Fractals

This review paper provides an overview of complexity-based analysis techniques in biomedical image analysis, examining their theoretical foundations, computational methodologies, and practical applications across various medical imaging modalities. Through a synthesis of relevant literature, we explore the utility of complexity-based metrics such as fractal dimension, entropy measures, and network analysis in characterizing the complexity of biomedical images (e.g. magnetic resonance imaging (MRI), computed tomography (CT) scans, X-ray images). Additionally, we discuss the clinical implications of complexity-based analysis in areas such as cancer detection, neuroimaging, and cardiovascular imaging, highlighting its potential to improve diagnostic accuracy, prognostic assessment, and treatment outcomes. The review concludes that complexity-based analysis significantly enhances the interpretability and diagnostic power of biomedical imaging, paving the way for more personalized and precise medical care. By elucidating the role of complexity-based analysis in biomedical image analysis, this review aims to provide insights into current trends, challenges, and future directions in this rapidly evolving field.

Mineral dissolution and precipitation reactions occur in a variety of porous rocks due to the deviations from a geochemical equilibrium between solid matrix and flow fluid, resulting in the alteration of the petrophysical properties of rocks. A physically-based theoretical model is proposed based on fractal theory for quantitatively assessing the temporal evolution of permeability for porous rocks undergoing mineral dissolution and precipitation reactions. The proposed model is associated with pore structure parameters, such as initial porosity $({\varphi }_0)$, the tortuosity fractal dimensions $(D_{T,t})$, the initial fractal dimension $(D_{f,0})$, and the overall dissolution rate $({\alpha }_t)$ considering the precipitation reactions in porous rocks. The proposed model provides a better prediction of the time dependence of permeability compared with the available experimental data. The essential effects of the overall dissolution rate and pore structural parameters of porous rocks on the dimensionless time-dependent permeabilities are investigated in detail. It is found that the overall dissolution rate has obvious effects on the dimensionless time dependence of permeabilities. The present model may provide useful insights into the understanding of permeability change during mineral dissolution and precipitation reactions in porous rocks.

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.

In this paper, a surface with fractal characteristics on the basis of a continuous function has been constructed. We conducted a study on the Box dimension and the Hausdorff dimension of this surface, building upon this foundation. We found that there exists a certain relationship between the dimensionality of the surface with fractal characteristics, which is obtained from the rotation of a fractal curve.

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order ($\alpha$) have minimal influence on the system when the noise intensity ($\sigma$) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain.

In this paper, the solutions of an extended form of the Fractional-order Neutron Point Kinetics (FNPK) equation in terms of Caputo-time derivatives of the same order are investigated. Instead of using a Caputo derivative, a distributed-order fractional derivative in the Caputo sense was employed in the term of the FNPK equation which is multiplied by the reactivity. This term plays an important role in the description of neutron kinetics during the start-up, shutdown, and steady-state processes in nuclear reactors. The extended (DFNPK) model was solved using the beta, normal, bimodal and Dirac delta distributions to investigate their effect on the transient state solutions of the neutron density. Regardless of the distribution used, the most significant finding is that a destabilizing effect on the neutron density is induced when the mode (or the instant of application of the Dirac delta) of the distribution tends to one while maintaining the orders of the Caputo-time derivatives constant. What defines the destabilizing effect are large magnitude oscillations, a rapid decay, and an oscillation-free steady state with a monotonic increase that is parallel to but somewhat above the trend determined by the FNPK equation. The extended model is anticipated to be effective for modeling neutron density dispersion in a highly heterogeneous medium that may be described using distributed derivatives.

In this paper, we construct a family of self-similar networks, named Coral networks, whose Iterated Function System (IFS) contains a rotation transformation. Using self-similarity, we obtain the weighted average geodesic distance of Coral fractal and networks.

In this paper, we present Katugampola fractional parameterized inequalities for s-convex function in the fourth sense. In the first place, with the help of integration by parts method, we provide a very important identity that will be of great significance for the next research. In addition, by comprehensively making use of some important concepts such as s-convex function in the fourth sense, Hölder’s integral inequality, and the well-known power-mean inequality, we establish three new parameterized inequalities. Finally, we provide some corollaries to help readers gain a deeper understanding of the theorems and the relevant theories.

Electrospinning is an amazing process that enables the production of primarily nanofiber membranes. Nevertheless, the mechanism underlying the formation of two-dimensional (2D) nonwoven structure remains a fascinating enigma. This paper presents a novel morphology of nanofiber membranes with a cluster structure that bears resemblance to the solar nebula. The mechanism of nanofiber cluster structures in the electrospinning process is elucidated through the geometric potential theory, which begins with a zero-dimensional receptor, progresses to a one-dimensional (1D) receptor, a 2D receptor, and ultimately culminates in a fractal-like lattice receptor. Moreover, a three-dimensional (3D) printing-like novel approach is presented for the manipulation of nanofiber membrane’s shape through the use of a lattice-structured receptor. The fabrication process of the cluster structure nanofiber membranes by electrospinning is presented, and their remarkable potential applications are discussed.

The edge-hyper-Wiener index is the sum of the distances and the square of distances between all pairs of edges. Using the technique of finite pattern, we obtain the exact formulas of edge-hyper-Wiener indices of fractal networks modeled by the classical Sierpinski gasket.

This paper investigates tangent measures in the sense of Preiss for non-homogeneous Cantor sets on ${ℝ}^d$ generated by specific iterated function systems that satisfy the strong separation condition. Through the dynamics of “zooming in” on any typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of non-homogeneous Cantor sets on ${ℝ}^d$.

The complexities of deep shale reservoirs are expressed in terms of multi-scale pore structure, abundant micro-fractures, strong stress dependence and multiple gas transport mechanisms. The current macro-scale numerical simulation models have not fully considered the influence of multi-scale pore-fracture structure on gas transport in deep shale reservoirs. In this work, we propose a general fractal-based multi-scale quin-medium model for gas transport in deep shale reservoirs. The quin-medium is composed of organic matter, inorganic matter, micro-fracture medium, induced hydraulic fracture network and hydraulic fracture. The shale matrix is modeled by multiple interacting continua (MINC) with intersected organic matter, inorganic matter and micro-fracture medium. Organic matter is distributed at the center of MINC while the micro-fracture medium is embedded into inorganic matter with layered distribution. The organic/inorganic pore structure and micro-fracture are characterized by fractal geometry, respectively. The apparent gas permeability in each medium is derived considering gas slippage and stress-dependence while the adsorbed gas surface diffusion and Langmuir volumetric strain are additionally considered in organic matter. The apparent gas permeability models are further substituted into the quin-medium continuity equations and the hydraulic fracture is described by an embedded discrete fracture model. The influences of pore structure fractal parameters, gas transport mechanisms and stress dependence on deep shale gas productivity are analyzed in detail. We found that the influence of stress dependence on deep shale gas productivity is more pronounced than the gas transport mechanism. Furthermore, the pore structure fractal parameter of inorganic matter has a more notable influence on deep shale gas productivity than that of organic matter and micro-fracture medium.

The Eccentric Connectivity Index (ECI) and Eccentric Distance Sum (EDS) are two important topological indices with wide applications in biology and chemistry. The Vicsek Fractal (VF) is a notable structure in both polymer physics and network science, and is also known as a regular hyperbranched polymer in the academic realm. In this study, the ECI and EDS of the VF are evaluated analytically by exploring the self-similarity of the structure. First, we derive the recurrence formulas for the ECI and EDS with respect to the generation $t$ of the VF. Then, exact results for the ECI and EDS of the VF are obtained by solving the recurrence relations. The results indicate that the ECI and EDS of the VF scale with a network size of $N_t$ are ECI $\sim N_t^{1+{log}_{p+1}^3}$ and $\text{EDS}\sim N_t^{2+2{log}_{p+1}^3}$, respectively. The results obtained not only reveal the biological activity and physical properties of the VF but also shed light on the control of the beamwidth in the azimuth plane and the design of the maglev centrifugal blood pump.

This paper focuses on synchronization control of the fractional-order unified chaotic system. First, the stability of equilibria is analyzed by the Routh–Hurwitz criteria. Then, three different controllers are designed to achieve chaos synchronization of the drive-response unified system. Further, the impact of the fractional order on the synchronization performance is illustrated. Numerical simulations are carried out to verify the feasibility of theoretical analysis.

The normalized fractional derivatives (NFDs) are a normalization of the existing fractional derivatives (FDs) which admit geometrical meanings. They have finite ordinary derivatives at the initial point, which cause smoothness of solutions for related fractional differential equations (FDEs). For FDs with non-singular kernels, the NFDs do not vanish at the starting point, in general, and therefore related FDEs admit solutions without the need to impose extra conditions. In this paper, we consider FDEs with the normalized Caputo–Fabrizio derivative of Caputo type. We show in closed forms the solutions of related FDEs and show that the Cauchy problem with the NFD admits a nontrivial solution. We also define the higher-order NFDs and show that related FDEs can be solved by transforming them to integro-differential equations with integer orders.

Based on the bifurcation theory, the exact solutions of fractional Date–Jimbo–Kashiwara–Miwa equation are constructed. In terms of $\beta$-derivative, a nonlinear integer-order ODE is obtained, the equation is transformed into a plane dynamical system. The phase portraits are obtained under different bifurcation conditions. According to the orbits of the phase portraits, new bright and dark solitary solutions, periodic solutions, periodic-singular solutions, and singular solutions are established, enriching the diversity of solutions. In addition, the dynamical behaviors of various types of solutions are shown by selecting appropriate parameters, which is useful for understanding the fluctuation behavior in physical models. Compared with previous literature, this paper focuses on the combination of qualitative analyses and qualitative calculations, which makes the paper more systematic and comprehensive, and fills the gap of using bifurcation theory to solve the FDJKM equation. The dynamical behavior of new solutions obtained will provide more and more attractive and complex physical phenomena to explore more mysteries.

Using the technique of finite pattern, for level-3 Sierpinski networks, we obtain their exact formulae of edge-hyper-Wiener index, which is the sum of the distances and the square of distances between all pairs of edges.

We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity $L_N(\omega )$ which describes the maximal length of string of zeros of the prefix among the first $N$ iterations of $\omega$ in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined.

Dynamical analysis of system of ordinary differential equations (ODEs) is an interesting topic among researchers and several tools have been discovered since so far to deal with its several other features. There are many built-in functions designed in MATLAB that researchers can use as a tool for the system of ODEs with integer order but in the similar systems have fractional-order derivative is a difficult task. Therefore, the purpose of this work is to use the Caputo fractional-order derivative to qualitatively analyze and then numerically simulate the tumor-immune system interaction model. Moreover, the Schauder fixed point theorem is used to establish the condition for the existence of at least one solution, while the Banach fixed point theorem is utilized to guarantee the existence of a unique solution. Moreover, the stability in the trajectories of considered system achieved with the aid of Ulam–Hyers (UH) stability. In addition, the numerical computations are performed using Variational Iteration Method (VIM) and the visualized results are shown using MATLAB.

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

A new fractal Rosenau–Burgers equation with He’s fractal derivative is proposed in this work. Exerting the semi-inverse method, two different kinds of the generalized fractal variational principles (GFVPs) of the fractal Rosenau–Burgers equation are constructed. The GFVPs extracted in this paper are expected to bring some new inspiration for the study and application of the variational method in the fractal space.

In this paper, we extend the results for connectivity and convex hulls of Sierpiński relatives whose convex hulls have right isosceles triangle boundaries to the relatives corresponding to equilateral triangle version. Furthermore, we use a particular subclass of the relatives called trapezoid-shaped relatives to build new hexagonal fractals, which helps to visualize and distinguish the subgroups of symmetries of the hexagon. These results contribute to identify the topologies of Sierpiński relatives.

This paper provides a comprehensive examination of fractal diffusion in an isotropic medium. This process has a fascinating temporal memory and an extremely high initial response, which are unquestionably characterized by the Caputo fractional derivative. Its spatio-symmetric property is described by the Riesz fractional derivative. To illustrate the diffusion mechanism, we present an example of a toxic gas leak in a subway station. The numerical results are extremely promising and challenging. They clearly indicate that a MEMS-based gas sensor can track a toxic gas leak immediately due to its extremely fast diffusion at the initial time. This is of great importance for public safety. Furthermore, the diffusion process is primarily contingent upon the density of the air, which we can easily control. The diffusion pattern in an isotropic medium exhibits fractal behavior, which is extremely helpful for designing an emergency system for evacuating people from the center of toxic gas leaks by changing air density or temperature. This paper presents a rigorous mathematical concept for monitoring systems and early warning systems for toxic gas leaks.

In recent years, there has been a considerable increase of interest in the sixth-generation (6G) communications. The advent of 6G wireless communications will facilitate a comprehensive upgrade and transformation of the industry, enabling higher data rates, lower latency, and ubiquitous connectivity. Research on microelectromechanical systems (MEMS) is a crucial element in the advancement of 6G wireless communications. In this paper, we examine the impact of geometric potential on MEMS within the context of fractal space, where a fractal geometric potential MEMS model is proposed and the critical conditions that govern the pull-in phenomenon are elucidated. Moreover, a periodic solution of the model is proposed and compared with other numerical methods.