This study employed multifractal analysis to investigate the changes in surface morphology of SiO2 anti-reflective coatings prepared on KDP substrates using the sol-gel method, under various conditions of ultraviolet (UV) irradiance. The coatings were successfully fabricated, and the chemical structure of the SiO2 sol was comprehensively characterized using Solid-State Nuclear Magnetic Resonance (SSNMR) technology. Under low UV irradiance (4 J/cm2), repeated experiments revealed a crack-induced mechanism of surface fatigue damage. Utilizing Scanning Electron Microscopy (SEM), the study discovered the induction effect of initial crack defects in UV-damaged coatings and established a damage model. Furthermore, Atomic Force Microscopy (AFM) was used to acquire images of the coatings\' surface morphology at different damage levels, which were analyzed using the multifractal spectrum f(α). This analysis confirmed the multifractal nature of the coatings both before and after damage. This study identified significant effects of UV irradiation on the width of the multifractal spectrum and Δf, indicating that the SiO2 anti-reflective coatings exhibit multifractal characteristics under various damage states. The coatings displayed a pattern of decreasing and then increasing singularity spectrum width, height distribution unevenness, and surface roughness with increasing damage. This study demonstrates that multifractal analysis is an effective tool for describing the complexity of the surface morphology of sol-gel-derived anti-reflective coatings for the first time and for validating their multifractal properties across different stages of UV damage. HIGHLIGHTS: Damage dynamic process of KDP crystal sol-gel coating was described by SEM&AFM; The crack propagation mechanism of sol-gel coating under UV radiation is proposed; The damage evolution of sol-gel coating was described by multifractal analysis.

Foaming pretreatment has been proven effective in promoting sludge drying, however, the variation in sludge properties significantly influences the foaming efficiency. Inspired by foam stabilizer of solid particles, Camellia oleifera shells (COS) was screened out from various biomasses as an additive incorporated with the CaO for promoting the sludge foaming. For the introduction of COS, this study analyzed the drying behaviors of foamed sludge, quantified the surface cracks information, characterized the combustion performance, and evaluated the energy consumption. The results indicated that 46.72-50.10% of time could be saved in foaming the sludge to 0.70 g/mL by addition of 3.0 wt% COS. Compared with the original sludge (OS), the 0.70 g/mL foamed sludge saved 47.43% of time for sludge drying at 80 °C, and this value further increased to 53.14% with 3.0 wt% COS addition. Combining the multifractal spectra and drying kinetics analysis, the foaming promoted the formation of complex surface cracks in the warm-up period, while COS further improved the complexity of cracks in the constant rate period, and the shrinkage of isolated sludge blocks in the falling rate period, thus enhanced the moisture diffusion and heat transfer. Furthermore, the appropriate porous structure and additional volatile matters promoted the combustion performance. The 0.90 g/mL foamed sludge with COS presented the lowest activation energy of 180.362 kJ/mol in combustion. Overall, compared with OS, the 0.70 g/mL foamed sludge with COS saved 40.65% energy consumption during the foaming, drying and combustion processes, providing an energy-efficient solution for the sludge treatment and disposal.

The spatial distribution of plant, soil, and microbial carbon pools, along with their intricate interactions, presents a great challenge for the current carbon cycle research. However, it is not clear what are the characteristics of the spatial variability of these carbon pools, particularly their cross-scale relationships. We investigated the cross-scale spatial variability of microbial necromass carbon (MNC), soil organic carbon (SOC) and plant biomass (PB), as well as their correlation in a tropical montane rainforest using multifractal analysis. The results showed multifractal spatial variations of MNC, SOC, and PB, demonstrating their adherence to power-law scaling. MNC, especially low MNC, exhibited stronger spatial heterogeneity and weaker evenness compared with SOC and PB. The cross-scale correlation between MNC and SOC was stronger than their correlations at the measurement scale. Furthermore, the cross-scale spatial variability of MNC and SOC exhibited stronger and more stable correlations than those with PB. Additionally, this research suggests that when SOC and PB are both low, it is advisable for reforestations to potentiate MNC formation, whereas when both SOC and PB are high some thinning can be advisable to favour MNC formation. Thus, these results support the utilization of management measures such as reforestation or thinning as nature-based solutions to regulate carbon sequestration capacity of tropical forests by affecting the correlations among various carbon pools.

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In the last 40 years, Chinese cities have seen extensive development across a wide array of spheres. In this study, we applied a power-law and multifractal analysis to characterise 22 indices for 288 prefecture-level cities in mainland China. The data used to characterise the indices is representative of the year 2012. The results show that the Gross domestic product (GDP) is regular and exhibits multifractal spatial characteristics. Specifically, most developmental fields exhibit coupling in conjunction with being chaotic. Furthermore, eight indices with multifractal characteristics clearly reflect the spatial complexity of the corresponding fields and the volatility between prefecture-level cities. Overall, the tertiary industry has undergone nongeneralised development. On the other hand, the construction industry resembles a bubble economy, and the spatial layout of traditional industries has a homogenising effect in Middle and East China. This study examines China\'s overall spatial characteristics and states based on the above-mentioned analyses; additionally, the study focuses on the early 2010s. The contribution of this study provides a quantitative analysis paradigm based on a multifractal approach.

Studying the evolution of the pore structure of coal during spontaneous combustion is of great value in further understanding the mechanism of coal spontaneous combustion (CSC) and its prevention. In this study, we selected three low-rank coals and used nuclear magnetic resonance (NMR) to visualize the macroscopic evolution of the pore structure of coal after heat treatment and to analyze the effect of temperature (25-500 °C) on the pore structure of coal, including porosity, permeability, and fractal dimensions. The obtained results show that the overall NMR signal in coal increases with increasing temperature, indicating that heat treatment can induce the enlargement, opening, and interconnection of pores and fractures in coal. The equivalent average pore radius (rm) of coal shows a positive correlation with temperature, with a substantial increase in rm, especially after temperatures above 200 °C. During heating, the porosity and permeability of all three coals tended to increase with temperature. At temperatures above 300 °C, the permeability of coal dramatically increases, predicting a higher fluid transport capacity. Furthermore, NMR multifractal theory was proposed for quantitative pore space dimensional characterization. The obtained results show that the fractal dimensions of the adsorption space of coal pores increase and then decrease with temperature during heating, while the fractal dimensions of percolation space are negatively correlated with temperature. In addition, the dimensions of adsorption space vary more strongly than those of percolation space, meaning that the adsorption capacity of low-rank coals is more significantly influenced by temperature.

Climate change and human activities have seriously degraded alpine grassland, potentially affecting soil particle size distribution (PSD) and further influencing the nutrient levels and erodibility of soil. Predicting the fertility and erodibility of alpine soil using multifractal dimensions of soil PSD could be used to enhance the management and restoration of degraded alpine grasslands. In the present study, we evaluated three types of alpine grasslands: alpine meadow (AM), alpine steppe (AS), and alpine desert steppe (ADS). Fencing and grazing management measures were conducted at sites containing each grassland type. Then, we analyzed the PSDs, erodibility, and other properties of soil in the 0-20 cm soil layer. Multifractal characterization of soil PSD was calculated using the fractal scale theory. The findings showed that grassland type significantly impacted soil nutrients and the multifractal dimensions of soil PSDs, whereas management measures affected soil erodibility significantly. The proportion of finer particles decreased as follows: AM > AS > ADS. Compared to grazing, fencing enhanced clay content and reduced the proportion of coarser particles under all three grassland types. AM had higher organic carbon and nitrogen levels than AS and ADS. Multifractal dimensions were highest for AM, with ADS having higher erodibility than AM and AS. Multifractal dimensions (except for correlation dimension) also had significantly positive relationships with soil organic carbon and available nutrient content and soil erodibility, but had significantly negative correlations with soil pH, bulk density, and electrical conductivity. Thus, the multifractal dimensions of soil PSDs could be used to characterize the erodibility and fertility characteristics of soil in alpine regions, providing a reference for assessing vegetation restoration measures in the Northern Tibet Plateau.

We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate the Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase-space-averaged multifractal dimensions, which are well described by random matrix theory in a strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.

A series of CuCe-modified TiO2-ZrO2 catalysts synthesized by stepwise impregnation method and ultrasonic-assisted impregnation method were investigated to research the removal of NO in the simulated flue gas. Results showed that the CuCe/TiO2-ZrO2 catalyst prepared by ultrasonic-assisted impregnation method exhibited the superior NO conversion, in which higher than 85% NO was degraded at the temperature range of 250-400 °C and the highest NO conversion of 94% at 350 °C. It proves that ultrasonic treatment can markedly improve the performance of catalysts. The effect of ultrasonic enhancement on CuCe/TiO2-ZrO2 was comprehensively studied through being characterized by physicochemical characterization. Results reveal that the ultrasonic cavitation effect improves the distribution of active species and the synergistic interaction between Cu with Ce components (Cu+ + Ce4+ ↔ Cu2+ + Ce3+) on the catalysts significantly, thus resulting in better dispersibility as well as a higher ratio of Cu2+ and Ce3+ of the catalysts. Moreover, it was found that the CuCe/TiO2-ZrO2 catalyst prepared by the ultrasonic-assisted impregnation method represented a higher degree of ultrafine metal particles and evenness. The above results were described with the generalized dimension and singularity spectra in multifractal analysis and validated by the comparative test. Therefore, it can be concluded that ultrasonic treatment facilitates the particle size and distribution of active sites on the catalysts.

Due to the COVID-19 pandemic, human activities are largely restricted in Shanghai, China and it is a valuable experiment to testify the correlation of air quality and human activities. In consideration of the complexity of air pollution, this study aims to compare the multifractal characteristics of air quality index (AQI) time series before and during COVID-19 partial lockdown, and analyze the correlations between multifractal parameters of AQI time series and human activities in Shanghai, China. The hourly AQI series in Shanghai from November 27, 2019 to March 23, 2020 is used for this study. Firstly, using the MF-DFA method, the multifractal characteristics of the AQI series are explored. Secondly, the causes of the multifractality of the AQI series are determined. Finally, the correlations between multifractal parameters of AQI time series and human activities are investigated. The multifractal analysis results reveal that the AQI series during COVID-19 partial lockdown also has multifractal characteristics, and the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained in comparison with the pre-lockdown phase. However, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. The results via the sliding window procedure indicate that the multifractal parameters ( Δ H , Δ α , Δ f ) show the similar fluctuations along with the fluctuations of passenger volume in Shanghai Metro. Furthermore, it is found that Δ H , Δ α and Δ f and adjusted passenger volume in Shanghai Metro are positively correlated. The possible trend is that the higher adjusted passenger volume is, the larger the value of Δ H , Δ α , Δ f becomes, which means the stronger multifractal characteristics and larger multifractal degree of air quality system.