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The main purpose of this special issue is to suggest strategies for meeting the mathematical needs of two-year college students in the decade of the 1990s. Since the 20 contributors were free to write about topics that they considered to be appropriate to the intellectual and personal development of all students their papers represent a broad overview of what leading mathematics educators think should be occurring in mathematics education in the 1990s. The articles are separated into the following five broad categories: (1) guiding thoughts about technological innovation, curriculum standards, and concomitant implementation; (2) outlines of curriculum content specific to calculus, discrete mathematics, statistics, and fractal geometry; (3) special considerations regarding remedial coursework and the needs of the learning disabled student; (4) utilization of computer algebra systems, videotapes, and multimedia presentations in the college mathematics classroom; and (5) the content and reasonable uses of placement tests. Each author supplies diagrams and references as appropriate. (JJK)

The fixed point theory is one of the most essential techniques of applicable mathematics for solving many realistic problems to get a unique solution by using the well known Banach contraction principle. It has paved the ways for numerous extensions, generalization and development of the theory of fixed points in very diverse settings. Our intention in the present paper is to study the Kannan contraction maps defined on a controlled metric space. The generalization of the fixed point theorem for Kannan contraction on controlled metric space is investigated in this paper. We construct an iterated function system called Controlled Kannan Iterated Function System (CK-IFS) with Kannan contraction maps in a controlled metric space and use it to develop a new kind of invariant set, which is called a Controlled Kannan Attractor or Controlled Kannan Fractal (CK-Fractal). Subsequently, the collage theorem for controlled Kannan fractal is also proved. The multivalued fractals are also constructed in the controlled metric space using Kannan and Reich-type contraction maps. The newly developing iterated function system and fractal set in the controlled metric space can provide a novel direction in the fractal theory. [ABSTRACT FROM AUTHOR]

This article from Linux Format explores the topic of fractals and complex numbers. Fractals are intricate geometric patterns found in nature and used in computer graphics and art. The article provides an overview of the Mandelbrot set, a famous fractal, and explains how to draw it using a simple formula. It also discusses the rotozoom effect, which involves rotating and zooming in on a texture. The article includes code snippets for implementing these effects and encourages readers to experiment with graphics programming. [Extracted from the article]

Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q (z) = z p + log c t for all z ∈ ℂ , where p ∈ N ∖ { 1 } , t ∈ [ 1 , ∞) , c ∈ ℂ ∖ { 0 } , using a four-step iteration scheme extended with s -convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized rational map to develop a theory of escape criterion via the SP-iteration process equipped with s -convexity. Furthermore, we develop algorithms for the exploration of unique kinds of Mandelbrot as well as Julia sets. We demonstrate graphically the change in colour, size, and shape of images with the change in values of the considered iteration's parameters. The new fractals thus obtained are visually very pleasing and attractive. Most of these newly generated fractals resemble natural objects around us. Moreover, we numerically study the dependence between the iteration's parameters and the set size. The experiments show that this dependency is non-linear. We believe that the obtained conclusions will motivate researchers who are interested in fractal geometry. [ABSTRACT FROM AUTHOR]

To date, a large number of research papers have been written on malware classification, identification, classification into different families, and the distinction between malware and goodware. These works have been based on captured malware samples and have attempted to analyse malware and goodware using various techniques like the analysis of malware using malware visualization. These works usually convert malware samples capturing the malware structure into image structures which are then subject to image processing. In this paper, we propose an unconventional and novel approach to malware visualization based on its dynamical analysis, subsequent complex network conversion and fractal geometry, e.g. Julia sets visualization. Very interesting images being subsequently used to classify as malware and goodware. The classification is done by deep learning network. The results of the presented experiments of fractal conversion and subsequent classification are based on a database of 6,589,997 goodware, 827,853 potentially unwanted applications and 4,174,203 malware samples provided by ESET. 1 1 https://www.eset.com. This paper aims to show a new direction in visualizing malware using fractal geometry and possibilities in analysis and classification. [Display omitted] • Introduction of a new method for malware visualization based on fractal geometry. • Show interesting results when visualy comparing classified samples using fractal geometry. • Test novel malware image classification using deep learning. • To point out and define new research directions in visual malware analysis opened by our method. [ABSTRACT FROM AUTHOR]

György Ligeti was very interested in many artistic and scientific fields and drew inspiration for his compositional work from all of them (his engagement with mathematics - particularly fractal geometry and chaos theory - is perhaps the best known). In this chapter I compare the concept of artistic research with Ligeti's practice and oeuvre. While the notion of artistic research was only appearing in embryonic form during the latter stages of Ligeti's career, many - though not all - of his statements seem to be suitable for describing his artistic processes. The benefit of this investigation is expected to be two-fold: applying the concept of artistic research to Ligeti's approaches and practices should yield new insights on the relationship between his work and his interest in other humanities and sciences. Yet this look at Ligeti may also help to refine the concept of artistic research as discussed and applied to the artistic output of today. [ABSTRACT FROM AUTHOR]

A recent article in New Scientist reports that researchers at the Max Planck Institute for Terrestrial Microbiology in Germany have discovered a bacterium that has evolved an enzyme capable of assembling in a fractal structure. Fractal patterns, which are mathematical patterns that repeat themselves, have been observed in nature on larger scales, but this is the first time they have been identified at the molecular level. The enzyme, called citrate synthase, in the cyanobacterium Synechococcus elongatus forms a Sierpiński triangle, a fractal pattern consisting of ever-smaller triangular gaps. This finding expands our understanding of fractal patterns in nature and their presence at the molecular level. [Extracted from the article]

Purpose: The purpose of this paper is to derive a new fractal active low-pass filter (LPF) within the local fractional derivative (LFD) calculus on the Cantor set (CS). Design/methodology/approach: To the best of the author's knowledge, a new fractal active LPF within the LFD on the CS is proposed for the first time in this work. By defining the nondifferentiable (ND) lumped elements on the fractal set, the author successfully extracted its ND transfer function by applying the local fractional Laplace transform. The properties of the ND transfer function on the CS are elaborated in detail. Findings: The comparative results between the fractal active LPF (for γ = ln2/ln3) and the classic one (for γ = 1) on the amplitude–frequency and phase–frequency characteristics show that the proposed method is correct and effective, and is expected to shed light on the theory study of the fractal electrical systems. Originality/value: To the best of the author's knowledge, the fractal active LPF within the LFD calculus on the CS is proposed for the first time in this study. The proposed method can be used to study the other problems in the fractal electrical systems, and is expected to shed a light on the theory study of the fractal electrical systems. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal calculus. Design/methodology/approach: For being generalized, an arbitrary kernel function has been adopted. The condition on order has been derived so that it is not related to the γ-dimension of the fractal set. The fractal Laplace transforms of our operators have been derived. Findings: Unlike the traditional power law kernel-based nonlocal fractal calculus operators, ours are generalized, consistent with the local fractal derivative and use higher degree of freedom. As intended, the proposed nonlocal fractal calculus is applicable to any kind of fractal electrical circuit. Thus, it has been found to be a more efficient tool for the fractal electrical circuit analysis than any previous fractal set dedicated calculus scheme. Originality/value: A fractal calculus scheme that is more efficient for the fractal electrical circuit analysis than any previous ones has been proposed in this work. [ABSTRACT FROM AUTHOR]

The article explores the various uses of handcards beyond making rolags and highlights their role in creating gradient and fractal yarns for spinning. Topics covered include techniques for loading and dizzing fiber from handcards, the preparation of gradient and fractal yarns, and the advantages of using handcards in fiber preparation.

The article focuses on creating fractal yarns, explaining the process of dividing a colorway into sections that retain the same colors in the same sequence as the original and then spinning and plying them to create a complex self-striping yarn. Topics include the use of blending boards to create unique fractal rolags, sampling to ensure fiber compatibility, and a step-by-step guide on creating a two-ply gradient fractal yarn.

Purpose: The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor. Design/methodology/approach: For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered. Findings: This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian. Originality/value: To the best of the author's knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out. [ABSTRACT FROM AUTHOR]

The authors would like to inform the readers that the conditions of Theorem 2.2 and Theorem 3.1 of Yadav et al. (2024) is not adequate. The statement and proof of Theorems 2.2 and 3.1 of Yadav et al. (2024) have been modified by Prof. Kiran Kumar Saha. We appreciate him for noticing this error. Throughout the article Yadav et al. (2024), Theorem 2.2 is replaced by Theorem 1.1, and Theorem 3.1 is replaced by Theorem 1.2. The statements of both Theorem 1.1 and 1.2 are given in this corrigendum. Due to this error found, Section 4 (Examples) of Yadav et al. (2024) needs to be updated. So, we have modified the examples. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new class of bi-quintic partially blended rational quartic zipper fractal interpolation surfaces (RQZFISs) tailored for surface data over a rectangular grid. The construction of these surfaces begins with the generation of a network of curves using univariable rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings. These fractal curves are then blended with quintic blended functions. The proposed RQZFISs encompass traditional rational surfaces and a class of fractal surfaces as particular cases. We demonstrate that the bivariable interpolant uniformly converges to the data-generating function. Additionally, the theory of positivity preservation for these interpolants is explored, with practical examples provided to illustrate positivity-preserving bivariable interpolants. • The class of rational splines is generalized using a binary vector. • A novel class of zipper fractal rational surface interpolants is developed. • The proposed surface interpolants unify traditional rational and fractal surfaces as special cases. • Shape preservation theory is explored, with practical examples provided. [ABSTRACT FROM AUTHOR]

A serious error exists in the above paper. [ABSTRACT FROM AUTHOR]

In the paper by Jalal et al. [1], the authors present phase portraits of a differential system exhibiting chaotic behavior with line equilibria. This commentary identifies inaccuracies in the provided figures and offers corrected versions. Specifically, discrepancies were found in Fig. 1 (phase portraits with parameters a = 15 and b = 1) and Fig. 2 (phase portraits with parameters a = 0 and b = 5). Corrected figures, along with MATLAB codes for verification, are provided. These corrections are discussed in the context of chaotic systems with line equilibria, referencing Jafari and Sprott's work [2], which explores similar systems. [ABSTRACT FROM AUTHOR]

In this corrigendum, we improve the statement of Theorem 3.2 in Bhalekar et al. (2022). We also provide the corrected version of Figure 6 in this paper. [ABSTRACT FROM AUTHOR]

Some years ago, a question was raised about whether an iterated function system consisting of a finite number of Reich (or Kannan) contractions (generally discontinuous contractive mappings) has a unique compact invariant set. In this article, we give remarkable results obtained by studying previous research articles that tried to answer this question positively and previous counterexamples that demonstrate that this question is negatively answered. Our results strongly encourage the development of fractal theory because the existence, uniqueness, compactness and boundedness of invariant sets with respect to functional families are one of the main research topics in the theory of fractals. • We provide counterexamples demonstrating that the question of whether an IFS consisting of a finite number of Kannan (or Reich) contractions (generally a discontinuous contractive mappings) with unique fixed points has a unique compact attractor is negatively solved. • We establish new theorems under weaker assumptions. • Explicit examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Two very serious errors exist in the above paper. [ABSTRACT FROM AUTHOR]

The article reports that a fractal is like an infinite version of a Russian nesting doll. It notes that ice crystallizes around a speck of dust in the atmosphere, with the shape of the water molecules leading to six-sided symmetry. It adds that a single equation comprises an entire mathematical universe.

In this work, we study the two-dimensional groundwater flow of fractured porous media with the help of analytic element method. A numerical solution formed by line elements called line-doublets based on a series expansion has been presented in the literature. In this solution, each fracture has an influence that may expand in a series that obeys Laplace's equation exactly. The unknown coefficients are found from the discharge potentials of all other elements that are related to the expansion coefficients in series expansion. This work aims to make comparison between the computational cost for fractals by the analytic element method with different techniques like: Iterative and Matrix method. • Two-dimensional groundwater flow of fractured porous media by the Analytic Element Method. • A numerical solution formed by line elements called line-doublets based on a series expansion. • Comparison between the computational cost (processing time) for fractals using line-doublets with different techniques like: Iterative and Matrix method. [ABSTRACT FROM AUTHOR]

The $k$ multiplicative and $k$ additive compounds of a matrix play an important role in geometry, multilinear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for the integer values of $k$. Here, we introduce generalizations called the $\alpha$ multiplicative and $\alpha$ additive compounds of a square matrix, with $\alpha$ real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterlé theorem. Our results lead to a generalization of contracting systems to $\alpha$ -contracting systems, with $\alpha$ real. Roughly speaking, the dynamics of such systems contracts any set with the Hausdorff dimension larger than $\alpha$. For $\alpha =1$ , they reduce to standard contracting systems. We demonstrate our theoretical results by designing a state-feedback controller for a classical chaotic system, guaranteeing the well-ordered behavior of the closed-loop system. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set. Design/methodology/approach: The fractal integrodifferential equations of resistor-inductor, resistor-capacitor, inductor-capacitor and resistor-inductor-capacitor circuits subjected to zero mean additive white Gaussian noise defined on fractal set have been formulated. The fractal time component has also been considered. The closed form expressions for crucial stochastic parameters of circuit responses have been derived from these equations. Numerical simulations of power spectral densities based on the derived autocorrelation functions have been performed. A comparison with those without fractal time component has been made. Findings: We have found that the Hausdorff dimension of the middle b Cantor set strongly affects the power spectral densities; thus, the average powers of noise induced circuit responses and the inclusion of fractal time component causes significantly different analysis results besides the physical measurability of electrical quantities. Originality/value: For the first time, the noise analysis of electrical circuit on fractal set has been performed. This is also the very first time that the fractal time component has been included in the fractal calculus-based circuit analysis. [ABSTRACT FROM AUTHOR]

Chaotic image encryption is widely used in the field of information security. This paper proposes a novel chaotic image encryption method with spiral-transform-based fractal sorting matrix (STFSM). First of all, the theory of STFSM with good scrambling effect is introduced, which has good irregularity and iterative. Then, the iterative algorithm and calculation example of STFSM are introduced. STFSM can be used as the map of spatial location transformations to implement the design of image encryption. Based on the complete STFSM theory and iterative algorithm, a chaotic image cryptosystem based on STFSM is proposed to achieve a good image encryption process. To test the security of the proposed algorithm, security tests and analyses such as entropy analysis, correlation analysis, resistance to differential attacks analysis, and robustness analysis are used for the proposed algorithm. The experimental analysis illustrates that the algorithm has a better encryption effect, whether using conventional tests or the attacks simulations described in this paper and can also effectively resist the attacks. [ABSTRACT FROM AUTHOR]

This paper creatively proposes a type of cubic fractal matrix and a new spatiotemporal chaotic system named logistic-logistic cascade coupled map lattice (L-LCCML). Furthermore, based on cubic fractal matrix and L-LCCML, this paper proposes a novel image encryption algorithm. The proposed cubic fractal matrix is three-dimensional, irregular, and self-similar. In particular, the proposed L-LCCML system adopts cascade coupling parameter to ensure the dynamic effect of coupling. Therefore, L-LCCML has excellent chaos and is suitable for information encryption. To provide a more secure approach, the proposed algorithm contains two diffusion operations, one permutation operation, and does not require multiple rounds of encryption. The diffusion operation is based on the cubic fractal matrix, which has good security and high encryption efficiency. In addition, the algorithm adopts sorting permutation based on L-LCCML, which provides good randomness for encryption. Experimental results show that the proposed algorithm has the characteristics of large key space, high sensitivity, fast encryption speed, good statistical properties of cipherd images, and etc. Therefore, the proposed algorithm is a usable alternative for practical secure communication. [ABSTRACT FROM AUTHOR]

This work is devoted to the fractal features of Bernoulli percolation in different space dimensions. The focus is made on the fractal attributes associated with the connectivity, ramification, and loopiness of percolation clusters and their substructures. In this way we ascertain a connection between the connectivity dimension and topological invariants. Consequently we elucidate the difference between the fractal dimensions of the minimum path and the geodesic on the percolation cluster. We also derive a relation between the topological Hausdorff dimension of percolation cluster and the correlation length exponent. Further we establish that the percolation cluster and its hull have the same topological Hausdorff dimension. These findings allow us to found the ranges for admissible values of dimension numbers characterizing the percolation cluster and their substructures in different space dimensions. Thus we scrutinize the data of numerical simulations. • The fractal attributes of percolation cluster are associated with its ramification, connectivity and loopiness. • The topological Hausdorff dimension of percolation cluster is a function of the correlation length exponent. • The connectivity dimension governs the scaling properties of topological invariants such as the Wiener index. • The ranges of admissible values of the connectivity, topological connectivity, and spectral dimensions are established. • The difference between the fractal dimensions of the minimum path and the geodesic is outlined. [ABSTRACT FROM AUTHOR]

The aesthetic patterns play a key role in the field of fractals. Due to self similarities in the nature of fractals, researchers used the fractals in many fields of sciences (i.e. in Mathematics, Computer Science, Physics, Image Encryption, Biology and Chemistry). The most studied fractals types are the Mandelbrot sets (MSs) and Julia sets (JSs). To generate fractals, escape criteria is required. In this work, a general escape criteria is proved via extended Jungck-Noor iteration with s -convexity. These results are used in algorithms to present the generation of fractals in extended Jungck-Noor orbit for general complex polynomial f (x) = ∑ i = 0 p a i x i with p ≥ 2 , where a i ∈ ℂ for i = 0 , 1 , 2 , ... , p. The graphics of MSs and JSs are demonstrated in the examples. The variations in MSs and JSs for different values of involved parameters are also shown. [ABSTRACT FROM AUTHOR]

Post-COVID Blues IN 2017, Wink and Stanton sold theirshares to Cirque du Soleil, which acquiredBlue Man Group for "tens of millions", perCirque's CEO. IN 1993, Wink, Goldman, and Stanton,until then the only Blue Men, trainedan understudy. Within months, the group of Blue Men had been winnowed downto three: Chris Wink, Matt Goldman, and Phil Stanton. [Extracted from the article]

Purpose: The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach: The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings: The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value: For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out. [ABSTRACT FROM AUTHOR]

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C 1 -rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form p i (θ) / q i (θ) , where p i (θ) and q i (θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C 3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C 1 -RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data. • A class of rational cubic trigonometric fractal interpolation functions (RCTFIFs) is proposed to preserve the shape of conic and periodic data. • Convergence analysis of the proposed RCTFIF is carried out. • The proposed RCTFIFs follow the constrained nature of data set when it lies above by a straight line or bounded by upper and lower bounds. • Sufficient conditions are derived to preserve the positivity and monotonicity features of interpolation data. [ABSTRACT FROM AUTHOR]

Recently a hybrid secure image encryption scheme based on Julia set and three-dimensional Lorenz chaotic system was suggested. The offered scheme was based on the multiplicative operation carried out by Julia set fractals and bitwise XOR which was implemented by the trajectories obtained from the Lorenz chaotic system. This paper presents the cryptanalysis of the scheme through a chosen-plaintext attack by taking three different chosen images. In this work, a practical chosen-plaintext attack is performed which reveals that the cryptosystem depending on only multiplication and bitwise XOR operation can be effortlessly attacked. The results of the analysis performed on the retrieved image reflect the effectiveness of the chosen-plaintext attack which fruitfully produces the original image. [ABSTRACT FROM AUTHOR]

Fractional calculus and entropy are two essential mathematical tools, and their conceptions support a productive interplay in the study of system dynamics and machine learning. In this article, we modify the fractional entropy and propose the cumulative permuted fractional entropy (CPFE). A theoretical analysis is provided to prove that CPFE not only meets the basic properties of the Shannon entropy but also has unique characteristics of its own. We apply it to typical discrete distributions, simulated data, and real-world data to prove its efficiency in the application. This article demonstrates that CPFE can measure the complexity and uncertainty of complex systems so that it can perform reliable and accurate classification. Finally, we introduce CPFE to support vector machines (SVMs) and get CPFE-SVM. The CPFE can be used to process data to make the irregular data linearly separable. Compared with the other five state-of-the-art algorithms, CPFE-SVM has significantly higher accuracy and less computational burden. Therefore, the CPFE-SVM is especially suitable for the classification of irregular large-scale data sets. Also, it is insensitive to noise. Implications of the results and future research directions are also presented. [ABSTRACT FROM AUTHOR]

On 3 January 2019, the Chinese spacecraft Chang'e-4 descended toward the moon. Countless craters came into view as the lander approached the surface, the fractal nature of the footage providing no sense of altitude. Su Yan, responsible for data reception for the landing at Miyun ground station, in Beijing, was waiting—nervously and in silence with her team—for vital signals indicating that optical, laser, and microwave sensors had combined effectively with rocket engines for a soft landing. “When the [spectral signals were] clearly visible, everyone cheered enthusiastically. Years of hard work had paid off in the most sweet way,” Su recalls. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to describe the Lane–Emden equation by the fractal derivative and establish its variational principle by using the semi-inverse method. The variational principle is helpful to research the structure of the solution. The approximate analytical solution of the fractal Lane–Emden equation is obtained by the variational iteration method. The example illustrates that the suggested scheme is efficient and accurate for fractal models. Design/methodology/approach: The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method. Findings: The variational iteration method is very fascinating in solving fractal differential equation. Originality/value: The author first proposes the variational iteration method for solving fractal differential equation. The example shows the efficiency and accuracy of the proposed method. The variational iteration method is valid for other nonlinear fractal models as well. [ABSTRACT FROM AUTHOR]

Researchers from Prince Sultan University in Riyadh, Saudi Arabia have conducted a study on infectious diseases and conditions using fractals and fractional analysis. The study focused on the use of fractals and fractional operators to model real-world problems, particularly the issue of reinfection. The researchers deduced qualitative and numerical aspects of a fractal-fractional hybrid model with reinfection, including equilibrium points, stability, and a numerical scheme. The study also included sensitivity analysis and presented comparisons of simulated and real data. The research has been peer-reviewed and provides valuable insights into the mathematical modeling of infectious diseases. [Extracted from the article]

A novel variable-distance magnetic pendulum model is proposed to explore its sensitivity to the initial conditions. In this study, equilibrium points and stability theorems were employed as preliminary quantitative methods to determine the existence of multiple coexisting attractors in the model. Furthermore, the trajectories of the pendulum and time histories were applied as qualitative methods to demonstrate the sensitivity to the initial conditions. Finally, the evolution of the basins of attraction was utilised to characterise their fractal nature. Qualitative and quantitative results consistently confirm that the mechanism behind the sensitivity to initial conditions in this model originates from fractal basins of attraction, thereby distinguishing it from the commonly held belief that attributes it to chaos. This investigation provides valuable insights into the global dynamic behaviour and engineering applications of magnetic pendulums. • A novel dynamic model of a variable-distance magnetic pendulum is proposed. • The number of equilibrium points and their stability are theoretically derived. • Multiple coexisting attractors are delineated through numerical simulations. • Mechanism of sensitivity to initial conditions originates from fractal attractors. • The evolution of the basins of attraction with changing magnet positions is explored. [ABSTRACT FROM AUTHOR]

The usual multifractal analysis for measures studies isotropic pointwise regularity. It does not take into account behaviors that may differ when measured in coordinate axes directions. In this paper, we focus on anisotropic and directional pointwise regularities for Borel probability measures on R 2. We will investigate general upper bound results about the dimension prints of the fractal sets of anisotropic regularities and irregularities, and iso-level directional regularity. We apply our results for selfaffine invariant measures on R 2 supported by selfaffine Sierpinski Sponges. We show different directional multifractal phenomena. We finally obtain lower bound results about the dimension prints of the fractal sets of anisotropic regularities and irregularities for the measure product of two Borel probability measures on R that have Frostman measures at states q 1 and q 2 respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the connection between a class of real number sequences and local fractal functions in terms of fractal dimensions. Under certain conditions, we show that the Box dimension of the graph of a local fractal function with a single unbounded variation point is equal to that of its zero points set plus one. Several concrete examples of such functions whose Box dimension can take any numbers belonging to [ 1 , 2 ] have also been given. This work may provide new approaches to the construction of various local fractal functions with the required Box dimension in the future. • The dimensional connection between local fractal functions and their zero points sets has been studied. • The calculation of the Box dimension of a class of real number sequences has been given. • The Box dimension of graphs of local fractal functions has been investigated. • Several concrete examples of local fractal functions have been constructed. [ABSTRACT FROM AUTHOR]

Complex networks have certain properties that distinguish them from their respective uniform or regular counterparts. One of these properties is the variation of topological properties along different hierarchical levels. In this work, we study how networks that are constructed by repeatedly incorporating a given motif exhibit this property. A motif is henceforth understood as a small subgraph with a reference node where the incorporation respectively occurs. We generate fractal networks using different motifs and observe how their structures change along the growth stages. Two regimes are respectively identified: transient and equilibrium. The former is characterized by significant topological changes that depend on the motif topology, while the equilibrium regime shows more stable, parallel trajectories. A more systematic analysis revealed that the betweenness centrality and the average shortest path lengths were the main topological properties accounting for the changes along the network growth. • A method for generating fractal networks from motifs with references is described. • Two dynamic regimes along the growth stages are identified: transient and equilibrium. • Markedly distinct networks are obtained from the considered motifs. [ABSTRACT FROM AUTHOR]

Many physical, biological, and social systems exhibit emergent properties arising from their components' interactions (cells). In this study, we systematically treat every-pair interactions (a) that exhibit power-law dependence on the Euclidean distance and (b) act in structures that can be characterized using fractal geometry. It can represent the two-body interaction potential, the heat flux between two parts of a structure, friendship strength between two people, etc.. We analytically derive the average intensity of influence that one cell has on the others or, conversely, receives from them. This quantity is referred to as the mean interaction field of the cells, and we find that (i) in a long-range interaction regime, the mean interaction field increases following a power-law with the size of the system, (ii) in a short-range interaction regime, the field saturates, and (iii) in the intermediate range it follows a logarithmic behavior. To validate our analytical solution, we perform numerical simulations. For long-range interactions, the theoretical calculations align closely with the numerical results. However, for short-range interactions, we observe that discreteness significantly impacts the continuum approximation used in the derivation, leading to incorrect asymptotic behavior in this regime. To address this issue, we propose an expansion that substantially improves the accuracy of the analytical expression. We discuss applications of the every-pair interactions system proposed, and one of them is to explore a framework for estimating the fractal dimension of unknown structures. This approach offers an alternative to established methods such as box-counting or sandbox methods. Overall, we believe that our analytical work will have broad applicability in systems where every-pair interactions play a role. • We derived an analytical solution for the mean interaction field between cells embedded in a fractal medium. • The results are validated through numerical simulations. • We examine the emergence of macroscopic effects based on the range of interaction, distinguishing between short and long ranges. • This study motivated the development of a framework to estimate the fractal dimension of structures, besides other applications. [ABSTRACT FROM AUTHOR]

A recent study conducted at National Textile University in Faisalabad, Pakistan, explored the dynamics of drug addiction transmission and the legal implications of criminal law within a population. The study proposed a nonlinear deterministic mathematical model that incorporated the fractal-fractional order derivative using the Atangana-Baleanu-Caputo (ABC) operator. The research found that controlling the effective transmission rate is crucial in effectively halting drug transmission. The study concluded that the proposed strategy provides a more nuanced understanding of the complex dynamics of drug addiction. [Extracted from the article]

In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if we consider the total equivalence relation, we recover the classical generalized iterated function systems. Our main result states that each compact subset of the underlying metric space generates, via a sequence of iterates, a fixed point of the associated fractal operator, called an attractor of the system. We also establish a structure result for the attractors and a theorem concerning the continuous dependence of the attractor on the associated compact set. Ultimately, we provide some examples which illustrate our main results. • We introduce the class of relational generalized iterated function systems. • We weaken the classical contraction condition using an equivalence relation. • We prove that each compact set generates an attractor of the system. • We establish a structure result for the attractors. • We prove a theorem concerning the continuous dependence of the attractors. [ABSTRACT FROM AUTHOR]

This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be fractals and the tiled set maybe a complicated unbounded subset of R M. • Generalize self-similar tiling to include tiles with non- integer dimension, in a graph directed setting. • Extension and a different point of view on well-known work of Anderson and Putnam. • Considers in detail and with care the address structure of these tilings. • The work adopts a direct and fresh point of view on the well worn topic of self-similar tiling theory. • The notion of "rigid" tilings is developed. • Earlier work that presages this include work by Bandt and Strichartz. [ABSTRACT FROM AUTHOR]

The goal of this article is to study the fractal surfaces and associated fractal operator on Lebesgue spaces with respect to fractal measures. First, we show that fractal surfaces belongs to Lebesgue spaces under certain conditions. Then, we define a fractal operator on Lebesgue spaces and discuss some analytical properties of it. Moreover, we show the existence of Schauder basis of the associated fractal functions for the space L q (I × J , μ p). In the end, we draw some graph of fractal surfaces for the various scaling factors and mention some future directions. • Fractal surfaces and associated fractal operator on Lebesgue spaces concerning fractal measures. • Analytical properties of associated fractal operators. • Schauder basis of the associated fractal functions for the Lebesgue spaces. • Graphical illustrations of fractal surfaces on the Lebesgue spaces. • Related future research directions. [ABSTRACT FROM AUTHOR]

We employ Mittag-Leffler type kernels to solve a system of fractional differential equations using fractal–fractional (FF) operators with two fractal and fractional orders. Using the notion of FF-derivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a non-decreasing and compact mapping are used in order to prove the existence of a solution for the FF-model of polluted lake system. For this purpose, the Leray–Schauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on Adams–Bashforth and Newton polynomials methods. The effect of fractal–fractional differentiation is illustrated numerically. Moreover, the effect of the FF-derivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic. • Mittag-Leffler fractal–fractional (FF) modeling of polluted lakes. • Well posedness of the polluted lake system with nonsingular and nonlocal fading memory. • Stability of the polluted lakes system. • New numerical techniques based on Adams–Bashforth and Newton polynomials methods. • The effect of linear, exponentially decaying, and periodic pollutants. [ABSTRACT FROM AUTHOR]

Some serious errors exist in the above paper. [ABSTRACT FROM AUTHOR]