Fractals

The Hausdorff derivative has been recognized as an efficient tool to characterize anomalous diffusion phenomena. This paper makes the first attempt to numerically investigate three-dimensional Hausdorff derivative diffusion equation by the method of fundamental solutions. The fundamental solution of the three-dimensional Hausdorff derivative diffusion equation is closely related to scaling transform and non-Euclidean Hausdorff fractal distance. The used method, as a meshless technique, is simple, accurate and efficient for solving the partial differential equations with fundamental solutions. Three numerical experiments have been conducted to reveal the effectiveness and rationality of Hausdorff derivative anomalous diffusion models with various temporal and spatial fractal orders, as well as the accuracy of the developed methodology.

In this paper, we introduce the $\alpha$-type concept of generalized $h$-convex function on real linear fractal sets ${ℝ}^{\alpha }(0<\alpha \le 1)$, from which the known definitions of generalized convex functions and generalized $s$-convex functions are derived, and from this, we obtain generalized Godunova–Levin functions and generalized $P$-functions. Some properties of generalized $h$-convex functions are discussed. Lastly, some generalized Hadamard-type inequalities of these classes functions are given.

This work theoretically studies the effects of wall velocity slip on the hydraulic resistance and convective heat transfer of laminar flow in a microchannel network with symmetric fractal treelike branching layout. It is found that the slip can reduce the hydraulic resistance and enhance the Nusselt number of laminar flow in the network; furthermore, the slip can also affect the optimal structure of the fractal treelike microchannel network with minimum hydraulic resistance and maximum convective heat transfer. Under the size constraint of constant total channel surface area, the optimal diameter ratio of microchannels at two successive branching levels of the symmetric fractal treelike microchannel network with a minimized hydraulic resistance is only dependent on branching number $N$ in the manner of $N^{-2/5}$ for no slip condition, but decreases with the increasing slip length, the increasing branching number and the increasing length ratio of microchannels at two successive branching levels for slip condition. The convective heat transfer of the treelike microchannel network is independent on the diameter ratio for no slip condition, but displays an increasing after decreasing trend with the increasing diameter ratio for slip condition. The symmetric treelike microchannel network with the worst convective heat transfer performance is the network with diameter ratio equaling one for slip condition.

A fractal basin boundary is a Wada fractal basin boundary if it contains at least three different basins. The corresponding basin is called a Wada fractal basin. Previous results show that some oscillators possess Wada fractal basins with common basin boundaries. Here we find that a nonlinear vibration isolation system can possess abundant coexisting basins and every basin is a Wada fractal basin. These Wada fractal basin boundaries separate different basins in the different regions. A proper classification of these Wada fractal basins is provided according to the order of saddles and Wada numbers. Basin organization is systematic and all basins spiral outward toward the infinity. The entangled basin boundaries are described by the manifolds of saddles and basins (tongues) accumulation.

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

Hydraulic tortuosity is one of the key parameters for evaluating effective transport properties of natural and artificial porous media. A pore-scale model is developed for fluid flow through porous media based on fractal geometry, and a novel analytical tortuosity–porosity correlation is presented. Numerical simulations are also performed on two-dimensional Sierpinski carpet model. The proposed fractal model is validated by comparison with numerical results and available experimental data. Results show that hydraulic tortuosity depends on both statistical and morphological characteristics of porous media. The exponents for the scaling law between tortuosity and porosity depend on pore size distribution and tortuous fractal dimension. It has been found that hydraulic tortuosity indicates evident anisotropy for asymmetrical particle arrangements under the same statistical characteristics of porous media. The present work may be helpful to understand the transport mechanisms of porous materials and provide guidelines for the development of oil and gas reservoir, water resource and chemical engineering, etc.

For $x\in [0,1)$, let $x=O^1(q_1(x),q_2(x),\dots ,q_k(x),\dots )$ be its first Ostrogradsky series with digits ${\{q_n(x)\}}_{n\ge 1}$, ${\{\frac{P_n(x)}{Q_n(x)}\}}_{n\ge 1}$ be its $O^1$-convergents. We calculate the Hausdorff dimensions of the sets

The two-point resistance of fractal network has been studied extensively by mathematicians and physicists. In this paper, for a class of self-similar networks named sailboat networks, we obtain a recursive algorithm for computing resistance between any two nodes, using elimination principle, substitution principle and local sum rules on effective resistance.

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

In this paper, the Fractal-Monte Carlo has been employed to simulate the effective thermal conductivity of porous media with roughened surfaces. The proposed probability model for the effective thermal conductivity of porous media with roughened surfaces can be expressed as a function of the relative roughness, porosity, minimum and maximum diameter of pores, fractal dimensions, and random number. The proposed model is validated by a satisfying agreement of our Fractal-Monte Carlo simulations and the experimental data. In our Fractal-Monte Carlo model, there is no extra empirical constant and each parameter has clear physical meaning. Then, the effects of micro-structural parameters of porous media on the effective thermal conductivity of porous media with roughened surfaces have been analyzed in detail. It can be found that the effective thermal conductivity of porous media with roughened surfaces decreases with the increase of relative roughness and tortuosity fractal dimension. Our results demonstrate that the proposed Fractal-Monte Carlo model can be used to characterize other transport properties such as mass transfer of porous media.

In this work, we consider fractal dimension such as Box dimension, of Weyl fractional integral of certain continuous functions. Upper Box dimension of Weyl fractional integral of continuous functions satisfying $\alpha$-order Hölder condition of positive order $v$ is no more than both $2-v$ and $2-\alpha$. Furthermore, it is no more than $2-\frac{v}{1-\alpha }$ which means strictly less than $2-v$. Meanwhile, $\alpha +v\ge 1$, Box dimension of Weyl fractional integral of continuous functions satisfying $\alpha$-order Hölder condition must be one.

In this paper, we investigate the Hölder continuity and the estimate for the box dimension of the Weyl fractional integral of some continuous function $f(x)$, denoted by $W_{\alpha }f(x)$. We obtain that if $f(x)$ is $\mu$-order Hölder continuous, then $W_{\alpha }f(x)$ is $\frac{\alpha +\mu }{\alpha +1}$-order Hölder continuous. Moreover, if $\mu$ belongs to $(0,1-\alpha )$, then $W_{\alpha }f(x)$ is $\lambda$-order Hölder continuous with $\lambda =\frac{\alpha }{(1-\mu )(1+\alpha )-{\alpha }^2}$.

The paper approaches the construction of fractal surfaces of interpolation and approximation on the basis of a fractal perturbation of any mapping defined on a rectangle. Conditions for the differentiability of these elements are also provided. The fractal surfaces obtained may be used for the approximation of real-world data. The method proposed does not require any restriction on the type of data. Furthermore, the present approach does not imply the solution of large systems of equations. The paper considers both the continuous and the discontinuous case.

Let $x=[a_1(x),a_2(x),\dots ]$ be the continued fraction expansion of $x\in [0,1)$. The growth rate of the product of the partial quotients $a_n(x)a_{n+1}(x)$ is closely connected with the improvability of Dirichlet’s theorem in the sense that the faster $a_n(x)a_{n+1}(x)$ grows, the less possibility the improvement of Dirichlet’s theorem has. In this paper, we study the size of the points for which $a_n(x)a_{n+1}(x)$ grows in a given speed. We call the points of this type as extremely Dirichlet non-improvable points.

We used the multifractal detrended cross-correlation analysis (MFDCCA) method based on ensemble empirical mode decomposition (EEMD) to study the 5-min high-frequency data of two Chinese stocks and two US stocks. Using EEMD method to decompose the original high-frequency stock data can effectively reduce the interference of noise on the series, which helps to reveal the internal characteristics of the stock system and extract more accurate and rich information. We first conducted a cross-correlation test and cross-correlation coefficient analysis on the reconstructed stock data of two groups, and found that there is a cross-correlations between them. Then we used the EEMD-based MFDCCA method to analyze the cross-correlation between the data and found that there are significant cross-correlations between DJI and NASDAQ and between SSEC and SZSE. The cross-correlation of the two Chinese stocks is stronger than that of the two US stocks. The MFDCCA results of the comparison of the original series with the reconstructed series after decomposition by the EEMD method show that the reconstructed series can display more internal details of the multifractal cross-correlation metrics compared with the original series.

In this paper, we present the construction of new nonlinear recurrent hidden variable fractal interpolation surfaces (RHVFISs) with function vertical scaling factors. We use Rakotch’s fixed point theorem which is a generalization of Banach’s fixed point theorem to get new nonlinear fractal surfaces. We construct recurrent vector-valued iterated function systems (IFSs) with function vertical scaling factors on rectangular grids and generate flexible and diverse RHVFISs which are attractors of the IFSs. We also give an explicit example to show the effectiveness of obtained results.

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as $𝜃$ tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as $𝜃$ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.

In spite of the advances in the state of the art in semantic artificial intelligence applications, there is still a long way to go to bring it to a level of mass adoption. Thus, in order to contribute to the advancement of this topic, this study develops a feasible model with a potential scalability for semantic applications’ mass adoption, specifically for news or statement cluster attribute identification, either positive, negative or neutral. This paper proposes a disruptive system based on Blockchain using a Semantic Browser Expert System Bot with artificial intelligence called Blockchain Semantic Browser Expert System (BSBES) to look for and analyze relevant information that significantly represents the cryptocurrencies adoption patterns. The artificial intelligence in this study consists of a deep learning neural network to process the input information to identify the news pattern in a semantic way using deep learning based on two aspects of the news: technical aspect and adoption aspect of the cryptocurrencies. BSBES performance is achieved based on deep learning tools, and scalability is supported by a blockchain system including a stability study.

In this work, the scaling behavior of the sleep process is evaluated by using detrended fluctuation analysis based on wavelets. The analysis is carried out from arrivals of short and recurrent cortical events called A-phases, which in turn build up the Cyclic Alternating Pattern phenomenon, and are classified in three types: A1, A2 and A3. In this study, 61 sleep recordings corresponding to healthy, nocturnal frontal lobe epilepsy patients and sleep-state misperception subjects, were analyzed. From the A-phase annotations, the onsets were extracted and a binary sequence with one second resolution was generated. An item in the sequence has a value of one if an A-phase onset occurs in the corresponding window, and a value of zero otherwise. In addition, we consider other different temporal resolutions from 2s to 256s. Furthermore, the same analysis was carried out for sequences obtained from the different types of A-phases and their combinations. The results of the numerical analysis showed a relationship between the time resolutions and the scaling exponents; specifically, for higher time resolutions a white noise behavior is observed, whereas for lower time resolutions a behavior towards to $1/f$-noise is exhibited. Statistical differences among groups were observed by applying various wavelet functions from the Daubechies family and choosing the appropriate sequence of A-phase onsets. This scaling analysis allows the characterization of the free-scale dynamic of the sleep process that is specific for each sleep condition. The scaling exponent could be useful as a diagnosis parameter in clinics when sleep macrostructure does not offer enough information.