Fractals最新文献

Diabetic retinopathy (DR) is one of the leading causes of blindness in a significant portion of the working population, and its damage on vision is irreversible. Therefore, rapid diagnosis on DR is crucial for saving the patient’s eyesight. Since Transformer shows superior performance in the field of computer vision compared with Convolutional Neural Networks (CNNs), it has been proposed and applied in computer aided diagnosis of DR. However, a large number of images should be used for training due to the lack of inductive bias in Transformers. It has been demonstrated that the retinal vessels follow self-similar fractal scaling law, and the fractal dimension of DR patients shows an evident difference from that of normal people. Based on this, the fractal dimension is introduced as a prior into Transformers to mitigate the adverse influence of lack of inductive bias on model performance. A new Transformer method pretrained with Masked Autoencoders and fractal dimension (MAEFD) is developed and proposed in this paper. The experiments on the APTOS dataset show that the classification performance for DR by the proposed MAEFD can be substantially improved. Additionally, the present model pretrained with 100,000 retinal images outperforms that pretrained with 1 million natural images in terms of DR classification performance.

In this work, a new gas transport model for shale reservoirs is constructed by embedding randomly distributed roughened tree-like bifurcation networks into the matrix porous medium. We constructed apparent permeability models for different shale gas flow mechanisms based on fractal theory, taking into account the effects of relative roughness and surface diffusion. The effects of bifurcation structure parameters as well as shale gas parameters on different apparent permeabilities are systematically analyzed. It is found that the permeability generally shows a decreasing trend with the increase in pore pressure, but the effect on viscous flow is inconsiderable. In addition, larger porosity, fractal dimension and bifurcation levels lead to increased permeability of different mechanisms. Whereas, the increase in the bifurcation levels implies a greater flow resistance, resulting in a decreased permeability. In addition, The relative roughness hinders the development of total permeability but favors surface diffusion permeability. Moreover, larger length ratios and diameter ratios are beneficial to the shale gas flow.

This paper proposes a methodology for the diagnosis of electrical system conditions using fractional-order integral transforms for feature extraction. This work proposes three feature extraction algorithms using the Fractional Fourier Transform (FRFT), the Fourier Transform combined with the Mittag-Leffler function, and the Wavelet Transform (WT). Each algorithm extracts data from an electrical system to obtain a set of features that are classified by an Artificial Neural Network to determine the system’s condition. The algorithms are utilized in diagnosing two types of electrical machine faults, one in a photovoltaic system, and another in classifying the power quality disturbances (PQDs). An optimization algorithm is suggested to find the optimal orders of integral transforms. The obtained results demonstrate the system’s effective diagnosis, displaying superior performance in classifying the faults and PQDs with complex signals.

This paper represents a pioneering effort to investigate multifractal dynamics that exclusively encompass the return time series of USA Education Group Stocks concerning two non-overlapping periods (before, during, and after COVID-19). Given this, we employ the Multifractal Detrended Fluctuations Analysis (MF-DFA). In this sense, we investigate the generalized Hurst exponent $h(q)$ and the Rényi exponent $\tau (q)$ for each asset and quantify their statistical properties, which allowed us to observe separately the contributing small scale (primarily via the negative moments $q$) and the large scale (via the positive moments $q$). We perform a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. Also, we shall apply the inefficiency multifractal index to assess the COVID-19 shock for both periods. Our findings show that for both periods, the majority of these assets are marked by multifractal dynamics associated with persistent behavior $({\alpha }_0>0.5)$, a higher degree of multifractality and the dominance of large fluctuations. At the same time, most of these assets show asymmetry parameter $(R>1)$ for both periods, indicating that large fluctuations contributed more to multifractality in the time series of returns.

Many complex networks have scale-free and small-world effects. In this paper, a family of evolving networks is constructed modeled by a non-symmetric self-similar planar fractal, using the encoding method in fractal geometry. Based on the self-similar structure, we study the degree distribution, clustering coefficient and average path length of our evolving network to verify their scale-free and small-world characteristics.

In this paper, we will study the geometric structure on the Sierpinski networks which are skeleton networks of a connected self-similar Sierpinski carpet. Under some suitable condition, we figure out that the renormalized shortest path distance is comparable to the planar Manhattan distance, and obtain the Hausdorff dimension of Sierpinski networks.

In this paper, we propose a fresh conception about convexity, known as the multiplicative $(s,P)$-convexity. Along with this direction, we research the properties of such type of convexity. In the framework of multiplicative fractional Riemann–Liouville integrals and under the ${}^{\ast }$differentiable $(s,P)$-convexity, we investigate the multiplicative fractional inequalities, including the Hermite–Hadamard- and Newton-type inequalities. To further verify the validity of our primary outcomes, we give a few numerical examples. As applications, we proffer a number of inequalities of multiplicative type in special means as well.

Fractional $q$-calculus is considered to be the fractional analogs of $q$-calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) $q$-integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving $\hslash$-convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF $q$-integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided.

Finger seal is a new flexible dynamic sealing technology, and its heat transfer characteristics and seepage characteristics are one of the main research hotspots. In this paper, based on the fractal theory, a fractal model of the total thermal conductance of the finger seal considering the heat transfer effect of the contact gap of the rough surface is established, a fractal model of the effective gas permeability of the adjacent finger seals annulus considering the gas slip effect and the temperature change is established, and a finite element calculation method of the two-way thermo-mechanical coupling for the finger seal is proposed. The results show that the solid-phase thermal conductance decreases with the increase of the scale coefficient. When the axial pressure difference is greater than 0.4$$MPa, the rotor speed is greater than 11,000$$r/min, the radial displacement excitation is [0.03$$mm, 0.09$$mm], and the temperature is less than 600$$K, the gas-phase thermal conductance between the finger seal and the rotor shows an increasing trend. The gas-phase thermal conductance of the finger seal and the rotor is always the main position under different working conditions. Under different fractal dimensions, the solid-phase thermal conductance gradually occupies the dominant position. Temperature has a certain effect on the effective gas permeability, and fractal dimension, scale coefficient, and axial pressure difference have less effect on the effective gas permeability. At an axial pressure difference of 0.08$$MPa, the numerical calculation results of the two-way thermo-mechanical coupling calculation method for finger seal are closer to the experimental results, with a maximum error rate of 1.96%. The above results further improve the theoretical research system of the heat transfer characteristics of the finger seal.