Narrow Results

There are several methods given for constructing the Sierpinski Triangle (ST). In particular, the ST can be created as a countably infinite intersection of contained figures, or as the countably infinite union of triangles. At first glance they both appear to create the same object. In this paper, we show that they are not the same. In addition, we analyze and compare with the previous two methods, a third method of constructing the ST which entails generating a countable number of points using an algorithm.

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct $\alpha$-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the $\alpha$-fractal rational quartic spline when the original function is in ${{𝒞}}^4(I)$. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the $\alpha$-fractal rational quartic spline to ${{𝒞}}^2$. The elements of the iterated function system are identified befittingly so that the class of $\alpha$-fractal function $Q^{\alpha }$ incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ $Q$. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

The construction of an abstract expressionist artwork is driven by chaotic mechanisms that sculpt multifractal characteristics. Jackson Pollock’s paintings, for example, arise due to the random process of depositing drops and jets of paint on a canvas. However, most of the paintings and drawings try to recreate with fidelity common forms, natural landscapes, and the human figure. Accordingly, in the context of the formation of statistically self-similar objects, a question persists: will it be possible to find some vestige of multifractal structure in drawings or paintings whose elaboration process tries to avoid chaos? In this work, we scrutinize into several artistic drawings in sand to answer this intriguing question. These pieces of art are elaborated using craters, furrows, and sand piles; and some of them are inscribed on the Representative List of the Intangible Cultural Heritage of Humanity. We prove that the sand drawings analyzed here are multifractal objects. This finding suggests that a piece of visual art, which may initially appear ordered, contains many components distributed at different degrees of self-similarity that substantially increase the structural complexity.

The paper approaches the construction of fractal surfaces of interpolation and approximation on the basis of a fractal perturbation of any mapping defined on a rectangle. Conditions for the differentiability of these elements are also provided. The fractal surfaces obtained may be used for the approximation of real-world data. The method proposed does not require any restriction on the type of data. Furthermore, the present approach does not imply the solution of large systems of equations. The paper considers both the continuous and the discontinuous case.

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical examples are also included to verify the conclusions.

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard diffusion mathematical equation, that considers the myocardium as a continuum. However, the modeling of structural properties and their influence on electrical propagation still reveal several limitations. In this paper, a model of cardiac electrical propagation is proposed based on complex order derivatives. By assuming that the myocardium has an underlying fractal process, the complex order dynamics emerges as an important modeling option. In this perspective, the real part of the order corresponds to the fractal dimension, while the imaginary part represents the log-periodic corrections of the fractal dimension. Indeed, the imaginary part in the derivative implies characteristic scales within the cardiac tissue. The analytical and numerical procedures for solving the related equation are presented. The sinus rhythm and persAF conditions are implemented using the Courtemanche formalism. The electrophysiological properties are measured and analyzed on different scales of observation. The results indicate that the complex order modulates the electrophysiology of the atrial system, through the variation of its real and imaginary parts. The combined effect of the two components yields a broad range of electrophysiological conditions. Therefore, the proposed model can be a useful tool for modeling electrical and structural properties during cardiac conduction.

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the ${{ℒ}}^p$ convergence of this type of interpolants for $1\le p<\infty$ extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

In recent years, the application of fractal theory in construction materials has drawn tremendous attention worldwide. This special issue section containing seven papers publishes the recent advances in the investigation and application of fractal-based approaches implemented in construction materials. The topics covered in this introduction mainly include: (1) the fractal characterization of construction materials from nano- to micro-scales; (2) combining fractals methods with other theoretical, numerical and/or experimental methods to evaluate or predict the macroscopic behavior of construction materials; (3) the relationship of fractal dimension with the macro-properties (i.e. mechanical property, shrinkage behavior, permeability, frost resistance, abrasion resistance, etc.) of construction materials.

Fractal Calculus is designed to reveal the study of waves. Most of the waves are very grim to model correctly. Fractal representation of waves helps to better understand complex wave phenomena. In this paper, a Maclaurin series method (MSM) is proposed to obtain the exact and approximate solutions of fractal nonlinear differential-difference models produced by coupled nonlinear optical waveguides. The method is very simple, easy to understand, and minimizes calculation compared to existing approximate methods.

In this paper, a new algorithm to select the relevant nodes — those that maintain the cohesion of the network — of the complex network is presented. The experiments on most of the real complex networks show that the proposed approach outperforms centrality measures as node degree, PageRank algorithm and betweenness centrality. The rationale of the algorithm for extracting relevant nodes is to discover the self-similarity of the network. As seen in the algorithm, throughout the extraction sequence of relevant nodes, differences are advised with node degree, PageRank algorithm and betweenness centrality. Finally, empirical evidence is considered to show that complex network robustness is a nonlinear function of the small-worldness measure.

This paper is devoted to the study of self-similar fractal rep tiles of the Euclidean planes ${ℝ}^2$ and ${ℝ}^3$ using integer matrices. We construct many examples appearing in the literature, as well as new examples by integer matrices. Several exotic and radially symmetric self-similar rep tiles and their variations are obtained with change of bases extending the work by Bandt and others. We also present a few examples of polyhedral self-similar rep tiles including variations.

We present an in-depth analysis of the fractal nature of 21 classical music pieces previously shown to have scale-free properties. The musical pieces are represented as networks where the nodes are musical notes and respective durations, and the edges are its chronological sequence. The node degree distribution of these networks is analyzed, looking for self-similarity. This analysis is done in the full network, in its fractal dimensions, and its skeletons. The assortativeness of the pieces is also studied as a fractal property. We show that two-thirds of these networks are scale-invariant, i.e. scale-free in some dimension or their skeleton. In particular, two pieces were given attention because of their exceptional tendency for fractality.

It is of crucial significance to study a class of complex dynamic planar and spatial systems with disparate coefficients by taking advantage of thoughts of Julia set. In this paper, we present theoretical control methods for the stable domain and Julia sets of systems. Then, we utilize symmetry to exhibit the forms of compositional Julia sets of complex dynamic planar systems. Furthermore, by selecting different coefficients, the location, size, area and shape of Julia set can be controlled, the spatial Julia sets also have certain characteristics, this supplies a possible reference for how to control the stable domain of specific complex dynamic systems. The simulation results are in good agreement with the effectiveness of the methods.

The impact of crack–cocaine dependence on the quality and microarchitecture of the mandibular bone tissue requires further investigation. This cross-sectional study evaluated the fractal dimension and panoramic radiomorphometric indices in crack–cocaine-addicted men. Panoramic radiographs were obtained from 24 addicted and 24 nonaddicted men (control individuals) between the ages of 18 years and 60 years. The fractal dimensions of four different regions, along with the cortical mandibular, mental, and panoramic mandibular indices, were evaluated bilaterally. Significance level of 5% ($p=0.05$) was adopted in the statistical analysis. Mean fractal dimension value of all the four different regions of the jaw in the addict group ($1.27\pm 0.05$) was lower than that of the nonaddict group ($1.32\pm 0.03$; $p<0.001$). Furthermore, the inter-group panoramic radiomorphometric indices were not significantly different ($p\ge 0.16$). Mean fractal dimension value was associated with the duration of addiction ($R=0.47$; $p=0.01$), contrary to the indices ($p\ge 0.10$). Crack–cocaine addiction and longer duration of addiction were associated with lower fractal dimension values in the mandibular bone. Therefore, patients and dentists should be aware of this condition while planning periodontal, implant, and orthodontic therapies. Furthermore, crack–cocaine patients should be referred to a specialist for the evaluation of osteoporosis and osteopenia.

In this paper, we study shape preserving aspects of bivariate $\alpha$-fractal functions. Its specific aims are: (i) to solve the range restricted problem for bivariate fractal approximation (ii) to establish the fractal analogue of lionized Weierstrass theorem of bivariate functions (iii) to study the constrained approximation by ${{𝒞}}^r$-bivariate $\alpha$-fractal functions (v) to investigate the conditions on the parameters of the iterated function system in order that the bivariate $\alpha$-fractal function $f^{\alpha }$ preserves fundamental shapes, namely, positivity and convexity (concavity) in addition to the smoothness of $f$ over a rectangle (vi) to establish fractal versions of some elementary theorems in the shape preserving approximation of bivariate functions.

Many techniques are currently available to test time series data for fractal patterns, but concerns have been raised about the fact that they do not always yield consistent results. Furthermore, fractal and nonfractal patterns, such as seasonal variation, tend to be heavily correlated in time series data. It is therefore essential that fractal patterns are clearly distinguished from nonfractal ones in the estimation process. The study is concerned with this capability in the following techniques: detrended fluctuation analysis (DFA), re-scaled range analysis (R/S), Higuchi’s fractality dimension, smoothed spectral regression, Geweke & Porter Hudak’s (GPH) estimator, the spectral periodogram (Sperio) estimate, Whittle’s estimate and the fractional differencing parameter, as well as the conventional time series diagnostics (unit root test, autocorrelation function (ACF) plots). A brief overview is provided of each fractal estimation method and results are presented of their application to three existing time series data sets: annual recordings of the water discharge level of the River Nile (663 years), daily number of births to teens in the state of Texas (5 years) and daily attendance rates in one urban school (4 years). Results indicate that all techniques indicate fractality in two of the three data sets (River Nile flow, school attendance). The results for the births to teens data were equivocal with a strong suggestion of both seasonal and fractal patterns in the ACF plot, but only seasonality in the modeling results. On the other hand, the ACF plot of the school attendance data hints at seasonality that is not picked up in the estimation. For a better perspective on these results, a simulation study was undertaken testing the ability of these techniques to distinguish seasonal and fractal processes under variable input conditions. The results of these simulations show largely accurate fractality estimates in the absence of seasonality. However, fractality estimates tend to be inflated when seasonal variability is present in the data. An integrated analytical approach is therefore recommended for these types of data, triangulating the results from the time series diagnostics with the parameter estimates from formal modeling and making seasonal adjustments to the data if necessary.

The current COVID-19 pandemic mainly affects the upper respiratory tract. People with COVID-19 report a wide range of symptoms, some of which are similar to those of common flu, such as sore throat and rhinorrhea. Additionally, COVID-19 shares many clinical symptoms with severe pneumonia, including fever, fatigue, dry cough, and respiratory distress. Several diagnostic strategies, such as the real-time polymerase chain reaction technique and computed tomography imaging, which are more costly than chest radiography, are employed as diagnostic tools. The purpose of this paper is to describe the role of the d-summable information dimension of X-ray images in differentiating several lesions and lung illnesses better than both fractal and information dimensions. The statistical analysis shows that the d-summable information dimension model better describes the information obtained from the X-ray images. Therefore, it is a more precise measure of complexity than the information and box-counting dimension. The results also show that the X-ray images of COVID-19 pneumonia reveal greater damage than those of tuberculosis, pneumonia, and various lung lesions, where the damage is minor or much focused. Because the d-summable information dimension increases as the image complexity decreases, it could pave the way to formulate a new measure to quantify the lung damage and assist the clinical diagnosis based on the area under the d-summable information model. In addition, the physical meaning of the $\nu$ parameter in the d-summable information dimension is given.

The smart wearable devices that can track the fitness activities are getting famous these days due to their easy-to-use features. The fitness trackers can work for an individual in a promising manner, provided that the user is well familiar with the device and is committed with the timelines. Several reports have provided evidence that these smart wearable devices have not showed promising results and in most of the cases, people have stopped using them, few weeks after the purchase. There are several reasons linked with this response. During this research, we have worked on the correlations of weight loss via smart device with the age, gender, body mass index (BMI) and ideal body weight (IBW), with the aid of gradient boosted decision trees (XGBoost) and support vector machine (SVM) learning tools. XGBoost and SVM are capable of dealing with complex datasets, with higher frequencies, and for data emerging from multiple sources. These machine learning tools use kernel functions for the clustering and other classification measures, and are thus better as compared to the logistic methods. Next, the time series forecasting tools are discussed with the Bayesian hyperparametric optimization. The time series of the weight loss monitoring of each individual, depicted in this manner, provided complex fractal patterns, with reduction in amplitude, with the passage of time.

In this paper, the convergence of the Nash-Q-Learning algorithm will be studied mainly. In the previous proof of convergence, each stage of the game must have a global optimal point or a saddle point. Obviously, the assumption is so strict that there are not many application scenarios for the algorithm. At the same time, the algorithm can also get a convergent result in the two Grid-World Games, which do not meet the above assumptions. Thus, previous researchers proposed that the assumptions may be appropriately relaxed. However, a rigorous theoretical proof is not given. The convergence point is a fractal attractor from the view of Fractals, general proof of convergence of the Nash-Q-Learning algorithm will be shown by the mathematical method. Meanwhile, some discussions on the efficiency and scalability of the algorithm are also described in detail.

The focus of this paper is to study unbounded variation functions from the perspective of Hölder conditions. Three special unbounded variation functions have been constructed. The first is a continuous function of unbounded variation that satisfies the Hölder condition of a given order and the second is a continuous function of unbounded variation that does not satisfy the Hölder condition of any order. The third is a continuous function of unbounded variation defined on any sub-interval of the interval $I$. Then, specific fractal dimension analysis of the above functions and relevant conclusions have been investigated. Finally, combining functional analysis and reinforcement learning, the convergence of reinforcement learning algorithms can be proved in unified framework.