Fractals最新文献

In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized affine fractal interpolation surfaces (FISs). By using these matrices, we establish relationships between oscillation vectors of different levels, which enables us to obtain the box dimension of generalized affine FISs under certain constraints.

Fractal pore structure exists widely in natural reservoir and dominates its transport property. For that, more and more effort is devoted to investigate the control mechanism on mass transfer in such a complex and multi-scale system. Apparently, effective characterization of the fractal structure is of fundamental importance. Although the newly emerged concept of complexity assembly clarified the complexity types and their assembly mechanism in a fractal system, equivalent extraction of the complexity types is the key for effective characterization. For these, we proposed a deep learning-based method to extract the original and behavioral complexity assembled in bed-packing fractal porous media for simplification and without loss of generality. In detail, the UNeXt network model was trained to obtain the independent connected regions of scaling objects with different scales, the edge detection and clustering analysis algorithms were employed to extract the number-size relationship between two successive scaling objects, and the unique inversion of fractal behavior was realized by taking the number-size model and fractal topography together. Consequently, an equivalent characterization method for fractal complex pore structure was developed based on the concept of complexity assembly. Our investigation provides a theoretical guidance and method reference for the quantitative characterization of fractal porous media that will guarantee the fundamental requirement for the accurate evaluation of the transport properties of natural reservoir.

An in-depth analysis of the attack vulnerability of fractal scale-free networks is of great significance for designing robust networks. Previous studies have mainly focused on the impact of fractal property on attack vulnerability of scale-free networks under static node attacks, while we extend the study to the cases of various types of targeted attacks, and explore the relationship between the attack vulnerability of fractal scale-free networks and the fractal dimension. A hierarchical multiplicative growth model is first proposed to generate scale-free networks with the same structural properties except for the fractal dimension. Furthermore, the fractal dimension of the network is calculated using two methods, namely, the box-covering method and the cluster-growing method, to exclude the possibility of differences in conclusions caused by the methods of calculating the fractal dimension for the subsequent relationship analysis. Finally, four attack strategies are used to attack the network, and the network performance is quantitatively measured by three structural indicators. Results on model networks show that compared to non-fractal modular networks, fractal scale-free networks are more robust to both static and dynamic targeted attacks on nodes and links, and the robustness of the network increases as the fractal dimension decreases. However, there is a cost in that as the fractal dimension decreases, the network becomes less efficient and more vulnerable to random failures on links. These findings contribute to a deeper understanding of the impact of fractal property on scale-free network performance and may be useful for designing resilient infrastructures.

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

The challenges of modeling shale oil transport are numerous and include strong solid-fluid interactions, fluid rheology, the multi-scale nature of the pore structure problem, and the different pore types involved. Until now, theoretical studies have not fully considered shale oil transport mechanisms and multi-scale pore structure properties. In this study, we propose a fractal-based oil transport model with uncertainty reduction for a multi-scale shale pore system. The fractal properties of the shale pore system are obtained using high-resolution scanning electron microscope (SEM) imaging combined with laboratory core sample gas permeability measurements to reduce the model uncertainty. This fractal-based oil transport model accounts for boundary slippage, fluid rheology, the adsorption layer, and different pore types. We further pinpoint the effects of the fractal properties (pore fractal dimension, tortuosity fractal dimension), the shale pore properties (pore type, pore size, total organic carbon in volume), and the fluid properties (yield stress, liquid slippage, adsorption layer) on the shale oil permeability and mobile oil saturation using the proposed model. The results reveal that the size of the inorganic pores has the largest influence on the shale oil transport properties, followed by the yield stress, tortuosity fractal dimension, and the fractal dimension of the inorganic pores.

The research object of this paper is the mixed $(\kappa ,s)$-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed $(\kappa ,s)$-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under $\sigma =({\sigma }_1,{\sigma }_2)$ order of the mixed integral is $3-min\{\frac{{\sigma }_1}{\kappa },\frac{{\sigma }_2}{\kappa }\}$ where $\kappa >0$.

The temperature effect on the permeability of porous rocks continues to be a considerable controversy in the area of reservoirs since the thermal expansion of mineral grains exhibits complicated influence on pore geometries in them. To investigate the degree of effect of pore structures on the hydro-thermal coupling process, a study of the thermal evolution of permeability and porosity of porous rocks is performed based on fractal theory and on thermal as well as stress effects. This work can provide a general physical explanation on some arguments in this area. The proposed models for permeability and porosity can be associated with temperature and the pore-structural parameters as well as physical parameters of porous rocks, such as the initial porosity (${\varphi }_0)$, the initial fractal dimension ($D_{f,0})$, the fractal dimension for tortuosity ($D_{T,T})$ and the thermal expansion coefficient of pore volume (${\alpha }_T)$. The validity of the proposed models for temperature-dependent permeability and temperature-dependent porosity is validated by comparing them with the available experimental results. The investigations are performed in detail considering the essential effects of pore-structural parameters and physical parameters of porous rock on the dimensionless temperature-dependent permeability and temperature-dependent porosity as well as the fractal dimensions for pore areas and tortuosity. It is found that the pore distribution scale range ratio (${\lambda }_{min,T}/{\lambda }_{max,T})$, and pore thermal expansion coefficient (${\alpha }_T)$ have significant effects on the dimensionless temperature-dependent permeability and temperature-dependent porosity of porous rock as well as the fractal dimensions for pore areas and tortuosity. The proposed models may provide a fundamental understandi

Analysis of the brain activity in different mental tasks is an important area of research. We used complexity-based analysis to study the changes in brain activity in four mental tasks: relaxation, Stroop color-word, mirror image recognition, and arithmetic tasks. We used fractal theory, sample entropy, and approximate entropy to analyze the changes in electroencephalogram (EEG) signals between different tasks. Our analysis showed that by moving from relaxation to the Stroop color-word, arithmetic, and mirror image recognition tasks, the complexity of EEG signals increases, respectively, reflecting rising brain activity between these conditions. Furthermore, only the fractal theory could decode the significant changes in brain activity between different conditions. Similar analyses can be done to decode the brain activity in case of other conditions.

In this paper, we explore upper box dimension of continuous functions on $[0,1]$ and their Riemann–Liouville fractional integral. Firstly, by comparing function limits, we prove that the upper box dimension of the Riemann–Liouville fractional order integral image of a continuous function will not exceed $2-\upsilon$, the result similar to [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Appl. Sin. E32 (2016) 1494–1508]. Secondly, we prove that upper box dimension of multiple algebraic sums of continuous functions does not exceed the largest box dimension among them, backing up our conclusion with an appropriate example. Finally, we draw the same conclusions for the product of multiple continuous functions.

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.