Fractals

Fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions $\frac{p_{i,j}(\theta )}{q_{i,j}(\theta )}$, where $p_{i,j}(\theta )$ and $q_{i,j}(\theta )$ are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane $\mathrm{\Pi }$, and consequently the proposed partially blended RCTFIS lies above the plane $\mathrm{\Pi }$. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.

We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the F${}^{\alpha }$-Calculus.

Tailings from mining and milling of uranium ores potentially are large volumes of low-level radioactive materials. A typical environmental problem associated with uranium tailings is radon exhalation, which can significantly pose risks to environment and human health. In order to reduce these risks, it is essential to study the dynamical nature and underlying mechanism of radon exhalation from uranium mill tailings. This motivates the conduction of this study, which is based on the fractal and chaotic methods (e.g. calculating the Hurst exponent, Lyapunov exponent and correlation dimension) and laboratory experiments of the radon exhalation rates. The experimental results show that the radon exhalation rate from uranium mill tailings is highly oscillated. In addition, the nonlinear analyses of the time series of radon exhalation rate demonstrate the following points: (1) the value of Hurst exponent much larger than 0.5 indicates non-random behavior of the radon time series; (2) the positive Lyapunov exponent and non-integer correlation dimension of the time series imply that the radon exhalation from uranium tailings is a chaotic dynamical process; (3) the required minimum number of variables should be five to describe the time evolution of radon exhalation. Therefore, it can be concluded that the internal factors, including heterogeneous distribution of radium, and randomness of radium decay, as well as the fractal characteristics of the tailings, can result in the chaotic evolution of radon exhalation from the tailings.

To construct symmetrical patterns on the unit sphere from the planar iterative function systems (IFSs), we present a method of constructing IFSs with $D_3$ symmetry which is composed of three-fold rotational symmetries together with reflections. An algorithm is developed to generate strange attractors with $D_3$ symmetry on a triangular face and then project it onto the surface of the unit sphere to form aesthetics patterns with spherical symmetry. As an illustrative example, we consider the regular inscribed icosahedron in the unit sphere which contains 20 triangular faces. This method is valid to randomly generate aesthetic spherical patterns using planar IFSs.

Given metric spaces $E$ and $F$, it is well known that

Heterogeneities affecting reservoirs often develop at different scales. Previous studies have described these heterogeneities using different parameters depending on their size, and there is no one comprehensive method of reservoir evaluation that considers every scale. This paper introduces a multiscale fractal approach to quantify consistently the hierarchical heterogeneities of sandstone reservoirs. Materials taken from typical depositional pattern and aerial photography are used to represent three main types of sandstone reservoir: turbidite, braided, and meandering river system. Subsequent multiscale fractal dimension analysis using the Bouligand–Minkowski method characterizes well the hierarchical heterogeneity of the sandstone reservoirs. The multiscale fractal dimension provides a curve function that describes the heterogeneity at different scales. The heterogeneity of a reservoir’s internal structure decreases as the observational scale increases. The shape of a deposit’s facies is vital for quantitative determination of the sedimentation type, and thus enhanced oil recovery. Characterization of hierarchical heterogeneity by multiscale fractal dimension can assist reservoir evaluation, geological modeling, and even the design of well patterns.

In this paper, we investigate the vertical-affiliation-free (VAF) evolving networks whose node set is the basic squares in the process of generating the Sierpinski carpet and edge exists between any two nodes if and only if the corresponding basic squares intersect just on their boundary. Although the VAF networks gets rid of the hierarchial organizations produced naturally by the self-similar structures of fractals, we still prove that they are scale-free and have the small-world effect.

Gas diffusion in dry porous media has been a hot topic in several areas of technology for many years. In this paper, a diffusivity model for gas diffusion in dry porous media is developed based on fractal theory and Fick’s law, which incorporates the effects of converging–diverging pores and tortuous characteristics of capillaries as well as Knudsen diffusion. The effective gas diffusivity model is expressed as a function of the fluctuation amplitude of the capillary cross-section size variations, the porosity, the pore area fractal dimension and the tortuosity fractal dimension. The results show that the relative diffusivity decreases with the increase of the fluctuation amplitude and increases with the increase of pore area fractal dimension. To verify the validity of the present model, the relative diffusivity from the proposed fractal model is compared with the existing experimental data as well as two available models of Bruggeman and Shou. Our proposed diffusivity model with pore converging–diverging effect included is in good agreement with reported experimental data.

Spontaneous imbibition in porous media is common in nature, imbibition potential is very important for understanding the imbibition ability, or the ability to keep high imbibition rate for a long time. Structure parameters have influence on imbibition potential. This work investigates the process of spontaneous imbibition of liquid into a fractal tree-like network, taking fractal structure parameters into consideration. The analytical expression for dimensionless imbibition rate with this fractal tree-like network is derived. The influence of structure parameters on imbibition potential is discussed. It is found that optimal diameter ratio $\beta$ is important for networks to have imbibition potential. Moreover, with liquid imbibed in more sub-branches, some structures of parameter combinations will show the characteristic of imbibition potential gradually. Finally, a parameter plane is made to visualize the percentage of good parameter in all possible combinations and to evaluate the imbibition potential of a specific network system more directly. It is also helpful to design and to optimize a fractal network with good imbibition potential.

This paper studies the gap sequences of graph-directed sets satisfying the strong separation condition. An interesting application is to investigate the gap sequences of self-similar sets with overlaps.

The box dimension of the graph of non-affine, continuous, nowhere differentiable function $f^{\alpha }$ which is a fractal analogue of a continuous function $f$ corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of $\alpha$-fractal function $f^{\alpha }$ for equally spaced as well as arbitrary data sets are found.

In this paper, based on the un-weight hierarchical networks, a family of weighted hierarchical networks are introduced, the weight factor is denoted by $r$. The weighted hierarchical networks depend on the number of nodes in complete bipartite graph, denoted by $n_1$, $n_2$ and $n=n_1+n_2$. Assume that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them. We deduce the analytical expression of the average receiving time (ART). The obtained remarkable results display two conditions. In the large network, when $nr>n_1{n}_2$, the ART grows as a power-law function of the network size $|V(G_k)|$ with the exponent, represented by $\theta ={log}_n(\frac{nr}{n_1{n}_2})$, $0<\theta <1$. This means that the smaller the value of $\theta$, the more efficient the process of receiving information. When $nr\le n_1{n}_2$, the ART grows with increasing order $|V(G_k)|$ as ${log}_n|V(G_k)|$ or ${({log}_n|V(G_k)|)}^2$.