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An accurate short‐term tidal current prediction is essential to optimize the operation mode of offshore microgrids. According to the correlation characteristics of tidal current speed and direction in time series, a tidal current prediction model is proposed by using fractal theory, dragonfly optimization algorithm and least squares support vector machine (DA‐LSSVM). First, the Rescaled range analysis (R/S analysis) in fractal theory is used to construct the tidal current analytic model with moving Hurst index of time series characteristics and Von‐Mises statistics (V statistic). The model evaluates the stability and autocorrelation of the tidal current and reveals the persistence of tidal current fluctuation. It also shows the orderliness hidden behind the randomness, indicating that the tidal current characteristics can be extended. Second, the least squares support vector machine is used to train the samples in high‐dimensional space, and the dragonfly algorithm is used to optimize the parameters of the least squares support vector machine to obtain the best prediction model. Finally, based on the empirical data recorded in the straits in North America, the model was used for the analysis and prediction of tidal current speed and tidal current direction, which verified the validity and accuracy of the proposed method.

In this work, the topographical effect of the scratching trajectory and the feed direction on the formation of lithographed lines on the (001) InP surface was investigated using an atomic force microscope (AFM) tip-based nanomachining approach. Nanoscratching tests were carried out using the sharp face of a diamond AFM tip in contact mode. From the topographic maps obtained by AFM, several morphological and fractal parameters were obtained and analyzed. Surface morphology presented a surface smoothing for surfaces with scratches produced in [011] and [001] directions. The height parameters confirmed this behavior because scratches in [001] direction exhibited lower roughness. Moreover, this scratch direction promoted the height distribution most symmetrical and platykurtic. The other morphological parameters revealed that this direction provided a more irregular surface (smaller Smc and Sxp ), peak distribution, denser and pointed, smaller portion of material in the core, less deep furrows, higher spatial frequency components, and high isotropy. Fractal parameters revealed that FRE90 has the highest spatial complexity, it is dominated by higher spatial frequencies, and has the lowest surface percolation. Furthermore, all samples exhibited high topographic uniformity.

Scar formation and chronic ulcers can develop following a skin injury. They are the result of the over- or underproduction of collagen. It is very important to evaluate the quality and quantity of the collagen that is produced during wound healing, especially with respect to its structure, as these factors are very important to a complicated outcome. However, there is no standard way to quantitatively analyse dermal collagen. As prior work characterised some potentially fractal properties of collagen, it was hypothesised that collagen structure could be evaluated with fractal dimension analysis. Small-angle X-ray scattering technology (SAXS) was used to evaluate the dermis of rats exposed to graft harvest, burn, and diabetic pathologic states. It was found that almost all collagen structures could be quantitatively measured with fractal dimension analysis. Further, there were significant differences in the three-dimensional (3-D) structure of normal collagen versus that measured in pathologic tissues. There was a significant difference in the 3-D structure of collagen at different stages of healing. The findings of this work suggest that fractal analysis is a good tool for wound healing analysis, and that quantitative collagen analysis is very useful for assessing the structure of dermal collagen.

An important problem in analysis on fractals is the existence of a self-similar energy on finitely ramified fractals. The self-similar energies are constructed in terms of eigenforms, that is, eigenvectors of a special nonlinear operator. Previous results by C. Sabot and V. Metz give conditions for the existence of an eigenform. In this paper, I give a different and probably shorter proof of the previous results, which appears to be suitable for improvements. Such a proof is based on a fixed-point theorem for anti-attracting maps on a convex set., Comment: 11 pages

This study illustrates the application of a geometric average integration of aeromagnetic, radiometric and satellite imagery data over a region prone to Cu‐bearing mineralization at Chahargonbad area in Kerman province of Iran. Processing aeromagnetic, radiometric and Advanced Spaceborne Thermal Emission and Reflection Radiometer satellite data can provide exploratory insights about favourable zones in association with porphyry‐type ore occurrences, which can be synthesized through a combination of knowledge‐ and data‐driven approaches as a geometric average and be represented in a mineral prospectivity map. The existence of known deposits in a prospect region can facilitate the investigation of significant exploratory footprints extracted from airborne data by calculating each indicator layer's weight by plotting a prediction–area curve accompanied by a concentration–area fractal curve. Among various indicators, the most important ones are determined based on derived weights from the prediction–area plots to be synthesized in a single Cu favorability map. To fulfil this aim, indicator layers from airborne geophysics (magmatic bodies, magnetic lineaments and potassium radiometry) and remote‐sensing data (alterations such as argillic, phyllic, propylitic and iron oxide along with geological lineaments) were prepared and evaluated using the known porphyry Cu mineralization by the simultaneous plot of the concentration–area fractal model and the prediction–area curve to attain the ore prediction rate and the relevant occupied area of each map for weight assignment of indicators. The geometric average prospectivity model was applied to synthesize the leading indicators, and the result was compared with a multi‐class index overlay map. This study's significance lies in improvement of the performance of the mineral prospectivity/potential mapping after running a geometric average by a higher ore prediction rate of 79%, which has occupied 21% of the area as potential zones for further mining investigations.

BACKGROUND: A new type of three‐dimensional electroosmotic micromixer is designed based on the Koch fractal curve principle. According to the fractal principle, the influences of fractal structure, voltage size, electrode spacing and electrode configuration on the mixing performance were proposed and studied. RESULTS: The simulation results show that under the combined action of fractal and electric field, irregular alternating vortices are generated in the microchannel, which can disturb the flow of fluid and greatly improve the mixing of fluids in the electroosmotic micromixer. In particular, when the direct current voltage is 10 V, the electrode spacing is 0.15 mm and the electroosmotic micromixer has three groups of alternately placed electrode pairs, the mixing efficiency can reach 98%. CONCLUSIONS: This new type of electroosmotic micromixer is suitable for many occasions, including the mixing of biochemical reagents, and has high application value. © 2021 Society of Chemical Industry (SCI).

We present a class of higher-dimensional Hilbert-type inequalities on a fractal set $(\mathbb{; ; ; R}; ; ; _+^{; ; ; \alpha n}; ; ; )^{; ; ; k}; ; ; $. The crucial step in establishing our results are higher-dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.

The mechanical properties and failure modes of shale that contains bedding were analyzed under coupled seepage pressure and stress. In the numerical model, seven sets of shale samples with different dip‐bedding values were considered, and a seepage‐stress coupling numerical simulation was carried out. The results showed that the compressive strength of shale exhibited obvious anisotropy with increase in the bedding angle, and the compressive strength decreased with an increase in the osmotic pressure. The elastic modulus of shale showed an increasing trend with increase in the bedding angle, while the osmotic pressure had little effect on the elastic modulus of shale. The results also showed that weak bedding had a significant impact on the shale failure mode. When the osmotic pressure was 4 MPa, the shale fracture mode showed five failure modes. When the osmotic pressure was 8 MPa, the shale fracture mode exhibited four failure modes. When the osmotic pressure was 12 MPa, the shale fracture mode showed four failure modes. As the osmotic pressure increased, the osmotic pressure played a leading role in the shale rupture process. The spatial distribution of the acoustic emission demonstrated some self‐similarity patterns, and its fractal characteristics were analyzed. It was found that the fractal dimension better reflects the rupture damage to the shale. The larger the fractal dimension is, the more severe the rupture of shale.

International audience; Over the last decades, several authors have suggested that multifractal measures, that is, self‐similar measures defined on fractal or non‐fractal objects, could be useful to describe soil properties, to model soil processes, and to deal with their extreme microscale heterogeneity. In this context, a key question relates to the extent to which multifractal measures can indeed fulfill all the expectations they have generated. To address this question, we discuss the possibility of generating a synthetic soil image exhibiting multifractal porosity. To this end, a simple geometrical multifractal model in 2D is developed, which helps us to better understand the concept of multifractality and to generate images. We show that it is possible to generate synthetic binary images over a limited range of scales, but that a pure multifractal model for the distribution of the solid or pore mass cannot be developed due to physical constraints. Moreover, in the generated images mimicking multifractal solid space, a higher degree of multifractality corresponds to a larger porosity, rendering it difficult to tune model parameters to match actual soil properties. In addition, simple statistics relying on power‐law fits appear insufficient to characterize soil architecture even if they may capture some key multiscale indicators of observed spatial heterogeneity. We argue that the same conclusions would be reached in a three‐dimensional space, as well as for grey‐scales images.

International audience; Topographical complexity of coral reefs is of primary importance for a number of hydrodynamical and ecological processes. The present study is based on a series of high‐resolution seabottom elevation measurements along the Maupiti barrier reef, French Polynesia. Several statistical metrics and spectral analysis are used to characterize the spatial evolution of the coral geometrical structure from the reef crest to the backreef. A consistent fractal‐like power law exists in the spectral density of bottom elevation for length‐scales between 0.1 and 7m while, at larger scale, the reef structure shows a different pattern. Such a fine characterization of the reef geometrical structure provides key elements to reconstruct the reef history, to improve the representation of reef roughness in hydrodynamical models and to monitor the evolution of coral reef systems in the global change context.

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.

Coal seam is a sedimentary body with complex pore system. The gas adsorption property of coal is greatly determined by the adsorption‐pores ( 2.7‐2.8, the influence of the specific surface area is more prominent. The equilibrium moisture content of the coal samples also has a positive correlation with D2, except for low‐rank coal sample YZG2 due to the presence of a large amount of oxygen‐containing functional groups, which increases its water‐holding capacity.