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The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.

AimsThe fractal dimension and fractal abundance are important root architecture characteristics that can reflect the changes in the morphology of root systems in response to habitats. However, there are few studies on the root architecture of inland salt marsh plants from the perspective of fractal geometry.MethodsThe study was run on three sites representing a gradient of flooding intensity and salinity: (i) high-flooding and low-salinity area, (ii) intermediate-flooding and intermediate-salinity area, and (iii) low-flooding and high-salinity area. Saussurea salsa was the research object, and the responses of root geometric morphology traits and fractal structure to heterogeneous habitats were studied in the Sugan Lake wetland.ResultsThe results showed that there was a significant negative correlation (p < 0.01) between the fractal dimension and fractal abundance of Saussurea salsa in different habitats. Moreover, the rate of fractal abundance increase was greater than the rate of fractal dimension decrease from the high-flooding, low-salinity area to the low-flooding, high-salinity area.ConclusionsWith the gradient of environment change, Saussurea salsa tended to form a root architecture with a high fractal dimension and low fractal abundance in the high-flooding, low-salinity area, whereas a root architecture with a low fractal dimension and high fractal abundance tended to form in the low-flooding, high-salinity area. The roots of Saussurea salsa showed fewer branches and were more inclined to show an increase in root expansion capacity in the low-flooding, high-salinity area.

The fractal dimension (FD) is a classical nonlinear dynamic index that can effectively reflect the dynamic transformation of a signal. However, FD can only reflect signal information of a single scale in the whole frequency band. To solve this problem, we combine refined composite multi-scale processing with FD and propose the refined composite multi-scale FD (RCMFD), which can reflect the information of signals at a multi-scale. Furthermore, hierarchical RCMFD (HRCMFD) is proposed by introducing hierarchical analysis, which successfully represents the multi-scale information of signals in each sub-frequency band. Moreover, two ship-radiated noise (SRN) multi-feature extraction methods based on RCMFD and HRCMFD are proposed. The simulation results indicate that RCMFD and HRCMFD can effectively discriminate different simulated signals. The experimental results show that the proposed two-feature extraction methods are more effective for distinguishing six types of SRN than other feature-extraction methods. The HRCMFD-based multi-feature extraction method has the best performance, and the recognition rate reaches 99.7% under the combination of five features.

The pore structure characteristics of soil are closely related to soil engineering properties. For saline soil distributed in seasonally frozen areas, existing studies have focused on the influence of freeze–thaw cycles on pore structure, while the influence of soluble salt in the soil is not well understood. This study aims to explore the influence of salt content and salt type on the pore structure of freeze-thawed soil. Soil samples with different salt contents (0–2%) and types (bicarbonate salt and sulfate salt) were subjected to 10 freeze–thaw tests, and their pore size distributions (PSDs) were obtained by mercury intrusion porosimetry tests. In addition, the PSDs were quantitatively analyzed by fractal theory. For both salts, the PSDs of the tested soil samples were bimodal after the freeze–thaw cycles, and the porosity of saline soil samples increased with increasing salt content overall. However, the contents of various types of pores in soil samples with two salt types were quite different. The variation in bicarbonate salt content mainly affected the mesopore and macropore contents in the soil samples, and their change trends were opposite to each other. For soil samples with sulfate salt, the porosity and macropore content increased significantly when the salt content exceeded 1%. In addition, the pore structures in saline soil presented fractal characteristics after the freeze–thaw cycles, and the fractal dimension was positively correlated with macropore content. This study may provide references for understanding the engineering properties of saline soil in seasonally frozen areas at the microscale.

The most economical, environmental, and friendly method for recycling gangue is filling mining with cemented waste rock backfill (CWRB), which solves the environmental problems caused by gangue discharge and reduces the mining damages. Evaluating the mesoscopic structure of CWRB is of great significance for maximizing the utilization of gangue recycling and improving the economic benefits of filling mining. This paper constructed the particle flow model of cemented waste rock backfill (CWRB) considering particle size distribution (PSD) of aggregates and hydration of cementing material to investigate the effect of the PSD of aggregates on its mesoscopic structural evolution. The strain energy, crack, force chain, and particle fragment of CWRB during the whole loading were discussed. The binary processing and calculation on the crack image were performed to analyze the fractal dimension of crack distribution by compiling program. The influencing mechanism of the PSD of aggregates on the strength of CWRB is revealed from the mesoscopic levels of crack evolution, force chain structure, and particle fragment. The results show that the strain energy increases firstly and then decreases with the PSD fractal dimension, while the crack number decreases firstly and then increases with that. The cracks with less number and more uniform distribution present the smaller fractal dimension, CWRB with a low fractal dimension of crack distribution has higher strength, the fractal dimension of crack distribution exhibits a correlation with the PSD of aggregates. CWRBs with the PSD fractal dimensions of 2.4–2.6 have the largest strain energy and the smallest crack number, performing the superior structural evolution during loading. This study presents the huge potential of optimizing PSD in CWRB application from a new perspective, it is of great significance for strengthening the internal structure of CWRB and reducing engineering cost.

The evolution of form and function of trees of diverse species has taken place over hundreds of millions of years, while urban environments are relatively new on an evolutionary time scale, representing a novel set of environmental constraints for trees to respond to. It is important to understand how trees of different species, planted in these anthropogenically-structured urban ecosystems, are responding to them. Many theories have been advanced to understand tree form and function, including several that suggest the fractal-like geometry of trees is a direct reflection of inherent and plastic morphological and physiological traits that govern tree growth and survival. In this research, we analyzed the “fractal dimension” of thousands of tree crowns of many different tree species, growing in different urban environments across the United States, to learn more about the nature of trees and their responses to urban environments at different scales. Our results provide new insights regarding how tree crown fractal dimension relates to balances between hydraulic- and light-capture-related functions (e.g., drought and shade tolerance). Our findings indicate that trees exhibit reduced crown fractal dimension primarily to reduce water loss in hotter cities. More specifically, the intrinsic drought tolerance of the studied species arises from lower surface to volume ratios at both whole-crown and leaf scales, preadapting them to drought stress in urban ecosystems. Needle-leaved species showed a clear trade-off between optimizing the fractal dimension of their crowns for drought vs. shade tolerance. Broad-leaved species showed a fractal crown architecture that responded principally to inherent drought tolerance. Adjusting for the temperature of cities and intrinsic species effects, the fractal dimension of tree crowns was lower in more heavily urbanized areas (with greater paved area or buildings) and due to crowns conflicting with utility wires. With expectations for more urbanization and generally hotter future climates, worldwide, our results add new insights into the physiological ecology of trees in urban environments, which may help humans to provide more hospitable habitats for trees in urbanized areas and to make better decisions about tree selection in urban forest management.

The evolution of fractures in overburden is quantitatively investigated to characterize the effect of mining activity. Scale model testing and numerical modeling were used based on the engineering geological and mining environments of Panel 11050 in the Quandian Coalmine in Henan Province, China. The maximum vertical displacement is 62.76 m, which is 140 m from the initial mining in the scale model test. Based on the fractal geometric theory, the fractal dimensions of the fractures in the overburden are calculated and visualized. The results reveal that if two coal seams are mined at the same time, the fractal dimension of the fracture network in the overburden increase with the progression of mining, but the rate of increase gradually slows. The relationships between the fractal dimension and the maximum height of the overburden failure and maximum overburden subsidence are nonlinear. The structural characteristics of the overburden are represented by the network of fractures. With increasing distance from the coal seam roof, the mining stress is gradually transferred upward to the overlying strata, and the scale of this stress transfer gradually reduces. The variation in the vertical stress gradually weakens and shows a delayed change with the mining process. The maximum principal stress is compressive stress and is distributed in an “arch” shape. The stress on both sides of the “arch” is high, and the intermediate stress is low. The stress within the “arch” shows an opposite trend.

Despite the intensity with which ecological information involving habitat complexity has been amassed to date, much remains to be revealed for a comprehensive understanding of the mechanics and implications of the structural complexity of habitats and its influences on ecological communities. This review examines the multi-faceted characteristics of habitat complexity, focusing in particular on aquatic ecosystems. Habitat complexity in aquatic systems is characterised by at least five different traits of physical structure: (1) spatial scales, (2) diversity of complexity-generating physical (structural) elements, (3) spatial arrangement of elements, (4) sizes of elements, (5) abundance/density of elements. Of these five traits, the concept of fractal dimension fully encompasses only the last one; in this sense, habitat complexity is more complex than what fractal measures represent. It is therefore important to investigate exactly which traits of habitat structure are exerting influences on organisms/communities. We hypothesise that, where an entire range of possible fractal dimension D is considered, intermediate levels of D are most likely to be associated with the highest level of biodiversity, to which the body size spectra of assemblages would have a close bearing. In most aquatic ecosystems, broadly two-dimensional structures of bottom substrate at the scale of 1–10 m mean that the addition of vertical, three dimensional structures almost always implies an increase in both the ‘diversity’ and ‘abundance’ components of structural elements, resulting in more habitats being made available to organisms of different sizes and functional designs. The conservation and management of aquatic ecosystems would be facilitated by rigorous assessments of linkages between habitat complexity and aquatic communities, for which an integrative approach to habitat complexity seems to offer a useful and versatile framework.

The pore structure of coal has a great influence on the exploitation of coalbed methane. The primary coal and tectonic coal of the Faer Coal Mine and Wenjiaba Coal Mine in Guizhou, China, were selected for low-temperature liquid nitrogen and high-pressure mercury injection experiments. Based on the Frenkel–Halsey–Hill fractal model and multifractal model, combined with nano-CT system, the difference between primary structural coal and tectonic coal and their relationships with fractal dimension were analyzed. According to the results, coal can increase its pore volume and surface area by tectonic action. From the perspective of pore volume and surface area, micropores occupy the main part. According to the pore volume and pore surface area of micropores, that of primary structural coal was lower than that of tectonic coal. For the studied coal samples, their pores have fractal characteristics and complex structure. It can be seen from the fractional dimension that that of tectonic coal was higher than that of primary structure coal, indicating that the microscopic pore structure of coal tends to be complex after tectonic action. According to the results of multifractal and nano-CT analysis, the Hausdorff dimension of tectonic coal was slightly smaller than that of primary structure coal, but the multifractal parameter value was higher than that of primary structure coal. After tectonic action, the connected pores of coal increased by 142.86% compared with the connected pores of primary coal, and the adsorption capacity was further improved. It is further proved that the structure can affect the adsorption and desorption capacity by changing the distribution of pore structure.

Background The chlorophyll content is a vital indicator for reflecting the photosynthesis ability of plants and it plays a significant role in monitoring the general health of plants. Since the chlorophyll content and the soil–plant analysis development (SPAD) value are positively correlated, it is feasible to predict the SPAD value by calculating the vegetation indices (VIs) through hyperspectral images, thereby evaluating the severity of plant diseases. However, current indices simply adopt few wavelengths of the hyperspectral information, which may decrease the prediction accuracy. Besides, few researches explored the applicability of VIs over rice under the bacterial blight disease stress. Methods In this study, the SPAD value was predicted by calculating the spectral fractal dimension index (SFDI) from a hyperspectral curve (420 to 950 nm). The correlation between the SPAD value and hyperspectral information was further analyzed for determining the sensitive bands that correspond to different disease levels. In addition, a SPAD prediction model was built upon the combination of selected indices and four machine learning methods. Results The results suggested that the SPAD value of rice leaves under different disease levels are sensitive to different wavelengths. Compared with current VIs, a stronger positive correlation was detected between the SPAD value and the SFDI, reaching an average correlation coefficient of 0.8263. For the prediction model, the one built with support vector regression and SFDI achieved the best performance, reaching R 2 , RMSE, and RE at 0.8752, 3.7715, and 7.8614%, respectively. Conclusions This work provides an in-depth insight for accurately and robustly predicting the SPAD value of rice leaves under the bacterial blight disease stress, and the SFDI is of great significance for monitoring the chlorophyll content in large-scale fields non-destructively.