Halenda (BJH) model. Meanwhile, changes in fractal dimensions of pore surface (D1) and pore structure (D2) were calculated based on Frenkel-Halsey-Hill (FHH) model. Relationship between pore volume, average pore diameter, specific surface area and two fractal dimensions were established. Results show that: (1) Hysteresis lines exist in all isothermal adsorption-desorption processes of coal. Pores of raw coal are mainly semi-open pores with poor connectivity. The types of pores do not change significantly after THF and CS2 treatment, while the number of pores rise. In contrast, affected by HCl treatment, ink-bottle pores and open pores increase and pore fracture network connectivity becomes better. (2) Pore sizes of treated coal mainly distribute in range of 0–10 nm. After HCl treatment, many closed pores are opened, with growth in proportion of micropores. Meanwhile, specific surface area (SSA) of pores increases by 55.21% and 31.36%, respectively. THF extraction results in an increase in pore volume and SSA. After CS2 extraction, dissolution of organic material increases pore size and further improve volume of transition pores and mesopores. (3) HCl treatment causes decrease in D1 value of pores. However, in organic-solvent-treatment cases, solvent residues retain on coal pores, resulting in growth in D1 value and pore surface becoming rougher. Proportion of total pore volume of transition pores and mesopores of acidized coal decreases and D2 value rises. Pore distribution is more uneven. Relevant results could provide theoretical references for revealing mechanism of chemical solvents effects on gas flow in coal.

Complexity Analysis in the PR, QT, RR and ST Segments of ECG for Early Assessment of Severity in Cardiac Autonomic Neuropathy

Early-stage detection of cardiac autonomic neuropathy (CAN) is important for better management of the disease and prevents hospitalization. This study has investigated the complex nature of PR, QT, RR, and ST time segments of ECG signals by computing the fractal dimension (FD) of all segments from 20 min ECG recordings of people with different severity of the disease and healthy individuals. The mean computed for each ECG time segment to distinguish between subjects was insufficient for an early diagnosis. Statistical analysis shows that the change of FD in various time segments of ECG throughout the recording was most suitable to assess the steps for severity in symptoms of CAN between the healthy and the subjects with early symptoms of CAN. The complexity of ECG features was evaluated using various classifier models, namely, support vector machine (SVM), naïve Bayes, random forest, K-nearest neighbor (KNN), AdaBoost, and neural networks. Performance measures were computed on all models, with a maximum neural network classifier having an accuracy of 96.9%. Feature ranking results show that fractal features have more significance than the time segments of ECG in differentiating the subjects. The results of statistical validation show that all the selected features based on ECG physiology proved to have an evident complexity change between normal and severity stages of CAN. Thus, this work reports the complexity analysis in all the selected time segments of ECG that can be an effective tool for early diagnostics for CAN.

The Influence of Pore Structure of the Core-Scale Fracture-Controlled Matrix Unit on Imbibition: Model Construction and Definition of the Fractal Coefficient

Abstract Imbibition oil recovery plays an important role in the development of low-permeability fractured reservoirs. However, the effects of the complex pore structure of the matrix on imbibition have not been considered comprehensively at different scales in the scientific literature. This paper reports a study of the mechanisms of influence of different matrix pores on imbibition recovery ratio and defines the concept of a fractal coefficient using the method of combining numerical simulation at the core scale with mathematical methods at a pore scale. A matrix model with different pore fractal dimensions and tortuosity of the capillary was established using Python language programming. Y-type and S-type were used to characterize the fractures, and a two-dimensional fracture-controlled matrix unit core-scale numerical model with complex pore structures was established. Based on phase field theory, an oil-water two-phase imbibition numerical simulation was conducted. Comparisons of numerical simulation results and microscopic analysis indicated that imbibition recovery ratio was 39.28% and 50.94%, respectively. The imbibition law revealed by the numerical simulation results is consistent with the results of the microscopic imbibition experiment, and the imbibition effect of bifurcated fractures was better than that of simple curved fractures. As the pore fractal dimension and tortuosity of the capillary increased, the imbibition recovery ratio decreased. A comparison of results of the mathematical model demonstrated that there was a difference between the pore scale and core scale because the pore fractal dimension and capillary tortuosity changed dynamically with pore structure at the core scale. Thus, a parameter that characterizes the relative change of pore fractal dimension and tortuosity was defined and called the fractal coefficient. When the skeleton particles are 200, 400, and 500, the fractal coefficient were calculated to be 0.625 and 0.6, respectively, indicating that when the pore structure changes, the imbibition recovery ratio should be dominated by capillary tortuosity.

Changes in fractal dimension and durability of ultra-high performance concrete (UHPC) with silica fume content

Changes in fractal dimension and durability of ultra-high performance concrete (UHPC) with silica fume content

Image analysis-based quantification of the visual attributes of fish, with emphasis on color and visual texture

Abstract Images of Red Sea goatfish Parupeneus forsskali were taken in a light box to perform color and visual texture analyses. The average L* and b* values did not change significantly during storage of 7 days, but the a* values decreased (P < 0.05). The change of visual texture parameters energy and entropy (calculated based on histograms, and based on co-occurrence matrices [COM]), box counting-based fractal results, and texture change index (TCI) values are presented. The appearance of fish became “smoother” over time. The entropy values calculated by histograms decreased with storage (P < 0.05), and the maximum range was 0.395. That for COM-based entropies was 71.96. TCI also decreased with storage (P < 0.05) with a maximum range of 10.67. However, energy values increased during storage. The maximum range of the energy values calculated by histograms over time for any color channel was 0.0036. That for COM-based energies was 5.7. There was no observable change in fractal dimension. These image analysis-based parameters were compared with sensory analysis. A trained sensory panel evaluated the visual texture of a sub-set of images. The R2 values for equation fit between sensory score and texture features were, in increasing order: COM based energy (0.185), COM-based entropy (0.313), histogram-based energy (0.375), histogram-based entropy (0.386), TCI values (0.575). Since TCI correlated better with sensory values, it is recommended to be used in this type of visual texture evaluation.

Dynamic Response and Damage Regularity of Sandstone with Different Moisture States under Cyclic Loading

In the process of geotechnical engineering excavation, wet and water-filled rock masses are inevitable. To obtain the mechanical properties of these rocks, indoor tests are required, and most of the rock tests rock tests are dry or nearly dry. They cannot really reflect the true nature of the rock, let alone its nature under a dynamic load. The rock was repeatedly impacted during the blasting excavation process. To determine the mechanical response characteristics and damage evolution of rocks with different moisture states under cyclic dynamic loads, rock samples with three saturation levels were prepared. In the experiment, the Hopkinson pressure bar equipment was utilized to perform five cycles of impact with the same incident energy, and the dynamic response of rocks with different impact times was recorded. Nuclear magnetic resonance technology was employed to obtain the change law of the pores of rock specimens after impact, and the cumulative damage rules of rock were combined with the fractal theory. From the experiments, it can be observed that the stress-strain curves of all rock samples are similar, in that they all have stress addition and unloading stages. The peak stress is proportional to the impact time and moisture content, whereas the opposite is true for the peak strain. After the impact, the small and large pores closed and increased, respectively. The porosity and porosity change rate increased with an increase in the impact time. With an increase in moisture content, this trend is more obvious. It can be observed via magnetic resonance imaging that the internal fractures of the water-bearing rock are obvious after multiple impacts. In particular, the saturated rock specimens exhibited severe damage. Fractal analysis of the NMR figures revealed that after three impact times, the fractal dimension change in the water-bearing rock samples was not obvious. This phenomenon indicated that a macro gap appeared. The fractal dimensions of the dry rock samples continued to increase, and the internal damage was less obvious.

Longitudinal study of the effect of a 5-year exercise intervention on structural brain complexity in older adults. A Generation 100 substudy

Physical inactivity has been identified as an important risk factor for dementia. High levels of cardiorespiratory fitness (CRF) have been shown to reduce the risk of dementia. However, the mechanism by which exercise affects brain health is still debated. Fractal dimension (FD) is an index that quantifies the structural complexity of the brain. The purpose of this study was to investigate the effects of a 5-year exercise intervention on the structural complexity of the brain, measured through the FD, in a subset of 105 healthy older adults participating in the randomized controlled trial Generation 100 Study. The subjects were randomized into control, moderate intensity continuous training, and high intensity interval training groups. Both brain MRI and CRF were acquired at baseline and at 1-, 3- and 5-years follow-ups. Cortical thickness and volume data were extracted with FreeSurfer, and FD of the cortical lobes, cerebral and cerebellar gray and white matter were computed. CRF was measured as peak oxygen uptake (VO2peak) using ergospirometry during graded maximal exercise testing. Linear mixed models were used to investigate exercise group differences and possible CRF effects on the brain's structural complexity. Associations between change over time in CRF and FD were performed if there was a significant association between CRF and FD. There were no effects of group membership on the structural complexity. However, we found a positive association between CRF and the cerebral gray matter FD (p < 0.001) and the temporal lobe gray matter FD (p < 0.001). This effect was not present for cortical thickness, suggesting that FD is a more sensitive index of structural changes. The change over time in CRF was associated with the change in temporal lobe gray matter FD from baseline to 5-year follow-up (p < 0.05). No association of the change was found between CRF and cerebral gray matter FD. These results demonstrated that entering old age with high and preserved CRF levels protected against loss of structural complexity in areas sensitive to aging and age-related pathology.

Evaluation of change in trabecular bone structure surrounding dental implants by fractal dimension analysis and comparison with radiomorphometric indicators: a retrospective study.

BackgroundThe trabecular bone surrounding dental implant show some structural changes during healing period. The purpose of this study is to observe the change in trabecular bone with fractal dimensional analysis from baseline to 3rd month of implant placement. It was also aimed to determine the correlation of fractal dimension (FD) change with morphometric indices, mandibular cortical index (MCI) and mandibular cortical width (MCW).MethodsDigital panoramic radiographs taken from 14 patients were evaluated in this study. A total of 30 implants which were placed on mandibular premolar or molar region were included. MCI and MCW assessments were made on baseline radiograph and FD were analyzed on baseline and 3rd month radiograph. FD change by time was recorded separately for every ROI. A paired sample t-test was used to evaluate the change between FD-baseline and FD-3rdmonth values. One-way ANOVA was used to determine the relationship between FD change and MCI. FD change and MCW measurements correlation was calculated by Pearson correlation analysisResultsFD value increased in 75 of 90 implant-adjacent regions evaluated in the 3rd month and it was observed that the FD values were statistically significantly increased in the mesial, distal and apical regions at the 3rd month measurements. There was no statistically significant relationship between mean FD change and MCI, and there was no correlation between mean FD change and MCW value.DiscussionThere are many studies evaluating the resorptive changes in bone. However, there are few studies investigating whether there is a correlation between pre- and post-operative trabecular pattern with MCI and MCW. The results of this study indicate that the implants have an improving effect on bone trabeculation in the region where they are placed.

Comparison of cone-beam computed tomography and digital panoramic radiography for detecting peri-implant alveolar bone changes using trabecular micro-structure analysis.

ObjectivesWe compared changes in fractal dimension (FD) and grayscale value (GSV) of peri-implant alveolar bone on digital panoramic radiography (DPR) and cone-beam computed tomography (CBCT) immediately after implant surgery and 12 months postoperative.Materials and MethodsIn this retrospective study, 16 patients who received posterior mandibular area dental implants with CBCT scans taken about 2 weeks after implantation and one year after implantation were analyzed. A region of interest was selected for each patient. FDs and GSVs were evaluated immediately after implant surgery and at 12-month follow-up to examine the functional loading of the implants.ResultsThere were no significant differences between DPR and CBCT measurements of FD values (P>0.05). No significant differences were observed between FD values and GSVs calculated after implant surgery and at the 12-month follow-up (P>0.05). GSVs were not correlated with FD values (P>0.05).ConclusionThe DPR and reconstructed panoramic CBCT images exhibit similar image quality for the assessment of FD. There were no changes in FD values or GSVs of the peri-implant trabecular bone structure at the 12-month postoperative evaluation of the functional loading of the implant in comparison to values immediately after implantation. GSVs representing bone mass do not align with FD values that predict bone microstructural parameters. Therefore, GSVs and FDs should be considered different parameters for assessing bone quality.

Spatiotemporal Patterns and Driving Force of Urbanization and Its Impact on Urban Ecology

Urbanization inevitably poses a threat to urban ecology by altering its external structure and internal attributes. Nighttime light (NTL) has become increasingly extensive and practical, offering a special perspective on the world in revealing urbanization. In this study, we applied the Normalized Impervious Surface Index (NISI) constructed by NTL and MODIS NDVI to examine the urbanization process in the Yangtze River Delta (YRD). Geographical detectors combined with factors involving human and natural influences were utilized to investigate the drive mechanism. Urban ecology stress was evaluated based on changes in urban morphological patterns and fractional vegetation cover (FVC). The results showed that the NISI can largely overcome the obstacle of directly coupling NTL data in performing urbanization and has efficient applicability in the long-term pixel scale. Built-up areas in the YRD increased by 2.83 times during the past two decades, from 2053.5 to 7872.5 km2. Urbanization intensity has saturated the city center and is spilling over into the suburbs, which show a “cold to hot” spatial clustering distribution. Economic factors are the primary forces driving urbanization, and road network density is becoming essential as factor that reflects urban infrastructure. Urban geometry pattern changes in fractal dimension (FD) and compactness revealed the ecological stress from changing urban external structure, and internal ecological stress was clear from the negative effect on 63.4% FVC. This impact gradually increased in urban expanded area and synchronously decreased when urbanization saturated the core area. An analysis of ecological stress caused by urbanization from changing physical structure and social attributes can provide evidence for urban management and coordinated development.

Evaluation of architectural changes in mandibular trabecular and cortical bone pattern after functional treatment

Abstract Purpose: To assess the changes in mandibular trabecular and cortical bone following functional appliance treatment through fractal dimension (FD) and cortical thickness (CorT) measurements. Material and methods The study was performed on 45 mandibular retrognathia cases. While the treatment group (T) comprised 30 patients treated using a functional appliance for one year during puberty, the control group (C) consisted of 15 patients who did not receive any orthodontic treatment due to insufficient oral hygiene. FD and CorT analyses were bilaterally evaluated for each mandible. FD measurements were performed on the condylar process, the antegonial notch, and ramus regions on panoramic radiographs. A paired t-test was applied for group analysis, before (T1) and after treatment (T2) and, correspondingly, before (C1) and after (C2) controls. In addition, an independent t-test was used to determine differences between the treatment (∆T = T2−T1) and the control group (∆C = C2−C1). Results Statistically significant differences were found in the treatment group between the initial and post-treatment FD values of the right (p &lt; 0.05) and left condyles (p &lt; 0.05), and the CorT measurements on the right (p &lt; 0.01) and the left sides (p &lt; 0.05). There was no statistically significant difference in FD parameters in the control group and a significant difference was only found in the left CorT (p &lt; 0.05). A comparison of the treatment and control groups revealed that the changes in FD and CorT measurements were not statistically significantly different between the groups (p &gt; 0.05). Conclusion: Mandibular protraction appliances do not produce trabecular bone alteration in the mandibular condyles, the antegonial notch and ramus over a 12-month time period.

Fractal Characteristic Analysis of Urban Land-Cover Spatial Patterns with Spatiotemporal Remote Sensing Images in Shenzhen City (1988–2015)

Understanding the urban land-cover spatial patterns is of particular significance for sustainable development planning. Due to the nonlinear characteristics related to the spatial pattern for land cover, it is essential to provide a new analysis method to analyze them across remote sensing imagery. This paper is devoted to exploring the fractals and fractal dimension properties of land-cover spatial patterns in Shenzhen city, China. Land-cover information was extracted using a supervised classification method with ArcGIS technology from cloud-free Landsat TM/ETM+/OLI imagery, covering 1988–2015. The box-counting method and the least squares regression method are combined to estimate fractal dimensions of the land-cover spatial pattern. The information entropy was used to verify our fractal dimension results. The results show the fractal dimension changes for each land cover type from 1988 to 2015: (1) the land-cover spatial form of Shenzhen city has a clear fractal structure, but fractal dimension values vary in different land cover types; (2) the fractal dimension of build-up land increases and reaches a stable value, while grassland and cultivated land decrease; The fractal structure of grassland and bare land showed a bifractals trend increasing year by year; (3) the information entropy dimension growth is approaching its maximum capacity before 2011. We integrated the information entropy index and fractal dimension to analyze the complexity in land-cover spatial evolution from space-filling, space balance, and space complexity. It can be concluded that driven by policies, the land-cover spatial form in Shenzhen experienced a process from a hierarchical spatial structure with a low evolution intensity to a higher evolution intensity with multiscale differential development. The fractal dimension has been becoming better through self-organization, and its land resources are reaching the growth limits.

Changes of cerebral cortical structure and cognitive dysfunction in \u201chealthy hemisphere\u201d after stroke: a study about cortical complexity and sulcus patterns in bilateral ischemic adult moyamoya disease

BackgroundMoyamoya disease (MMD) is an uncommon cerebrovascular disease which leads to progressive stenosis and occlusion of the bilateral internal carotid artery and main intracerebral arteries. Concerns are always on how the hemisphere with infarction affects cognitive function, while little attention is paid to the role that the non-infarcted hemisphere plays. Therefore, we aimed to detect cortical indexes, especially cortical complexity in the left or right hemisphere separately in patients with MMD after stroke.Methods28 patients with MMD (14 males, 14 females) and 14 healthy controls were included in this study. All participants underwent cognitive tests and magnetic resonance imaging (MRI) scan. The preprocessing of three-dimensional T1 weighted images were performed by standard surface-based morphometry. Surface-based morphometry statistical analysis was carried out with a threshold of False Discovery Rate (FDR) P < 0.05 and fractal dimension (FD) was used to provide a quantitative description of cerebral cortical complexity.ResultsWidespread cognitive dysfunctions were found in MMD patient with stroke. Extensive FD reduction in the left hemisphere with right-sided infarction, mainly in the superior temporal, inferior frontal, and insula, while the post central gyrus, superior parietal, and inferior parietal gyrus also showed a wide range of significant differences (FDR corrected P < 0.05). Meanwhile, FD changes in the right hemisphere with left-sided infarction are restricted to the precuneus and cingulate isthmus (FDR corrected P < 0.05).ConclusionsExtensive cognitive impairment was reconfirmed in Moyamoya disease with stroke, while wild and asymmetrical decrease of cortical complexity is observed on both sides. These differences could be relative to unbalanced cognitive dysfunction, and may be the result of a long-term chronic ischemia and compensatory of the contralateral hemisphere to the infarction.

Differential Entropy: An Appropriate Analysis to Interpret the Shape Complexity of Self-Similar Organic Islands.

Differential entropy, along with fractal dimension, is herein employed to describe and interpret the shape complexity of self-similar organic islands. The islands are imaged with in situ Atomic Force Microscopy, following, step-by-step, the evolution of their shape while deposition proceeds. The fractal dimension shows a linear correlation with the film thickness, whereas the differential entropy presents an exponential plateau. Plotting differential entropy versus fractal dimension, a linear correlation can be found. This analysis enables one to discern the 6T growth on different surfaces, i.e., native SiOx or 6T layer, and suggests a more comprehensive interpretation of the shape evolution. Changes in fractal dimension reflect rougher variations of the island contour, whereas changes in differential entropy correlates with finer contour details. The computation of differential entropy therefore helps to obtain more physical information on the island shape dependence on the substrate, beyond the standard description obtained with the fractal dimension.

Application of the Fractal Dimension Calculation Technique to Determine the Shape of Selected Monchepluton Intrusion Crystals (NE Fennoscandia)

Thirty-eight samples of minerals from Paleoproterozoic Layered PGE Intrusion Monchepluton, located in NE Fennoscandia, were tested. An automated computational technique was used which involved counting the sides superimposed on vectorized graphics using separated crystal boundaries. The results were obtained for olivine, orthopyroxene, clinopyroxene, and plagioclase. On this basis, an interpretation of the mineral box fractal dimension was made, along with an interpretation of its nature in the rock in which it was found. The performed calculations were applied to the sampling positions, and maps of changes in fractal dimensions were prepared. The nature of the minerals studied was correlated with the type of rocks present in Monchepluton. Then, the fractal dimensions were scaled to a percentage scale determining the mean value for the euhedral crystal as 100%, and a map was obtained representing the advancement of secondary processes after summing these data for all four investigated minerals. This method was analyzed and its advantages and limitations shown.