Fractals

We construct new examples of parametric iterated function systems converging to some fractal shapes. The main goal is the study of the continuous growth and the rate of change of the attractor of the corresponding parametrization.

Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let $L\ge 2$ be an integer and let $\alpha \in (0,1/L)$. Let $C_{\alpha ,L}$ be the uniform Cantor set defined by the following set equation:

A fractal modification of the telegraph equation with fractal derivatives is given, and its variational principle is established by the semi-inverse method. The two-scale transform method and He’s homotopy perturbation method are successfully adopted to solve the fractal equation.

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical examples are also included to verify the conclusions.

During coal and rock loading, a significantly large number of electromagnetic signals are generated as a result of fracture appearance and crack expansion. The generation of electromagnetic signal is the comprehensive embodiment of the coal rock failure behavior. Therefore, the generated signals contain complex and rich messages that can reflect the damage process and degree of coal and rock. In this work, the multifractal theory is applied to analyze the nonlinear characteristics of the electromagnetic wave and its spectrum induced during coal rock, which present good correlation with failure process. The failure process of coal rock is non-uniform, non-continuous and nonlinear, during which, there is a good synchronization and correlation between the electromagnetic pulses and the stress drop, rather than the stress. Both waveform and its spectrum of electromagnetic signal have multifractal characteristics, the larger the fracture scale is, the more significant the multifractal characteristic of electromagnetic signal is, and the multifractal characteristic of electromagnetic signal from coal is higher than that from sandstone. The difference of fracture energy and size can be represented by the maximum of the multifractal dimension $D_{q\mathrm{\text{max}}}$ of the electromagnetic wave and its spectrum during coal rock failure. In the electromagnetic spectrum, small signals are always dominant, and the dominant frequency is only a few isolated points. What is more, with the increase of fracture size, the difference between the dominant frequency and the non-dominant frequency is gradually enhanced.

In this paper, we investigate the exact solutions of the generalized time fractional foam drainage equation. The Lie-group scaling transformation method and improved $(G^{\prime }/G)$-expansion method are adopted here. The equation describes the evolution of the vertical density profile of a foam under gravity. New exact solutions and maple diagrams of the generalized time fractional foam drainage equation can help us better understand the physical phenomena.

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval $[-1,1]$ is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.

For any $x\in (0,1]$, let

Let $A=\left(\right.\begin{array}{cc}\hfill n\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m\hfill \end{array}\left)\right.$, where $n,m\ge 3$ are integers and ${𝒟}=\{0,1,\dots ,n-1\}\times \{0,1,\dots ,m-1\}$ be a digit set. Then the pair $(A,{𝒟})$ generates a fractal set $T:=T(A,{𝒟})$ satisfying $AT=T+{𝒟}$ which is a unit square. However, if we remove one digit from ${𝒟}$, then the structure of $T$ will become very interesting. A well-known example is the Sierpinski carpet. In this paper, we study the resulting self-affine sets of moving a digit in ${𝒟}$ to a different place. That is, we consider a digit set ${{𝒟}}^{\ast }={𝒟}\setminus \{d_0\}\cup \{d^{\ast }\}$, where $d_0\in {𝒟},d^{\ast }\in {\mathbb{Z}}^2$. We give a complete characterization for the connectedness of self-affine carpet $T(A,{{𝒟}}^{\ast })$ in terms of the domains of $d_0$ and $d^{\ast }$.

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence.

Because of the self-similarity properties of nature, fractals are widely adopted as generators of natural object multimedia contents. Unfortunately, fractals are difficult to control due to their iterated function systems, and traditional researches on fractal generating visual objects focus on mathematical manipulations. In Generative Adversarial Nets (GANs), visual object generators can be automatically guided by a single image. In this work, we explore the problem of guiding fractal generators with GAN. We assume that the same category of fractal patterns is produced by a group of parameters of initial patterns, affine transformations and random noises. Connections between these fractal parameters and visual objects are modeled by a Gaussian mixture model (GMM). Generator trainings are performed as gradients on GMM instead of fractals, so that evaluation numbers of iterated function systems are minimized. The proposed model requires no mathematical expertise from the user because parameters are trained by automatic procedures of GMM and GAN. Experiments include one 2D demonstration and three 3D real-world applications, where high-resolution visual objects are generated, and a user study shows the effectiveness of artificial intelligence guidances on fractals.

A generalized KdV equation with fractal derivatives is suggested, and a special function is introduced to establish a fractal variational principle. A detailed derivation is elucidated step by step by the semi-inverse method, and some special cases are discussed.

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

This study aims at combining the machine learning technique with the Hausdorff derivative to solve one-dimensional Hausdorff derivative diffusion equations. In the proposed artificial neural network method, the multilayer feed-forward neural network is chosen and improved by using the Hausdorff derivative to the activation function of hidden layers. A trial solution is a combination of the boundary and initial condition terms and the network output, which can approximate the analytical solution. To transform the original Hausdorff derivative equation into a minimization problem, an error function is defined, where the coefficients are approximated by using the gradient descent algorithm in the back-propagation process. Two numerical examples are given to illustrate the accuracy and the robustness of the proposed method. The obtained results show that the improved machine learning technique is efficient in computing the Hausdorff derivative diffusion equations both from computational accuracy and stability.

In the process of gas extraction, fracture-pore structure significantly influences the macroscopic permeability of coal seam. However, under the multi-field coupling, the mechanism of coal seam fracture-pore evolution remains to be clarified. In this paper, considering the effect of adsorption expansion, the fractal theory for porous media coupled with the multi-field model for coal seam is considered, and a multi-field coupling mechanical model is constructed by considering the influence of fracture-pore structure. Furthermore, the evolution mechanism of fractal dimension with physical and mechanical parameters of coal seam is studied. It is found that the fractal dimension for coal seam is inversely proportional to mining time and in situ stress, proportional to elastic modulus, Langmuir volume constant and Langmuir volume strain constant, and inversely proportional to Langmuir pressure constant. Compared with other factors, Langmuir pressure constant and Langmuir volume strain constant have the significance influence on the fractal dimension for the fracture length.

In this paper, in order to calculate the average path length for unweighted and weighted hierarchical networks, we define the sum of the distances from each node to the vertex and the bottom nodes. For the unweighted network, we show that the average path length grows with the size of $N_t$ as $ln(N_t)$. For the weighted network, we prove its weighted path length approaches a constant related to the weighting factor $r$ and parameter $m$. In particular, for $r=\frac{1}{m}$, the average weighted path length of the network tends to a specific value $\frac{4}{m}$.

In this work, we present a robotic arm assisted by a visual system to decide whether an object with different colors, parallel flat surfaces and other types of surfaces would be subject to be manipulated without a drop risk. This robotic arm is assisted with sensors such as temperature, humidity, artificial vision, etc. and monitored with a Blockchain Internet of Things (BIoT) expert system assistance, which is shared and accessed by the internet by the users. A prototype for industrial purpose is launched to start providing data for training the expert system, achieving in this way an expert system with machine learning. The variations derived from the identification of the reference points and the characteristics of the robotic arm are a limiting factor of the system, however, it was possible to correctly locate the robotic arm in the workspace to take the object and manipulate it using machine learning based on a BIoT expert system.