Fractal Function Research Articles

π‐shaped circularly polarized high‐gain Minkowski fractal antenna for radar applications

SummaryThis article investigates a compact π‐shaped circularly polarized Minkowski fractal antenna (MFA). The proposed radiator is taking a π‐shape with semicircular cuts along the opposite side of the patch with a V‐cut at the bottom. A second‐order Minkowski fractal curve is implemented on the bottom ground plane. The antenna element is powered by a coaxial probe. The asymmetric structure generates two orthogonal modes, which results in circular polarization. A maximum gain response of 7.1 dBic is observed in the impedance band width (IBW) covering 9.7 to 11.2 GHz. The patch is fabricated on substrate RT/Duroid 5880™ (𝜀𝑟=2.2). The antenna has a compact dimension of 0.90λ0 × 0.90λ0 × 0.05λ0 (fr = 10.45 GHz). A good match in the results validates the simulated and measured data in comparison with the existing models from literature with respect to diversity performance, thus making it suitable for radar applications.

To the Theory of Decaying Turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d>2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t−λ, with a continuous spectrum of indexes λ in the local limit μ→0. The spectrum is determined by a resolvent, which is represented as an infinite product of 3⊗3 matrices depending of the element of the Euler ensemble.

Multivariate Zipper Fractal Functions

A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions.

Organalysis: Multifunctional Image Preprocessing and Analysis Software for Cardiac Organoid Studies.

Due to a growing need in visualizing human pluripotent stem cell-derived organoids from recent advancements in the field, an efficient bulk-processing application is necessary to provide preprocessing and image analysis services. In this study, we developed Organalysis, a high-accuracy, multifunctional, and accessible application that meets these needs by providing the functionality of image manipulation and enhancement, organoid area and intensity calculation, fractal analysis, noise removal, and feature importance computation. The image manipulation feature includes brightness and contrast adjustment. The area and intensity calculation computes six values for each image: organoid area, total image area, percentage of the image covered by organoid, the total intensity of organoid, the total intensity of organoid-by-organoid area, and total intensity of organoid by total image area. The fractal analysis function computes the fractal dimension value for each image. The noise removal function removes superfluous marks from the input images, such as bubbles and other unwanted noise. The feature importance function trains a lasso-regularized linear regression machine learning algorithm to identify cardiac growth factors that are the strongest determinants for cell differentiation. The batch processing of this application further builds on existing services like ImageJ to provide a more convenient way to process multiple images. Collectively, the versatility and preciseness of Organalysis demonstrate novelty, since no other current imaging software combines the capability of batch processing and the breadth of feature analysis. Therefore, Organalysis provides unique functions in cardiac organoid research and proves to be invaluable in regenerative medicine.

Multifractal scaling analyses of the spatial diffusion pattern of COVID-19 pandemic in Chinese mainland

Revealing spatio-temporal evolution regularity in the spatial diffusion of epidemics is helpful for preventing and controlling the spread of epidemics. Based on the real-time COVID-19 datasets by prefecture-level cities, this paper is devoted to exploring the multifractal scaling in spatial diffusion pattern of COVID-19 pandemic and its evolution characteristics in Chinese mainland. The ArcGIS technology and box-counting method are employed to extract spatial data and the least square regression based on rescaling probability (μ-weight method) is used to calculate fractal parameters. The results show multifractal distribution of COVID-19 pandemic in China. The generalized correlation dimension spectrums are inverse S-shaped curves, but the fractal dimension values significantly exceed the Euclidean dimension of embedding space when moment order q«0. The local singularity spectrums are asymmetric unimodal curves, which slant to right. The fractal dimension growth curves are shown as quasi S-shaped curves. From these spectrums and growth curves, the main conclusions can be drawn as follows: First, self-similar patterns developed in the process of COVID-19 pandemic, which seems to be dominated by multifractal scaling law. Second, the spatial pattern of COVID-19 across China can be characterized by global clustering with local disordered diffusion. Third, the spatial diffusion process of COVID-19 in China experienced four stages, i.e., initial stage, the rapid diffusion stage, the hierarchical diffusion stage, and finally the contraction stage. This study suggests that multifractal theory can be utilized to characterize spatio-temporal diffusion of COVID-19 pandemic, and the case analyses may be instructive for further exploring natural laws of spatial diffusion.

Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor

The inventory model is used to determine the optimal inventory of a product. In certain cases, parameters in the inventory model are uncertain. Fractal interpolation techniques can be used to overcome parameter with uncertainty. Fractal interpolation results are affected by the fractal interpolation function and the vertical scaling factor. The vertical scaling factor is positive and less than 1. In this study, fractal interpolation techniques are introduced with variations in vertical scaling factor to overcome the uncertainty of demand data in inventory models. Furthermore, the interpolation results are used in fuzzy inventory models and expressed by Trapezoidal Fuzzy Number. This paper considers an inventory model with varying demand to optimize rice inventory. Based on the data obtained, the accuracy level will increase for the vertical scaling factor values close to 1. Optimal rice inventory of each successive fuzzy parameter is 1447963, 1013914, 504950, 215312. If the cost parameter is increased, then the amount of inventory is decreases.

New Type of Fractal Functions for the General Data Sets

New Type of Fractal Functions for the General Data Sets

Statement of Retraction: Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications

Statement of Retraction: Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications

A Geometric Based Connection between Fractional Calculus and Fractal Functions

A Geometric Based Connection between Fractional Calculus and Fractal Functions

Smoothness analysis and approximation aspects of non-stationary bivariate fractal functions

The present note aims to establish the notion of non-stationary bivariate α-fractal functions and discusses some of their approximation properties. We see that using a sequence of iterated function systems generalizes the existing stationary fractal interpolation function (FIF). Also, we show the existence of Borel probability fractal measures supported on the graph of the non-stationary fractal function. Further, we define a fractal operator associated with the constructed non-stationary FIFs, and many applications of this operator such as fractal approximation and the existence of fractal Schauder bases are observed. In the end, we study the constrained approximation with the proposed interpolant.

Fractal hypersurfaces, affine Weyl groups, and wavelet sets

In this expository paper, we present some fundamental connections between iterated function systems, in particular those whose attractors are the graphs of multivariate real-valued fractal functions, and foldable figures, affine Weyl groups, and wavelet sets.

Stage division of urban growth based on logistic model of fractal dimension curves

The time series of fractal dimension values of urban form always take on sigmoid curves. The basic model of these curves is logistic function. From the logistic model of fractal dimension curves, we can derive the speed formula and acceleration formula of urban growth. Using the inflexions of the fractal parameter curves, we can identify the different phases of city development. The main results are as follows. (1) Based on the curve of fractal dimension of urban form, urban growth can be divided into four stages: initial slow growth, accelerated fast growth, decelerated fast growth, and terminal slow growth. The three dividing points are 0.2113Dmax, 0.5Dmax, and 0.7887Dmax, where Dmax is the capacity of fractal dimension. When the fractal dimension reaches half of its capacity value, 0.5Dmax, the urban growth speed reaches its peak. (2) Based on the curve of fractal dimension odds, urban growth can also be divided into four stages: initial slow filling, accelerated fast filling, decelerated fast filling, terminal slow filling. The three dividing points are 0.2113Zmax, 0.5Zmax, and 0.7887Zmax, where Zmax=/(2-Dmax) denotes the capacity of fractal dimension odds (Dmax < 2). Empirical analyses show that the first scheme based fractal dimension is suitable for the young cities and the second scheme based on fractal dimension odds can be applied to mature cities. A conclusion can be drawn that logistic function is one of significant model for the stage division of urban growth based on fractal parameters of cities. The results of this study provide a new way of understanding the features and mechanism of urban phase transition.

Resonant barrier crossing in a modulated classical double well potential

We show, both numerically and analytically, that for the modulated double well potential given by −12mω2x2(1+ϵcos⁡(Ωt))+mλ4x4, a particle initially placed with zero momentum at the bottom of one of the wells can oscillate back and forth between the two wells for an infinitesimal value of ϵ when the modulating frequency Ω equals 2ω. The effect is also present, but in a less dramatic fashion where Ω=22ω. In the ϵ−Ω plane, around the point Ω=2ω, the boundary separating the region where the tunnelling occurs from the region where it does not, is a fractal curve whose fractal dimension is 1.45.

Applications of fractal interpolants in kernel regression estimations

Generating a function from its finite set of samples is widely studied in data fitting problems. The theory of nonparametric modeling has been developed by many researchers, and several types of estimators have been established in the literature. On the other hand, the fractal theory provides new technologies for making complicated curves and fitting experimental data, and fractal interpolation functions are established as generalizations of traditional interpolation techniques. The aim of this paper is to investigate applications of (smooth) fractal interpolation functions in nonparametric kernel regressions. We discuss a key sufficient condition in detail in the construction of (smooth) fractal interpolation functions. Then the Priestley–Chao estimator is considered, and we establish its fractal perturbation. The raw data we consider in our example is the Crude Oil WTI Futures daily highest price from 2021/10/13 to 2022/7/29. We take 11 samples from the raw data to construct a fractal perturbation of the Priestley–Chao regression function. SLSQP is applied to find the optimal values of hyperparameters so that the given empirical mean squared error is minimum. We see that nonparametric kernel regressions for finite sample data are widely used in applications, and we can decrease the empirical mean squared error further by considering fractal perturbations of these regression functions.

On the integral transform of fractal interpolation functions

This paper explores the integral transform of two distinct fractal interpolation functions, namely the linear fractal interpolation function and the hidden variable fractal interpolation function with variable scaling factors. Further, with a particular application of kernel functions, we investigate the integral transform of fractal functions, such as the Laplace transform and the Laplace Carson transform. Moreover, we show that the compositeness of two fractal interpolation functions, f1 in {tɛ,xɛ} and f2 in {xɛ,zɛ} remains a fractal interpolation function. It also generates iterated function system from given iterated function systems. In addition to this, the study is carried out on the composite linear fractal interpolation function of the integral transform, the Laplace transform, and the Laplace Carson transform.

Box dimension of generalized affine fractal interpolation functions

Let f be a generalized affine fractal interpolation function with a vertical scaling function S . In this paper, we study \mathrm{dim}_B \Gamma f , the box dimension of the graph of f , under the assumption that S is a Lipschitz function. By introducing vertical scaling matrices, we estimate the upper and the lower bounds of oscillations of f . As a result, we obtain an explicit formula of \mathrm{dim}_B \Gamma f under certain constraint conditions.

Generalized G-Hausdorff space and applications in fractals

In this paper, we introduce the concept of a G-Hausdorff space and show how the results established in the usual metric space can be generalized to the G-metric space. The proven results are used to propose an iterated function system (IFS) called G-IFS. Additionally, the existence of the fractal associated with this construction is demonstrated. This paper shows how non-affine transformations and fractal interpolation functions (FIFs) can be used to approximate fractals by G-IFS. This paper contributes to the understanding of fractal geometry and its applications in mathematics and other fields.

Bivariate Fractal Interpolation Functions on Triangular Domain for Numerical Integration and Approximation

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided such that the actual double integral coincides with the integral value calculated using fractal theory. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Adequate examples are presented to illustrate the method of construction and to verify the value of double integration.

Pore Structure and Fractal Characteristics of Mixed Siliciclastic-Carbonate Rocks from the Yingxi Area, Southwest Qaidam Basin, China

Summary Evaluating reservoir properties at the pore scale is vital to better estimate hydrocarbon reserves and plan field development. The lacustrine mixed siliciclastic-carbonate deposits of the Upper Paleogene Xiaganchaigou Formation in the west Yingxiongling area form one of the most important hydrocarbon reservoirs in the southwestern Qaidam Basin (China). In this study, we analyzed well samples with X-ray diffraction (XRD), nuclear magnetic resonance (NMR), and mercury injection capillary pressure (MICP) data in integration with scanning electron microscopy (SEM) images to decipher the mineral composition and pore structure characteristics of the Xiaganchaigou Formation. We also calculate the fractal dimensions using MICP, NMR T2 spectrum, and SEM images based on fractal theory models. The results indicate that the mixed siliciclastic-carbonate samples of the upper section of the Xiaganchaigou Formation are mainly formed by dolomite and clay minerals with low siliceous and calcite content. Porosity is relatively low (2.01−9.83%) and positively correlated with dolomite content, thus indicating that the dolomite intercrystalline pores formed by infiltration and reflux dolomitization control the reservoir characteristics. The size of dolomite intercrystalline pores varies between several and hundreds of nanometers. The porosity has a poor correlation with permeability, which indicates that the pores are mostly primary, which lack the transformation of late dissolution. Three types of mixed siliciclastic-carbonate reservoirs are identified according to pore size distribution (&amp;lt;20 nm, 20−300 nm and multiple distribution), calculated using the NMR T2 spectrum. Fractal curves calculated by combining the MICP and NMR data are characterized by multisegments. The number of segments depends on the degree of heterogeneity of pore structure: two segment for high heterogeneity and three segment for low heterogeneity, also indicating a multifractal feature in mixed rock reservoirs. There is a negative correlation trend between porosity and fractal dimensions, and larger pores often have larger fractal dimensions. These results show that MICP-based fractal values are higher than those of NMR-based, which result from unconnected pores that the MICP is unable to reach. Fractal dimensions obtained from SEM have a small and narrow distribution range and are negatively correlated with the number of pores with smaller sizes. In essence, this study shows that the fractal dimension can be a concise index to evaluate the heterogeneity of lacustrine mixed siliciclastic-carbonate reservoirs, which can serve as an important reference for hydrocarbon development plans.

Compaction-deformation behaviors and breakage characteristics of loose broken coal under various stress–temperature coupling loads

This study probed the compaction-deformation and breakage characteristics of loose broken coal under stress–temperature coupling loads. The constitutive model of loose broken coal under stress–temperature coupling loads was first constructed. Then, we quantificationally investigated the particle size distribution (PSD) evolution for the compacted loose coal using the fractal and Weibull functions. In addition, the relative breakage rate, energy evolution, and particle breakage degree of the loose coal compacted were quantificationally evaluated. Results suggest that stress is the prominent factor leading to the breakage of loose broken coal, while increased temperature exacerbates the particle breakage process in combination with the axial stress. Changes in temperature have a distinctive influence on particle breakage for loose broken coal under low-stress conditions, whereas less conspicuous under high-stress conditions. Finally, the evolution of PSDs and related breakage parameters for the entire loose broken coal compaction process were mathematically predicted utilizing the grey model.