Random Fractals Research Articles

Design of Random and Deterministic Fractal Surfaces from Voronoi Cells

We show a fractal surface generation method that, unlike other methods, generates both random and deterministic fractals that model natural and architectural elements. The method starts with a succession of sets of sites, which determine, by means of a metric, a succession of Voronoi tessellations of the region where the fractal is defined. For each element of the tessellation sequence we define a tessellation function which depends on each tile. This generates a succession of tessellation functions that will be the parameter of the same seed function. Finally, the fractal is generated by a weighted sum of the seed function evaluated on each value of the succession of parameters. If the sites used to generate the Voronoi tessellation are random, natural elements such as mountains, craters, lakes, etc. are generated; if they are deterministic, architectural and decorative elements are generated. In addition, the designers can control the morphology of the generated fractal by simply varying the metric.

Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.

Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors

We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain D⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}\\subset {\\mathbb {R}}^d$$\\end{document}, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.

Adaptive stochastic fractal search algorithm for multi-objective optimization

Stochastic fractal search algorithm, a relatively efficient algorithm has certain advantages in solving multi-objective optimization problems. Decomposition-based strategy is effective for solving multi-objective optimization, which guarantees the diversity of solutions. On the other hand, a challenge for decomposition-based strategy is to enhance the algorithm’s convergence while maintaining the algorithm’s distribution. This paper proposes a reference vector-guided stochastic fractal search algorithm for multi-objective optimization(RVSFS). In the proposed algorithm, the leader solution of each subspace is selected by reference vectors to assist the generation of new solutions and ensure the convergence of the algorithm. A new selection criterion length–angle distance (LAD) is proposed. Furthermore, the Gaussian walk and Levy flight are conducted in different searching stages to balance the exploration and exploitation ability. To speed up convergence and maintain the diversity of solutions, an adaptive dynamic proportion mechanism is designed when using the random fractals and Gaussian walks to generate offspring. Since information is shared and exchanged among different solutions in each iteration, the searching process is accelerated and the exploration capability can be enhanced contemporary. On top of this, a vector-guided enhanced selection operator is designed in the elite selection step to provide desirable distribution. Our experimental results on 45 benchmarks from UF and WFG test suits show prominent competitiveness compared with five state-of-the-art algorithms. RVSFS is very competitive on all 9 UF benchmarks in terms of the metric inverted generational distance and also performs well on 14 of 36 benchmarks from WFG in terms of the metric inverted generational distance and the metric hypervolume.

Mean Lipschitz–Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz–Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions, we show that rescaled limits of these mean values exist as the parallel radius tends to 0 . Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

Construct stochastic processes model to solve graphic and math problems

The use of probabilistic reasoning has been used to a variety of applications, such as image identification, computer diagnostics, stock price prediction, movie recommendation, and cyber intrusion detection. However, the range of probabilistic programming was limited until recently (partly due to the low computing capacity), and the bulk of inference techniques needed to be developed manually for each task. However, in 2015, using a 50-line probabilistic computer vision program, 3D representations of human faces were produced from 2D images of those faces.Use a computer to model various complex stochastic phenomena. All projects are programmed in java language. Find the actual number of 1s in a binary number. Formulate and analyze mathematical models to real life phenomena like Data Matrix and QR code. Those code mentioned above can provide a link to the real-life website or some certain message needed to be shown. using some random numbers to model uncertainty. To model some fractals. The first fractal maybe not self-avoiding. But the second one will never meet the same path for twice. To apply the concept stochastic processes. More java features and some random fractals and regression. Multiple factor analysis.

Langevin analogy between particle trajectories and polymer configurations.

A diffusive trajectory drawn by the generalized Langevin equation(GLE) for a colloidal particle evokes a random fractal of a static polymer configuration. This article proposes a static GLE-like description that enables the generation of a single configuration of a polymer chain with the noise formulated to satisfy the static fluctuation-response relation (FRR) along a one-dimensional chain structure but not along a temporal coordinate. A remarkable point is qualitative differences and similarities in the FRR formulation between the static and the dynamic GLEs. Guided by the static FRR, we further make analogous arguments in light of stochastic energetics and the steady-state fluctuation theorem.

Corrigendum to Crack detection based on deep learning: a method for evaluating the object detection networks considering the random fractal of crack

Corrigendum to Crack detection based on deep learning: a method for evaluating the object detection networks considering the random fractal of crack

Crack detection based on deep learning: a method for evaluating the object detection networks considering the random fractal of crack

The visual test for cracks is a tough work for human eyes. With the help of cameras, methods for object detection based on deep learning can detect the cracks automatically from digital images. However, an anomaly is observed that the models for crack detection are obviously underestimated by the widely used mean Average Precision (mAP) standard. In this study, by theoretical analyses about the topology of cracks, we attribute the failure of mAP standard to the random fractal feature. More deeply, this issue is due to the strict box matching during the calculation. To solve this problem, we construct a practical fractal-available evaluation method adopting the idea of covering box matching. Several metrics including Extended Recall (XR), Extended Precision (XP), and Extended F-score (Fext) are defined for scoring the crack detectors. Then, we conduct experiments on several mainstream models for object detection. Results show two advantages of the proposed evaluation method: (1) the proposed metrics evaluate cases better; (2) the proposed evaluation is non-maximum suppression-unrelated. Moreover, we present a case study to optimize a faster region-based convolutional neural networks model for crack detection adopting the proposed metrics. Recall (represented by the XR) of our best model achieves an industrial-level at 95.8%. These results denote that the proposed evaluation method removes the obstacle in evaluating the methods for object detection for cracks, which show great potential for automatic industrial inspections.

ARTIFICIAL INTELLIGENCE AND STOCHASTIC OPTIMIZATION ALGORITHMS FOR THE CHAOTIC DATASETS

Almost every natural process is stochastic due to the basic consequences of nature’s existence and the dynamical behavior of each process that is not stationary but evolves with the passage of time. These stochastic processes not only exist and appear in the fields of biological sciences but are also evident in industrial, agricultural and economical research datasets. Stochastic processes are challenging to model and to solve as well. The stochastic patterns when repeated result into random fractals and are very common in natural processes. These processes are usually simulated with the aid of smart computational and optimization tools. With the progress in the field of artificial intelligence, smart tools are developed that can model the stochastic processes by generalization and genetic optimization. Based on the basic theoretical description of the stochastic optimization algorithms, the stochastic learning tools, stochastic modeling, stochastic approximation and stochastic fractals, a comparative analysis is presented with the aid of the stochastic fractal search, multi-objective stochastic fractal search and pattern search algorithms.

Topological random fractals

The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of topological electronic states to random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from a two-dimensional site percolation process, we demonstrate that these topological random fractals exhibit the hallmarks of topological insulators. Specifically, our large-scale numerical studies reveal that topological random fractals display a robust mobility gap, support quantized conductance and represent a well-defined thermodynamic phase of matter. The finite-size scaling analysis further suggests that the critical properties are not consistent with the expectations of class D systems in two dimensions, hinting to the nontrivial relationship between fractal and integer-dimensional topological states. Our results establish topological random fractals as the most complex systems known to support nontrivial band topology with their distinct unique properties.

Large Intersection Property for Limsup Sets in Metric Space

We show that limsup sets generated by a sequence of open sets in compact Ahlfors s-regular $(0<s<\infty )$ space $(X,{\mathscr{B}},\mu ,\rho )$ belong to the classes of sets with large intersections with index λ, denoted by $\mathcal {G}^{\lambda }(X)$ , under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices γ2 and δ belong to $\mathcal {G}^{s-\delta -\gamma _{2}}(X)$ almost surely, and random covering sets with exponentially mixing property belong to $\mathcal {G}^{s_{0}}(X)$ almost surely, where s0 equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space.

Analysis of droplet size distribution and selection of spray parameters based on the fractal theory

Using multiple parameters to represent the droplet size distribution (DSD) always leads to various outcomes that make it hard to decide. The research combined theoretical derivation with experiments to study the random fractal of the DSD. The function of the DSD was deduced, and the fractal dimension (FD) was used as an independent parameter to characterize the DSD. The DSD was measured by the Malvern measuring apparatus under different spray pressure. The results showed that the FD of the five nozzles measured experimentally is between 1.5 and 3. The independent parameter FD has more advantages than multi-parameters (the Sauter mean diameter and the size distribution span). In the air cleaning technology by spray, the FD was used as an evaluation parameter. The optimal distance from the nozzle to the wetting fine wire plate was determined to be 75–85 cm, and the total dust efficiency of the wetting fine wire plate air cleaning device can reach 98.98%.

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces.

Unique Periodic Rings Composed of Fractal-Growth Dendritic Branching in Poly(p-dioxanone).

Amorphous poly(p-vinyl phenol) (PVPh) was added into semicrystalline poly(p-dioxanone) (PPDO) to induce a uniquely novel dendritic/ringed morphology. Polarized-light optical, atomic-force and scanning electron microscopy (POM, AFM, and SEM) techniques were used to observe the crystal arrangement of a uniquely peculiar cactus-like dendritic PPDO spherulite, with periodic ring bands not continuingly circular such as those conventional types reported in the literature, but discrete and detached to self-assemble on each of the branches of the lobs. Correlations and responsible mechanisms for the formation of this peculiar banded-dendritic structure were analyzed. The periodic bands on the top surface and interior of each of the cactus-like lobs were discussed. The banded pattern was composed of feather-like lamellae in random fractals alternately varying their orientations from the radial direction to the tangential one. The tail ends of lamellae at the growth front spawned nucleation cites for new branches; in cycles, the feather-like lamellae self-divided into multiple branches following the Fibonacci sequence to fill the ever-expanding space with the increase of the radius. The branching fractals in the sequence and the periodic ring-banded assembly on each of the segregated lobs of cactus-like dendrites were the key characteristics leading to the formation of this unique dendritic/ringed PPDO spherulite.

Conductivity Estimates of Fractal Models of Geological Media

AbstractGeological media are inherently fractal over multiple length scales, exhibiting a wide range of fractal dimensions . We study the size dependent conductivity of models of geological media based on so called regular random fractals. Specifically, we consider generalizations of the classic Menger sponges where the impermeable cells can take random locations. This results, for a given random fractal generator and a given construction order, in multiple realizations of the target media, each one maintaining the same . We use direct numerical simulation to estimate the conductivity, over a number of patterns, with orders . The conductivity bounds and estimates for regular fractal media are, in general, not applicable to these random fractal models. The conductivity of a specific realization may differ from the ensemble average by a few percents, which suggests a lack of self‐averaging; however, the self‐averaging is regained if the size of the domain of interest is beyond the maximal fractality length scale. We propose a low computational cost approach for estimating the ensemble average conductivity of a given random fractal at a given order, which matches the order of magnitude predicted from more intensive calculations. Through tabulation of conductivities of generators of random fractals with small scaling factors and fitting models, we explain how these fractals can model geological media in which fractal dimensions and conductivities (or effective diffusion coefficients) are measured.

An extensive survey on fractal structures using iterated function system in patch antennas

SummaryOver the last few years, fractal structure has been used in several engineering fields such as surface physics, telecommunication, fluid mechanics, and so forth. In telecommunication, often there is a demand for miniaturized patch antennas with fractal structures because it is capable of handling very high frequencies with its diminutive structure. The space filling capability of the fractal structures is used to effectively distribute the surface current all over the radiating material. Fractal structures have a better size reduction capability which aids a miniaturized antenna design. This article provides a generalized and detailed survey on fractal structures in a Euclidean space using iterated function system (IFS) in patch antennas. There are various techniques to design a fractal structure, namely, IFSs, escape‐time fractals, strange attractors, L‐systems, random fractals, and so forth. IFSs are widely used technique to construct a finite set of construction mapping on a complete metric space. The fractal geometries designed using the technique are Sierpinski carpet and gasket, Koch snowflake, deterministic tree/binary tree, Gosper island, Minkowski island, Cantor set, T‐square, and some space filling fractals. This paper deals with the above given fractal structures with their design expressions and shapes in recent developments and applications mentioned. The fractal geometries are used in MIMO antennas to abridge the mutually coupled radiators and improve the gain and efficiency of the antenna. This paper also analyzes the effects of fractal structures on MIMO antennas and discusses the ascendancy fractal structures by cross‐referencing the significant antenna parameters.

Random fractals and their intersection with winning sets

AbstractWe show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$.We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

Covering fractals with constant radius tiles: Distribution functions and their implications for resource management

The need to optimize and rationally exploit, both fractally distributed natural resources and civil infrastructures, has led us to study the covering of random fractals with circular tiles of a fixed radius. The study has been carried out for different fractal dimensions and radii of the tiles. From the study it is deduced that the fractal mass enclosed by the circular tiles follows a parabolic fractal distribution, which depends only on two parameters. In addition, the distribution parameters have been expressed in terms of fractal dimension and radius of the tiles. Therefore, it is possible to estimate the fractal mass enclosed by the circular tiles of a given radius, once the fractal dimension is known. Consequently, the distribution allows a rational and optimal management of the resources located on the fractal.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.