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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.

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.

Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity of multichannel time series, multichannel information processing was introduced, and multivariate Higuchi fractal dimension (MvHFD) was proposed. To further analyze the complexity at multiple scales, multivariate multiscale Higuchi fractal dimension (MvmHFD) was proposed by introducing multiscale processing algorithms as a technology that not only improved the use of fractal dimension in the analysis of multichannel information, but also characterized the complexity of the time series at multiple scales in the studied time series data. The effectiveness and feasibility of MvHFD and MvmHFD were verified by simulated signal experiments and real signal experiments, in which the simulation experiments tested the stability, computational efficiency, and signal separation performance of MvHFD and MvmHFD, and the real signal experiments tested the effect of MvmHFD on the recognition of multi-channel mechanical signals. The experimental results show that compared to other indicators, A achieves a recognition rate of 100% for signals in three features, which is at least 17.2% higher than for other metrics.

Knowledge of the behavior of highly compacted expansive clays, as an engineered barrier, in disposal of high-level nuclear waste (HLW) systems to prevent the pollution due to migration of radionuclide is extremely essential. The prominent properties of globally and widely used bentonites have been extensively studied during past two decades. In China, GaoMiaoZi (GMZ) bentonite is the first choice as a buffer or backfill material for deep geological repositories. This review article presents the recent progresses of knowledge on water retention properties, hydromechanical behavior, and fractal characteristics of GMZ bentonite-based materials, by reviewing 217 internationally published research articles. Firstly, the current literature regarding hydrogeochemical and mechanical characteristics of GMZ bentonite influenced by various saline solutions are critically summarized and reviewed. Then, the role of osmotic suction π alongside the application of surface fractal dimension D s is presented from the standpoint of fractal theory. Finally, the strength characteristics of GMZ bentonites using fractal approach have been discussed. Furthermore, this study sheds light on gaps, opportunities, and further research for understanding and analyzing the long-term hydromechanical characteristics of the designed backfill material, from the standpoint of surface fractality of bentonites, and implications of sustainable buffer materials in the field of geoenvironmental engineering.

Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is dubious, is mainly attributed to the strict definition of fractal dimension as a measure or index for characterizing the complexity of fractals. In this article, we propose an alternative, ht-index, to quantify the fractal or scaling structure of geographic features. A geographic feature has ht-index (h) if the pattern of far more small things than large ones recurs (h - 1) times at different scales. The higher the ht-index, the more complex the geographic feature. We conduct three case studies to illustrate how the computed ht-indexes capture the complexity of different geographic features. We further discuss how ht-index is complementary to fractal dimension and elaborate on a dynamic view behind ht-index that enables better understanding of geographic forms and processes.

In order to study gas transport properties of fractured shale gas reservoirs for the accurate estimation of shale gas production, a new multiscale fractal transport model with an effective porosity model was proposed based on the fractal theory and the multilayer fractal Frenkel–Halsey–Hill (FHH) adsorption. In shale matrix, both fractal microstructures of pores (such as pore size distribution, flow path tortuosity, and pore surface roughness) and multiscale flow mechanisms (including slip flow and Knudsen diffusion) were coupled. In fracture network, fractal fracture length distribution, stress compaction, and gas pressure were introduced to formulate a new fracture permeability model. These permeability and effective porosity models were then incorporated into the governing equations of gas flow and the deformation equation of reservoirs to form a numerical model. This numerical model was solved within COMSOL Multiphysics for shale gas recovery. Both transport models in shale matrix and fracture network were validated by experimental data or compared with other models. Finally, sensitivity analysis was conducted to identify key parameters to gas recovery enhancement. It was found that the multilayer gas adsorption and fractal microstructures have great impacts on gas production in shale reservoirs. The cumulative gas production can be increased by 26% after 8000 days when the multilayer adsorbed gas is considered. Larger surface fractal dimension and larger tortuosity fractal dimension represent more roughness pore surface, higher flow resistance, and lower cumulative gas production. Bigger pore diameter fractal dimension means more pores, higher permeability, and higher cumulative gas production. Our model with fractal FHH adsorption was in better agreements with field data from Marcellus and Barnett shale reservoirs than other models.

Aims The 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. Methods The 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. Results The 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. Conclusions With 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 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.

This study advances the theory of power-law fluid infiltration grouting by developing spherical and columnar diffusion models rooted in fractal porous media theory and power-law rheological equations. An analytical solution for determining the slurry diffusion radius is derived and validated through laboratory experiments and numerical simulations. Key findings include the following: (1) The fractal permeability constant demonstrates an exponential dependence on the rheological index (n), with a critical threshold at n = 0.4. Below this threshold, the constant asymptotically approaches zero (slope < 0.1), while beyond it, sensitivity intensifies exponentially, attaining 0.48 at n = 0.9. (2) Non-linear positive correlations exist between the slurry diffusion radius and both the grouting pressure (P) and the water–cement ratio (W/C). Spherical diffusion dominates over columnar diffusion, with their ratio shifting from 1:0.96 at P = 0.1 MPa to 1:0.82 at P = 0.5 MPa. The diffusion distance differential increases from 22 mm to 38 mm as the W/C rises from 0.5 to 0.7, attributable to reduced interfacial shear resistance from decreasing slurry viscosity and yield stress. (3) Experimental validation confirms exponentially decaying model errors: spherical grouting errors decrease from 21.54% (t = 5 s) to 8.43% (t = 15 s) and columnar errors from 25.45% to 10.17%, both within the 50% engineering tolerance. (4) Numerical simulations show that the meander fractal dimension (48 mm) demonstrates a higher sensitivity than the volume fractal dimension (37 mm), with both dimensions reaching maximum values. These findings establish a theoretical framework for optimizing grouting design in heterogeneous porous media.

Measures of predictability in physiological signals using entropy measures have been widely applied in many areas of research. Multiscale entropy expresses different levels of either approximate entropy or sample entropy by means of multiple factors for generating multiple time series, enabling the capture of more useful information than using a scalar value produced by the two entropy methods. This paper presents the use of different time shifts on various intervals of time series to discover different entropy patterns of the time series. Examples and experimental results using white noise, 1/ f noise, photoplethysmography, and electromyography signals suggest the validity and better performance of the proposed time-shift multiscale entropy analysis of physiological signals than the multiscale entropy.