Fractals

In this paper, a brief summary of some of the relevant models for non-Darcy flow is reviewed, they mainly include seven models, among them three models are based on the average hydraulic radius, two are based on fractal geometry theory and technique, one is based on first principle, one is based on experimental measurement. Each model has its own advantages and disadvantages.

A new self-similar multibarrier system is proposed and used to study transmission of Dirac electrons in graphene. Such system is based on the scaling of the length and energy of the barriers. The use of self-similar structures allows us to compare the transmission in graphene and gallium arsenide (GaAs). The transmission coefficient for charge carriers in graphene shows a surprising scaling behavior structure, which is not seen in GaAs. The scaling properties are established as a function of three parameters: barrier’s energy, the length and the generation of the system.

In this note, we investigate the regularity of Cantor’s one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by showing that its Hölder regularity lies between 0.35 and 0.72 almost everywhere (with respect to the Lebesgue measure).

A mixing time density of $A+B\to 0$ on a finite one-dimensional domain is defined for general initial and boundary conditions in which $A$ and $B$ diffuse at the same rate. The density is a measure of the number of $A$ and $B$ particles that mix through the center of the reaction zone. It also corresponds to the reaction density for the special case in which $A$ and $B$ annihilate upon contact. An exact expression is found for the generating function of the mixing time. The analysis is extended to multiple reaction fronts and finitely ramified fractals. The method involves using the kernel of the Laplace transform integral operator to map and analyze a moving homogeneous Dirichlet interior point condition.

In this paper, the network metrics are studied in a time series of the KOSPI and the KOSDAQ indices converting by the visibility graph algorithm. The degree distributions for the KOSPI and the KOSDAQ are proportional to a power law rather than the Poisson distribution. Since we mainly simulate and analyze the network metrics from the nodes and its links, our result cannot be found unambiguously to have universal and characteristic properties of statistical quantities via financial networks. Particularly, these topological properties may improve by implementing the statistical method and its technique from altered data of financial networks.

Self-similarity of random trees is related to the operation of pruning. Pruning ${ℛ}$ cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton–Strahler order of a vertex $v$ and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at $v$. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers ${{𝒩}}_k[K]$ and ${{𝒩}}_{ij}[K]$ are defined as the expected number of branches of order $k$, and the expected number of order-$i$ branches that merged order-$j$ branches, $j>i$, respectively, in a finite tree of order $K$. The Tokunaga coefficients are defined as $T_{ij}[K]={{𝒩}}_{ij}[K]/{{𝒩}}_j[K]$. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: $T_k:=T_{i,i+k}[K]$. We show that for self-similar trees, the condition ${limsup}_{k\to \infty }{T}_k^{1/k}<\infty$ is necessary and sufficient for the existence of the strong Horton law: ${{𝒩}}_k[K]/{{𝒩}}_1[K]\to R^{1-k}$, as $K\to \infty$ for some $R>0$ and every $k\ge 1$. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.

The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

Roughness measurement of knee implant surfaces is investigated. The study of roughness measurement show that the topography of knee implant surface is multi-scale and surface spectra follows a power law behavior. A magnification of rough surface topography implies that there is no difference between original and magnified profile of implant surface. For implant surface, statistical parameters such as variance of height, curvature, and slope are found to be scale-dependent. Fractal representation of implant surface shows that the size-distribution of the multi-scale contacts spots follows a power law and is characterized by the fractal dimension of implant surface.

Fractal surface description of the rough surfaces of knee implant is used to obtain force–displacement relationship of the contact force. Using an approximate function through the fusion of two piecewise functions, energy absorption of a knee implant in a single cycle of load–unload is obtained.

The concept of local fractional derivative was introduced in order to be able to study the local scaling behavior of functions. However it has turned out to be much more useful. It was found that simple equations involving these operators naturally incorporate the fractal sets into the equations. Here, the scope of these equations has been extended further by considering different possibilities for the known function. We have also studied a separable local fractional differential equation along with its method of solution.

One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.

We investigate bond percolation on the iterated barycentric subdivision of a triangle, the hexa-carpet, and the non-p.c.f. Sierpinski gasket. With the use of known results on the diamond fractal, we are able to bound the critical probability of bond percolation on the non-p.c.f. gasket and the iterated barycentric subdivision of a triangle from above by 0.282. We then show how both the gasket and hexacarpet fractals are related via the iterated barycentric subdivisions of a triangle: the two spaces exhibit duality properties although they are not themselves dual graphs. Finally, we show the existence of a non-trivial phase transition on all three graphs.

Transport in porous media is common in nature, attracting many attentions for a long time. Tree-like network model is often used as a simplification for porous space, expressing the complexity of pore spaces instead of capillary bundle. To investigate spontaneous imbibition characteristics in this network, a dynamic asymmetric branch-like capillary model is used to represent basic network structure, using fractal method to represent tortuosity. This work investigates the influence of parameters on imbibition process in the branch-like capillary model. An analytical equation for the imbibition mass versus time is derived. Parameters from capillary structures to liquid properties are taken into account and analyzed based on the numerical solution of the equation. It is found that the imbibition process in asymmetric branch-like capillary model can be recognized by four sections and brunching tubes are positive for imbibition process. Concomitantly, meniscus arrest event is simulated and discussed. Moreover, the influence of parameters on imbibition process is discussed. These parameters can be classified as static and dynamic. Static parameters mainly change the capillary force, which are related to the ultimate imbibition mass or imbibition ability, while dynamic parameters mainly have influence on resistance of flowing fluid, which are related to the imbibition speed in the imbibition process.

Detrended fluctuation analysis (DFA) is used to examine long-range dependence in variations and volatilities of American treasury bills (TB) during periods of low and high movements in TB rates. Volatility series are estimated by generalized autoregressive conditional heteroskedasticity (GARCH) model under Gaussian, Student, and the generalized error distribution (GED) assumptions. The DFA-based Hurst exponents from 3-month, 6-month, and 1-year TB data indicates that in general the dynamics of the TB variations process is characterized by persistence during stable time period (before 2008 international financial crisis) and anti-persistence during unstable time period (post-2008 international financial crisis). For volatility series, it is found that; for stable period; 3-month volatility process is more likely random, 6-month volatility process is anti-persistent, and 1-year volatility process is persistent. For unstable period, estimation results show that the generating process is persistent for all maturities and for all distributional assumptions.

Combine Chebyshev systems with fractal interpolation, certain continuous functions have been approximated by fractal interpolation functions unanimously. Local structure of these fractal interpolation functions (FIF) has been discussed. The relationship between order of Riemann–Liouville fractional calculus and Box dimension of FIF has been investigated.