Fractals

For an iterated function system (IFS) with contractive similarities, the post-critically finite (p.c.f.) condition and H-condition are two important separation conditions in the literature. But their relationship is not clear. In this paper, we show that if the similarity matrices of the IFS are commensurable, then the p.c.f. condition implies H-condition.

Inspired by the snowflake structure, an innovative Koch-fractal fin is proposed to optimize the fin geometry of the latent heat thermal energy storage (LHTES) units. A model of unsteady heat transfer accompanied with phase change is developed and numerically analyzed to investigate the effect of fin structure on the discharging process of a LHTES unit. The dynamic response of heat release, the dynamic temperature response and solidification front evolution of a LHTES unit with Koch-fractal fins are discussed and compared with the corresponding radial fins. Furthermore, a comprehensive evaluation of thermal performance of LHTES units is conducted in terms of the TES capacity, TES rate and solidification time. The results indicate that the heat release rate of a LHTES unit with Koch-fractal fins is faster than that with radial fins. Moreover, because the Koch-fractal fins have advantages of higher specific surface area, faster heat flow path from point to surface and smaller thermal resistance arising from the reasonable spatial layout, the evolution of solidification front is faster and the temperature distribution is more uniform. The results of quantitative evaluation show that although the TES capacity is identical, the TES rate of a LHTES unit with Koch-fractal fins is six times that with radial fins.

The research on the ultimate crushing state of coarse aggregates is beneficial to analyze and predict the evolutionary process of crushing. The Growing Path method uses the two-dimensional fractal geometry structure to simulate the size variation of particle size fraction during the particle breakage of coarse aggregates and it serves to investigate the ultimate fractal dimension corresponding to the ultimate crushing state of coarse aggregates. This method manifests the self-growing characteristics of particle size distribution in the process of particle crushing. This study found that the two-dimensional image of ultimate fractal model was precisely similar to that of the Sierpinski gasket of fractal theory when the ultimate crushing state was reached. The results from the model analysis show that the theoretically ultimate fractal dimension is about 2.585, which is consistent with the existing results calculated from the three-dimensional ultimate fragmentation model of cataclastic rock located in the fault zones. The relationship between two fractal models was analyzed. Furthermore, the application of fractal geometry presented in this study will also serve as a reference for the analysis of the other chaos phenomena observed in geotechnical engineering.

This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector.

A two-dimensional model is developed to numerically study the water flow boiling through a tree-shaped microchannel by VOF method. In this work, the bubble dynamics and flow patterns along the channel are examined. Additionally, the pressure drop, heat transfer performance and the effects of mass flow rate and heat flux on the heat transfer coefficient are analyzed and discussed. The numerical results indicate that, there are three main bubble dynamic behaviors at the wall, namely coalesce-lift-off, coalesce-slide and coalesce-reattachment. At the bifurcation in high branching level, the slug bubbles may coalesce or breakup. The flow patterns of bubbly, bubbly-slug flows occur at low branching level and slug flow occurs at high branching level. The passage of bubbles causes the increasing of fluid temperature and local pressure. Additionally, the pressure drop decreases with the branching level. The flow pattern and channel confinement effect play a vital role in heat transfer performance. The nucleate boiling dominant heat transfer is observed at low branching level, the heat transfer performance is enhanced with increasing branching level from $k=0$ to 2. While, at high branching level, the heat transfer performance becomes weaker due to the suppression of nucleate boiling. Moreover, the heat transfer coefficient increases with the mass flow rate and heat flux.

Intuitively, edge weight has an effect on the dynamical processes occurring on the networks. However, the theoretical research on the effects of edge weight on network dynamics is still rare. In this paper, we present two weighted network models by adjusting the matching relationship between weights and edges. Both network models are controlled by the weight factor $r\in (0,1]$. They have the same connection structure and weight sequence when $r$ is fixed. Based on their self-similar network structure, we study two types of biased walks with a trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. For both weighted scale-free networks, we obtain exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process in both network models. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule and the spectral dimension of the fractal network, and not all ATT for the weighted networks are affected by the weight factor $r$. In other words, not all weight adjustments can change the trapping efficiency in the network. We hope that the present findings could help us get deeper understanding about the influence factor of biased walk in complex systems.

In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

In this paper, we consider the non-spectrality of the planar self-affine measures ${\mu }_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a four-element digit set

In this paper, fluid transport through fibrous porous media is studied by the fractal theory with a focus on the effect of surface roughness of capillaries. A fractal model for Kozeny–Carman (KC) constant and dimensionless permeability of fibrous porous media with roughened surfaces is derived. The determined KC constant and dimensionless permeability of fibrous porous media with roughened surfaces are in good agreement with available experimental data and existing models reported in the literature. It is found that the KC constant of fibrous porous media with roughened surfaces increases with the increase of relative roughness, porosity, area fractal dimension of pore and tortuosity fractal dimension, respectively. Besides, it is seen that the dimensionless permeability of fibrous porous media with roughened surfaces decreases with increasing relative roughness and tortuosity fractal dimension. However, it is observed that the dimensionless permeability of fibrous porous media with roughened surfaces increases with porosity. With the proposed fractal model, the physical mechanisms of fluids transport through fibrous porous media are better elucidated.

First, we introduce a generalized $m$-convexity concept defined on the real linear fractal set ${ℝ}^{\alpha }$ $(0<\alpha \le 1)$ and discuss the relation between generalized $m$-convexity and $m$-convexity. Second, we present several important properties of the generalized $m$-convex mappings. Meanwhile, via local fractional integrals, we also derive certain estimation-type results on generalizations of Hadamard-type, Fejér-type and Simpson-type inequalities. As applications related to local fractional integrals, we construct several inequalities for generalized probability density mappings and $\alpha$-type special means.

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

The essence of Huber fractal image coding (HFIC) is to predict the fractal code of a noiseless image as accurately as possible from its corrupted observation with outliers by adopting Huber M-estimation technique. However, the traditional HFIC is not quite satisfactory mainly due to the absence of contractivity restriction for the estimate of the fractal parameters (actually, it is a fundamental requirement in the theory of fractal image coding). In this paper, we introduce a primal-dual algorithm for robust fractal image coding (PD-RFIC), which formulates the problem of robust prediction of the fractal parameters with contractivity condition as a constrained optimization model and then adopts a primal-dual algorithm to solve it. Furthermore, in order to relieve using the corrupted domain block as the independent variable in the proposed method, instead of using the mean operation on a $2\times 2$ subblock in the traditional HFIC, we apply a median operation on a larger subblock to obtain the contracted domain blocks for achieving the robustness against outliers. The effectiveness of the proposed method is experimentally illustrated on problems of image denoising with impulse noise (specifically, salt & pepper noise and random-valued noise). Remarkable improvements of the proposed method over conventional HFIC are demonstrated in terms of both numerical evaluations and visual quality. In addition, a median-based version of Fisher classification method is also developed to accelerate the encoding speed of the proposed method.

This paper concerns the average distances of evolving networks modeled by Sierpinski tetrahedron. We express the limit of average distances on reorganized networks as an integral of geodesic distance on Sierpinski tetrahedron. Based on the self-similarity and renewal theorem, we obtain the asymptotic formula on the average distance of our evolving networks.

Local fractional calculus is used to derive fractional partner of the KdV–Burgers–Kuramoto equation, its physical understanding is elucidated and its solution properties are revealed. Adomian’s decomposition method (ADM) and the variational iteration method (VIM) are applied to obtain its approximate analytical solution, the results have both theoretical importance and practical applications.

For any Lalley–Gatzouras self-affine carpet, we find out that the length of any curves from $(x_1,x_2)$ to $(y_1,y_2)$ in the carpet is no less than $|x_1-y_1|+|x_2-y_2|$.

In this work, we propose a new model for the dielectric constant and thermal conductivity, and apply it to the design of composites with low dielectric constant and high thermal conductivity based on fractal theory and effective medium theory. In particular, we use this model to prepare porous AlN/BN/SiOC composites with low dielectric constant and high thermal conductivity in different component fractions. We successfully synthesize ceramic samples with low dielectric constants $(<3)$ and high thermal conductivity ($\ge 2\mathrm{\text{W}}\cdot {\mathrm{\text{m}}}^{-1}\cdot {\mathrm{\text{K}}}^{-1}$). These results indicate that the proposed fractal design is valid.