Fractals

We present the hierarchical graph for the growth of weighted networks in which the structural growth is coupled with the edges’ weight dynamical evolution. We investigate consensus problems of the graph from weighted Laplacian spectra perspective, focusing on three important quantities of consensus problems, convergence speed, delay robustness, and first-order coherence, which are determined by the second smallest eigenvalue, largest eigenvalue, and sum of reciprocals of each nonzero eigenvalue of weighted Laplacian matrix, respectively. Unlike previous enquiries, we want to emphasize the importance of weight factor in the study of coherence problems. In what follows, we attempt to study that the weighted Laplacian eigenvalues of the weighted hierarchical graphs, which are determined through analytic recursive equations. We find in our study that the value of convergence speed and delay robustness in weighted hierarchical graphs increases as weight factor increases and the value of first-order coherence decreases as weight factor increases. Moreover, as is expected, weight factor affects the performance of consensus behavior and can be regarded as a leverage in the problem of consensus problems. This paper puts forward the proposal and the countermeasure for stability optimization of networks from the perspective of weight factor for future researchers.

Drilling is a famous and widely used machining operation to make holes in the workpiece. The size and surface quality of drilled hole are two factors that should be considered mainly. In this research, we examine the effect of different machining parameters and conditions on the surface quality of generated hole in drilling operation. For this purpose, we employ fractal theory and investigate how the variations of depth of cut and spindle speed affect the complexity of surface texture of drilled holes in wet and dry machining conditions. Based on the obtained results, the increment of depth of cut and spindle speed in case of wet and dry machining causes lower complexity on the generated surface from drilling. In addition, the generated surface from dry machining is more complex than the generated surface from wet machining. The obtained method in this research can be applied to other machining operations in order to investigate the effect of machining parameters and conditions on the surface quality of machined workpiece.

The oil flow behaviors in inorganic and organic nanopores are still unclear due to the effects of complex surface properties on boundary slip velocity and effective oil viscosity, such as surface roughness and various wettability. In this work, the oil flow enhancement factor and apparent permeability model are derived by coupling slip length and effective viscosity based on the no-slip Hagen–Poiseuille equation. The effects of the oil–wall molecular interactions, pore radius, fractal surface roughness and heterogeneous wettability are considered, and the heterogeneous wettability is caused by the trapped gas between rough elements. The results show that (1) the enhancement factor ranges from 10${}^{-1}$ to 10⁸ with contact angle ranging from 0${}^{\circ }$ to 180${}^{\circ }$, and gradually tends to 1 with an increasing pore radius. Additionally, the flow enhancement factor and apparent permeability increase with an increasing contact angle; (2) The surface roughness can increase or decrease apparent contact angles of oil droplet on water-wetting or oil-wetting surface, respectively, and the increasing or decreasing value of contact angle increases with an increasing fractal dimension. Therefore, as rough fractal dimension increases, the enhancement factor and apparent permeability decrease in oil-wetting pores and increase in water-wetting pores; (3) With an increasing trapped gas content, because the surface becomes more oil-nonwetting that leads to the larger slip length and smaller effective viscosity, the enhancement factor and apparent permeability increase. This model has certain significance for the study of liquid flow behaviors in nanopores with rough surface and heterogeneous wettability, and can be used to fit the results measured by experiments by changing the fractal dimension and trapped liquid content.

Intuitively, link weight could affect the dynamics of the network. However, the theoretical research on the effects of link weight on network dynamics is still rare. In this paper, we present two heterogeneous weighted pseudo-fractal webs controlled by two weight parameters $a$ and $b$ ($0<a,b\le 2$). Both graph models are scale-free deterministic graphs, and they have the same weight sequence when $a$ and $b$ are fixed. Based on their self-similar graph structure, we study the effect of heterogeneous weight on the random walks in graph with scale-free characteristics. We obtain analytically the average trapping time (ATT) for biased random walks in graphs with a trap located at a fixed node. Analyzing and comparing the obtained solutions, we find that in the large graph limit, the ATT for both graph models all grow as a power function of the graph size (number of nodes) with the exponent $f(a,b)$ dependents on the ratio of parameters $a$ and $b$, but their exponents $f(a,b)$ are not the same, one gets the minimum when $a=b$, while the other gets the maximum. Furthermore, the average weighted shortest path length (AWSPL) to the trap is calculated for both graph models, respectively. We show that when the graph size tends to infinity, their AWSPL grows unbounded with the graph size for most parameters. We hope that these results could help people understand the impact of heterogeneous weight on network dynamics.

One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in $ℝ$. These fractal maps, the so-called $\alpha$-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of $\alpha$-fractal functions, the current study targets to record the continuous dependence of the $\alpha$-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the $\alpha$-fractal function.

Relative permeability (RP) plays a critical role in fluid flow and transport in fracture systems, with considerable advances in theoretical and computational methods over the last few decades. Nevertheless, there is no standard procedure for measuring RP in fractures, and the essential controls on RP in rough fractures are not yet definitive. In this study, we developed a theoretical model to investigate the RP in inclined rough fractures under stress dependence based on fractal theory. The topography of the rough fracture surfaces is well addressed by fractal theory and lubrication theory, and thin bending theory is used to characterize permeability evolution characteristics of fractures under stress conditions. The model accounts for multiple key factors, including the microstructure of the rough fractures (e.g. the fractal dimension, the area ratio, the length ratio, and the maximum and minimum base radii), rock lithology (e.g. elastic modulus and Poisson’s ratio), gravity and effective stress. The predicted results agree well with the available experimental data. It is inferred that the effect of gravity on RP decreases as pressure gradient increases. In general, the theoretical fractal model reveals the coupled flow-deformation mechanisms in fractures, and tends to improve the efficacy of reservoir development strategies. With this new analytical solution, it can help to reduce the uncertainty in flow through fractures and obtain data with high accuracy.

The main purpose of this paper is to investigate the weak singularity of solutions for some nonlinear Hadamard fractional differential systems (HFDSs). By constructing proper Banach space and employing lower and upper solutions technique, we prove the existence of the blow-up solutions for a class of HFDSs. In addition, we establish a more general condition than the classical Lipschitz condition which is employed to guarantee the equivalence between the solutions to HFDS and the corresponding functional operator. Also several examples are presented to verify our theoretical results.

Obtaining the optimum surface finish is one of the key factors in machining operations. For this purpose, engineers apply a set of machining parameters to obtain the desired surface quality. On the other hand, tool faces wear during machining operation that itself affects the surface quality of machined surface. Therefore, tool wear and surface finish of machined workpiece should be related to each other. In this research, we employ fractal analysis in order to investigate the correlation between variations of complex structure of machined surface and tool wear in turning operation. In fact, we changed the machining parameters between different experiments and investigated how the machined surface is correlated with the tool wear. Based on the obtained results, we can see the correlation between the complexity of machined surface and tool wear by increasing the depth of cut, spindle speed and feed rate in different experiments. The method of analysis employed in this research can be widely applied to other machining operations in order to find the correlation between the surface quality of machined surface and tool wear.

As a kind of microchannel layout with good transport performance, tree-like branching microchannel network has been widely used for microfluidic systems, however, the optimal analysis of the tree-like branching microchannel network for electroosmotic flow (EOF) to reach a minimized fluidic resistance still needs a deep study. In this work, the EOF in tree-like branching microchannel network is theoretically and numerically studied. It is found that there is an optimal structure of the tree-like branching network for the EOF to achieve a minimum fluidic resistance under the size constraint of constant total channel volume. This work found that the optimal channel radii of the tree-like network for EOF to reach a minimum fluidic resistance satisfy the relationship of $r_k^2={\sum }_{i=1}^N{r}_{k+1,i}^2$, where $r_k$ is the radius of the parent channel, $r_{k+1,i}$ is the radius of the child channels and $N$ is the total number of child channels. This formula can be regarded as an extended Murray’s law for EOF and is helpful for the optimization design of tree-like branching microchannel network for EOF to reach maximum transport efficiency under the constant applied driven voltage.

Micro-displacement behavior during hydraulic fracturing greatly influences damage formation. Formation temperature and the non-Newtonian behavior of fracturing liquid are the major influencing factors in the micro-displacement process. In this work, to investigate the influencing factors on micro-displacement process, we built a mathematical model for the micro-displacement of a fracturing liquid on the basis of the fractal theory. Then, the analytical expression for penetration equation was obtained according to the model, and explicit solution for the relationship between the volume driven into formation and time was derived. Finally, the influences of non-Newtonian behavior, formation temperature, wettability, and pore size complexity on the micro-displacement process were discussed. Results show that the decrease in formation temperature or decrease in the non-Newtonian degree of fracturing liquid reduce fracturing liquid leakage.

The substitution network is a deterministic model of evolving self-similar networks. For normalized substitution networks, the limit of metric spaces with respect to networks is a self-similar fractal and the limit of average distances on networks is the integral of geodesic distance of the fractal on the self-similar measure. After some technical handles, we establish the finiteness of integrals and obtain a linear equation set to solve the average distance on the fractal.

When the entries of Pascal’s triangle which are congruent to a given nonzero residue modulo a fixed prime are mapped to corresponding locations of the unit square, a fractal-like structure emerges. In a previous publication, Bradley, Khalil, Niemeyer and Ossanna [The box-counting dimension of Pascal’s triangle $r$ mod $p$, Fractals 26(5) (2018) 1850071] showed that this mapping yields a nonempty compact set which can be realized as a limit of a sequence of sets representing incrementally refined approximations. Moreover, it was shown therein that for any fixed prime, the sequence converges to the same set, regardless of the nonzero residue or combination of nonzero residues considered. Consequently, the fractal (box-counting) dimension of the limiting set is independent of the residue. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first $p$ rows of Pascal’s triangle which are congruent to a given nonzero residue $r$ modulo the prime $p$. Unfortunately, the numerical evidence presented in this paper supports the contention that there is no such formula. Nevertheless, the evidence indicates that for sufficiently large primes $p$, the number of entries congruent to $rmodp$ for $r=1$, $1<r<p-1$, and $r=p-1$ is well approximated by the respective linear functions $3p$, $p/2$, and $p$. In particular, for large primes $p$ there are approximately six times as many occurrences of the residue $1$ in the first $p$ rows of Pascal’s triangle reduced modulo $p$ than there are of any other residue $r$ in the range $1<r<p-1$, and three times as many as $r=p-1$. On the other hand, if we let the nonnegative integer $k$ vary while keeping the prime $p$ fixed, and look at the relative frequency of various residue classes that occur in the first $p^k$ rows, the seemingly substantial differences in frequency between $r=1$, $1<r<p-1$, and $r=p-1$ when $k=1$ are increasingly dissipated as $k$ grows without bound. We show that in the limit as $k$ tends to infinity, all nonzero residues are equally represented with asymptotic proportion $1/(p-1)$.

We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics of Newton maps $N_f$ associated to real polynomial maps $f:{ℝ}^2\to {ℝ}^2$ with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map $N_f$ has no wandering domain, almost every point under $N_f$ is asymptotic to a fixed point and there is some non-empty open set of points whose $\alpha$-limit equals the set of non-regular points of the Julia set of $N_f$. The first two points were proved by B. Barna in the real one-dimensional case.

Ultrasonic receiving wave can reflect physical properties and damage degree of coal samples. Therefore, it is of great significance to deeply study the parameters of ultrasonic. In this paper, time-domain characteristics of receiving wave are analyzed systematically, which present good correlation with stress. The frequency spectrum of receiving wave is obtained using Fast Fourier Transform (FFT), and peak frequency and centroid frequency are calculated. During the entire loading process, peak frequency fluctuates around 110kHz, but corresponding centroid frequency decreases obviously at the end stage of loading. According to multifractal theory, the multifractal spectrum of wavelet packet energy characteristics is calculated. The results show that wavelet packet energy distribution has obvious multifractal characteristics, and multifractal parameter $\mathrm{\Delta }\alpha$ presents downward trend before coal samples buckling failure. Based on damage process of the coal samples, the reason of change in $\mathrm{\Delta }\alpha$ value is related to damage degree of coal samples. This research is of great significance for understanding the deformation and failure process of coal samples using ultrasonic technology.

In this paper, we prove that a large class of Moran sets on the line with packing dimension 1 is quasisymmetrically packing-minimal.

The structure of network has a significant impact on the stability of the network. It is useful to reveal the effect of fractal structure on the vulnerability of complex network since it is a ubiquitous feature in many real-world networks. There have been many studies on the stability of the small world and scale-free models, but little has been down on the quantitative research on fractal models. In this paper, the vulnerability was studied from two perspectives: the connection pattern between hubs and the fractal dimensions of the networks. First, statistics expression of inter-connections between any two hubs was defined and used to represent the connection pattern of the whole network. Our experimental results show that statistic values of inter-connections were obvious differences for each kind of complex model, and the more inter-connections, the more stable the network was. Secondly, the fractal dimension was considered to be a key factor related to vulnerability. Here we found the quantitative power function relationship between vulnerability and fractal dimension and gave the explicit mathematical formula. The results are helpful to build stable artificial network models through the analysis and comparison of the real brain network.

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

The heat and mass transfer in the ablation process is of great importance for the ablation protection engineering. Accurate temperature assessment can provide effective support for the design of thermal protection structure and ablative material of reentry hypersonic vehicle. In this work, we studied the effect of permeability on heat and mass transfer and ablation thermal protection of gases produced during ablation. Since the carbide formed by ablation of material is a typical porous medium, its structure has self-similarity and can be described by fractal theory. Taking into account the angle of global coordinates and the local coordinates, this paper derived the permeability in any direction as a function of three different fractal dimensions in $x$, $y$ and $z$ directions. In order to verify the correctness of this method, the new model is introduced into the ablation process. Aiming at the ablation process and the diffusion equation of pyrolysis gas in the carbide layer, the temperature, material density and pyrolysis gas density distribution of three-dimensional spherical head under different permeability were simulated numerically. It is found that the permeability of carbides formed by ablative reaction of ablative materials related to fractal structure has an effect on ablation process. From our preliminary results, the higher the permeability, the faster the ablation speed, and the more obvious the overall temperature rise is.

The fractal geometry provides a new insight for the description for actual pore structure of metal foams. In this paper, the fractal grids are introduced to describe the pore structure of metal foams. A two-dimensional unsteady melting model of phase change material (PCM) in the cross-fractal metal foam is developed and numerically simulated. The transient temperature variation and melting front evolution during the charging process in the cross-fractal metal foam are analyzed. The effects of initial temperature difference, porosity and fractal dimension on the melting heat transfer in fractal metal foam are examined and analyzed. The results indicate that the temperature distribution is more uniform and the melting rate is faster in the cross-fractal metal foam compared with that in the corresponding cavity structure. Interestingly, the fractal metal foam with smaller fractal dimension provides a faster heat flow path and hence enhances the melting performance though the porosity is identical. The melting performance in fractal metal foam can be enhanced when the metal foams have a lower porosity, a smaller fractal dimension and a larger initial temperature difference.

The transfer of fluids from porous matrix to fracture is a key issue to accurately predict the fluid flow behavior in porous–fracture media. In this work, to take into account the transfer of fluids, the analytical model of dimensionless permeability is proposed based on the fractal geometry theory for porous media. The proposed model is expressed as a function of microstructural parameters of the porous matrix and fracture, such as the pore area fractal dimension $D_p$, fractal dimension $D_T$ for tortuosity of tortuous capillaries, the ratio $\delta$ of the maximum pore size in porous matrix to fracture aperture, as well as the ratio $\mathrm{\Delta }{p}_m/\mathrm{\Delta }{p}_f$ of the pressure difference along the fracture to that along the porous matrix layers. The model reveals that the ratios $\delta$ and $\mathrm{\Delta }{p}_m/\mathrm{\Delta }{p}_f$ have significant influences on the permeability contribution from the porous matrix to the seepage behavior of the fracture. While the contribution of porosity of leak-wall porous surface of the fracture to the permeability is less than 10%. The present results may provide an important theoretical foundation for exploration of petroleum, gas and geothermal energy extraction systems.